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3.411 \(\int \frac {\cos ^{10}(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=252 \[ -\frac {\left (\sqrt {a}-\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}-\frac {(a+3 b) \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac {4 x (a+b)}{b^2}-\frac {x (a+3 b)}{2 b^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 b d}-\frac {17 \sin (c+d x) \cos ^3(c+d x)}{24 b d}-\frac {17 \sin (c+d x) \cos (c+d x)}{16 b d}-\frac {17 x}{16 b} \]

[Out]

-17/16*x/b-4*(a+b)*x/b^2-1/2*(a+3*b)*x/b^2-17/16*cos(d*x+c)*sin(d*x+c)/b/d-1/2*(a+3*b)*cos(d*x+c)*sin(d*x+c)/b
^2/d-17/24*cos(d*x+c)^3*sin(d*x+c)/b/d-1/6*cos(d*x+c)^5*sin(d*x+c)/b/d-1/2*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(
d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))^(9/2)/a^(3/4)/b^(5/2)/d+1/2*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4
))*(a^(1/2)+b^(1/2))^(9/2)/a^(3/4)/b^(5/2)/d

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Rubi [A]  time = 0.44, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3224, 1170, 199, 203, 1166, 205} \[ -\frac {\left (\sqrt {a}-\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}-\frac {(a+3 b) \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac {4 x (a+b)}{b^2}-\frac {x (a+3 b)}{2 b^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 b d}-\frac {17 \sin (c+d x) \cos ^3(c+d x)}{24 b d}-\frac {17 \sin (c+d x) \cos (c+d x)}{16 b d}-\frac {17 x}{16 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^10/(a - b*Sin[c + d*x]^4),x]

[Out]

(-17*x)/(16*b) - (4*(a + b)*x)/b^2 - ((a + 3*b)*x)/(2*b^2) - ((Sqrt[a] - Sqrt[b])^(9/2)*ArcTan[(Sqrt[Sqrt[a] -
 Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/2)*d) + ((Sqrt[a] + Sqrt[b])^(9/2)*ArcTan[(Sqrt[Sqrt[a] + Sq
rt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/2)*d) - (17*Cos[c + d*x]*Sin[c + d*x])/(16*b*d) - ((a + 3*b)*Co
s[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (17*Cos[c + d*x]^3*Sin[c + d*x])/(24*b*d) - (Cos[c + d*x]^5*Sin[c + d*x])
/(6*b*d)

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
 FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
 q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

Rule 3224

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(m/2 + 2
*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cos ^{10}(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{b \left (1+x^2\right )^4}-\frac {2}{b \left (1+x^2\right )^3}+\frac {-a-3 b}{b^2 \left (1+x^2\right )^2}-\frac {4 (a+b)}{b^2 \left (1+x^2\right )}+\frac {5 a^2+10 a b+b^2+4 \left (a^2-b^2\right ) x^2}{b^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {5 a^2+10 a b+b^2+4 \left (a^2-b^2\right ) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {(4 (a+b)) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {(a+3 b) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}\\ &=-\frac {4 (a+b) x}{b^2}-\frac {(a+3 b) \cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 b d}-\frac {\left (\left (\sqrt {a}-\sqrt {b}\right )^5 \left (\sqrt {a}+\sqrt {b}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt {a} b^{5/2} d}+\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right )^5\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt {a} b^{5/2} d}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{6 b d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{2 b d}-\frac {(a+3 b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}\\ &=-\frac {4 (a+b) x}{b^2}-\frac {(a+3 b) x}{2 b^2}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}-\frac {3 \cos (c+d x) \sin (c+d x)}{4 b d}-\frac {(a+3 b) \cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {17 \cos ^3(c+d x) \sin (c+d x)}{24 b d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 b d}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{8 b d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{4 b d}\\ &=-\frac {3 x}{4 b}-\frac {4 (a+b) x}{b^2}-\frac {(a+3 b) x}{2 b^2}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}-\frac {17 \cos (c+d x) \sin (c+d x)}{16 b d}-\frac {(a+3 b) \cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {17 \cos ^3(c+d x) \sin (c+d x)}{24 b d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 b d}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{16 b d}\\ &=-\frac {17 x}{16 b}-\frac {4 (a+b) x}{b^2}-\frac {(a+3 b) x}{2 b^2}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{9/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/2} d}-\frac {17 \cos (c+d x) \sin (c+d x)}{16 b d}-\frac {(a+3 b) \cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {17 \cos ^3(c+d x) \sin (c+d x)}{24 b d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 b d}\\ \end {align*}

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Mathematica [A]  time = 0.91, size = 233, normalized size = 0.92 \[ -\frac {36 b (24 a+35 b) (c+d x)+3 b (16 a+95 b) \sin (2 (c+d x))-\frac {96 \sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^5 \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {96 \sqrt {b} \left (\sqrt {a}-\sqrt {b}\right )^5 \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}-a}}+21 b^2 \sin (4 (c+d x))+b^2 \sin (6 (c+d x))}{192 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^10/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/192*(36*b*(24*a + 35*b)*(c + d*x) - (96*(Sqrt[a] + Sqrt[b])^5*Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d
*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - (96*(Sqrt[a] - Sqrt[b])^5*Sqrt[b]*ArcTa
nh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 3*b*
(16*a + 95*b)*Sin[2*(c + d*x)] + 21*b^2*Sin[4*(c + d*x)] + b^2*Sin[6*(c + d*x)])/(b^3*d)

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fricas [B]  time = 1.96, size = 2948, normalized size = 11.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/48*(6*b^2*d*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*
a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*
b^5*d^2))*log(9/4*a^8 + 12*a^7*b - 39*a^6*b^2 + 143/2*a^4*b^4 - 52*a^3*b^5 - 3*a^2*b^6 + 8*a*b^7 + 1/4*b^8 - 1
/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 +
 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 924
0*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) + (9*a^7*b^2 + 138*a^6*b^3
 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^5*d^2*
sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*
b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 -
4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7
 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 154
8*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) - 6*b^2*d*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2
 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3
*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*b^5*d^2))*log(9/4*a^8 + 12*a^7*b - 39*a^6*b^2 + 143/2*a^4*b^4 - 52*a^3
*b^5 - 3*a^2*b^6 + 8*a*b^7 + 1/4*b^8 - 1/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^
2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 - 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b
^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)
*sin(d*x + c) + (9*a^7*b^2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos
(d*x + c)*sin(d*x + c))*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b
^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 -
9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6
*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*
b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) + 6*b^2*d*sqrt(-(a*b^5*d^2
*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a
*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2))*log(-9/4*a^8 - 12*a
^7*b + 39*a^6*b^2 - 143/2*a^4*b^4 + 52*a^3*b^5 + 3*a^2*b^6 - 8*a*b^7 - 1/4*b^8 + 1/4*(9*a^8 + 48*a^7*b - 156*a
^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^4*b^7 + a^3*b^8)*
d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 7
2*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) - (9*a^7*b^2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5
 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b +
 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) +
 a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a
^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a
^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)
/(a^3*b^9*d^4))) - 6*b^2*d*sqrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a
^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^
3 + 9*b^4)/(a*b^5*d^2))*log(-9/4*a^8 - 12*a^7*b + 39*a^6*b^2 - 143/2*a^4*b^4 + 52*a^3*b^5 + 3*a^2*b^6 - 8*a*b^
7 - 1/4*b^8 + 1/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*c
os(d*x + c)^2 - 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 2194
2*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) - (9*a^7*b^
2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*s
qrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548
*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2)) + 1/
4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4
*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 92
40*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) - 9*(24*a + 35*b)*d*x - (8*b*cos(d*x + c)^5 + 34*b
*cos(d*x + c)^3 + 3*(8*a + 41*b)*cos(d*x + c))*sin(d*x + c))/(b^2*d)

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giac [B]  time = 1.16, size = 896, normalized size = 3.56 \[ \frac {\frac {24 \, {\left (15 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b - 62 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} - 16 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{4} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{5} - 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} - 24 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b + 46 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} + 40 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} + 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} + \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{5} b^{3} - 12 \, a^{4} b^{4} + 14 \, a^{3} b^{5} - 4 \, a^{2} b^{6} - a b^{7}} + \frac {24 \, {\left (15 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b - 62 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} - 16 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{4} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{5} + 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} + 24 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b - 46 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 40 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} - \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{5} b^{3} - 12 \, a^{4} b^{4} + 14 \, a^{3} b^{5} - 4 \, a^{2} b^{6} - a b^{7}} - \frac {9 \, {\left (d x + c\right )} {\left (24 \, a + 35 \, b\right )}}{b^{2}} - \frac {24 \, a \tan \left (d x + c\right )^{5} + 123 \, b \tan \left (d x + c\right )^{5} + 48 \, a \tan \left (d x + c\right )^{3} + 280 \, b \tan \left (d x + c\right )^{3} + 24 \, a \tan \left (d x + c\right ) + 165 \, b \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3} b^{2}}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

1/48*(24*(15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 62*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 - 16*s
qrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^5 - 3*sqrt(a^2 - a*b + sqrt(a
*b)*(a - b))*sqrt(a*b)*a^4 - 24*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 46*sqrt(a^2 - a*b + sqrt
(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 40*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + 5*sqrt(a^2 - a*b +
 sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 + sqrt(a^2*
b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^3 - 12*a^4*b^4 + 14*a^3*b^5 - 4*a^2*b^6 - a*
b^7) + 24*(15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 62*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 - 16*
sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^5 + 3*sqrt(a^2 - a*b - sqrt(
a*b)*(a - b))*sqrt(a*b)*a^4 + 24*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 46*sqrt(a^2 - a*b - sqr
t(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 40*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 5*sqrt(a^2 - a*b
- sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 - sqrt(a^2
*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^3 - 12*a^4*b^4 + 14*a^3*b^5 - 4*a^2*b^6 - a
*b^7) - 9*(d*x + c)*(24*a + 35*b)/b^2 - (24*a*tan(d*x + c)^5 + 123*b*tan(d*x + c)^5 + 48*a*tan(d*x + c)^3 + 28
0*b*tan(d*x + c)^3 + 24*a*tan(d*x + c) + 165*b*tan(d*x + c))/((tan(d*x + c)^2 + 1)^3*b^2))/d

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maple [B]  time = 0.67, size = 880, normalized size = 3.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^10/(a-b*sin(d*x+c)^4),x)

[Out]

-1/2/d/b^2/(tan(d*x+c)^2+1)^3*tan(d*x+c)^5*a-41/16/d/b/(tan(d*x+c)^2+1)^3*tan(d*x+c)^5-1/d/b^2/(tan(d*x+c)^2+1
)^3*tan(d*x+c)^3*a-35/6/d/b/(tan(d*x+c)^2+1)^3*tan(d*x+c)^3-1/2/d/b^2/(tan(d*x+c)^2+1)^3*tan(d*x+c)*a-55/16/d/
b/(tan(d*x+c)^2+1)^3*tan(d*x+c)-9/2/d/b^2*arctan(tan(d*x+c))*a-105/16/d/b*arctan(tan(d*x+c))+2/d/b^2/(((a*b)^(
1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*a^2+1/2/d/b^2/(a*b)^(1/2)/(((a*b
)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*a^3+5/2/d/b/(a*b)^(1/2)/(((a*
b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*a^2-5/2/d*a/(a*b)^(1/2)/(((a
*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+2/d/b^2/(((a*b)^(1/2)+a)*(a
-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^2-1/2/d/b^2/(a*b)^(1/2)/(((a*b)^(1/2)+a)*(
a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^3-5/2/d/b/(a*b)^(1/2)/(((a*b)^(1/2)+a)*(a
-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^2+5/2/d*a/(a*b)^(1/2)/(((a*b)^(1/2)+a)*(a-
b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-2/d/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a
+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-1/2/d*b/(a*b)^(1/2)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)
*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-2/d/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(
1/2)+a)*(a-b))^(1/2))+1/2/d*b/(a*b)^(1/2)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+
a)*(a-b))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/192*(192*b^2*d*integrate(-4*(4*(a^2*b + 10*a*b^2 + 5*b^3)*cos(6*d*x + 6*c)^2 + 4*(72*a^3 + 53*a^2*b - 54*a*
b^2 + 9*b^3)*cos(4*d*x + 4*c)^2 + 4*(a^2*b + 10*a*b^2 + 5*b^3)*cos(2*d*x + 2*c)^2 + 4*(a^2*b + 10*a*b^2 + 5*b^
3)*sin(6*d*x + 6*c)^2 + 4*(72*a^3 + 53*a^2*b - 54*a*b^2 + 9*b^3)*sin(4*d*x + 4*c)^2 + 2*(8*a^3 + 113*a^2*b + 5
0*a*b^2 - 27*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a^2*b + 10*a*b^2 + 5*b^3)*sin(2*d*x + 2*c)^2 - ((a^2*
b + 10*a*b^2 + 5*b^3)*cos(6*d*x + 6*c) + 2*(9*a^2*b + 10*a*b^2 - 3*b^3)*cos(4*d*x + 4*c) + (a^2*b + 10*a*b^2 +
 5*b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (a^2*b + 10*a*b^2 + 5*b^3 - 2*(8*a^3 + 113*a^2*b + 50*a*b^2 - 27*
b^3)*cos(4*d*x + 4*c) - 8*(a^2*b + 10*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 2*(9*a^2*b + 10*a*b^
2 - 3*b^3 - (8*a^3 + 113*a^2*b + 50*a*b^2 - 27*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a^2*b + 10*a*b^2 + 5
*b^3)*cos(2*d*x + 2*c) - ((a^2*b + 10*a*b^2 + 5*b^3)*sin(6*d*x + 6*c) + 2*(9*a^2*b + 10*a*b^2 - 3*b^3)*sin(4*d
*x + 4*c) + (a^2*b + 10*a*b^2 + 5*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((8*a^3 + 113*a^2*b + 50*a*b^2 -
 27*b^3)*sin(4*d*x + 4*c) + 4*(a^2*b + 10*a*b^2 + 5*b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))/(b^4*cos(8*d*x +
8*c)^2 + 16*b^4*cos(6*d*x + 6*c)^2 + 16*b^4*cos(2*d*x + 2*c)^2 + b^4*sin(8*d*x + 8*c)^2 + 16*b^4*sin(6*d*x + 6
*c)^2 + 16*b^4*sin(2*d*x + 2*c)^2 - 8*b^4*cos(2*d*x + 2*c) + b^4 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*cos(4*d*x
 + 4*c)^2 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c)*sin(2
*d*x + 2*c) - 2*(4*b^4*cos(6*d*x + 6*c) + 4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*d*x + 4*c))
*cos(8*d*x + 8*c) + 8*(4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) -
 4*(8*a*b^3 - 3*b^4 - 4*(8*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*b^4*sin(6*d*x + 6*c) + 2*b
^4*sin(2*d*x + 2*c) + (8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*b^4*sin(2*d*x + 2*c) + (8*a
*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + 36*(24*a + 35*b)*d*x + b*sin(6*d*x + 6*c) + 21*b*sin(4
*d*x + 4*c) + 3*(16*a + 95*b)*sin(2*d*x + 2*c))/(b^2*d)

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mupad [B]  time = 18.80, size = 10319, normalized size = 40.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^10/(a - b*sin(c + d*x)^4),x)

[Out]

(atan(((((tan(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 +
780960*a^4*b^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6)
+ (3*(((9873*a*b^13)/16 + 8*b^14 + (198963*a^2*b^12)/16 - (13467*a^3*b^11)/8 - (240165*a^4*b^10)/8 + (68805*a^
5*b^9)/16 + (307047*a^6*b^8)/16 - 1929*a^7*b^7 - 2766*a^8*b^6 - 152*a^9*b^5)/b^8 + (3*((3*((64*a*b^15 + 1216*a
^2*b^14 - 2064*a^3*b^13 - 480*a^4*b^12 + 1968*a^5*b^11 - 704*a^6*b^10)/b^8 - (3*tan(c + d*x)*(a*24i + b*35i)*(
49152*a^2*b^13 - 49152*a^3*b^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(2048*b^8))*(a*24i + b*35i))/(32*b^2) - (t
an(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 4
98944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*(a*24i + b*35i))/(32*b^2))*(a*24i + b*35i))/(32*b^2)
)*(a*24i + b*35i)*3i)/(32*b^2) + (((tan(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826
*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 -
 74000*a^9*b^2))/(64*b^6) - (3*(((9873*a*b^13)/16 + 8*b^14 + (198963*a^2*b^12)/16 - (13467*a^3*b^11)/8 - (2401
65*a^4*b^10)/8 + (68805*a^5*b^9)/16 + (307047*a^6*b^8)/16 - 1929*a^7*b^7 - 2766*a^8*b^6 - 152*a^9*b^5)/b^8 + (
3*((3*((64*a*b^15 + 1216*a^2*b^14 - 2064*a^3*b^13 - 480*a^4*b^12 + 1968*a^5*b^11 - 704*a^6*b^10)/b^8 + (3*tan(
c + d*x)*(a*24i + b*35i)*(49152*a^2*b^13 - 49152*a^3*b^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(2048*b^8))*(a*2
4i + b*35i))/(32*b^2) + (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^10 - 919536*a^
4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*(a*24i + b*35i))/(32*b^2))
*(a*24i + b*35i))/(32*b^2))*(a*24i + b*35i)*3i)/(32*b^2))/(((92769*a*b^11)/64 - (39*a^11*b)/8 + 9*a^12 - (1186
5*b^12)/64 - (76467*a^2*b^10)/16 + (133839*a^3*b^9)/16 - (243927*a^4*b^8)/32 + (58743*a^5*b^7)/32 + (50967*a^6
*b^6)/16 - (52227*a^7*b^5)/16 + (61119*a^8*b^4)/64 + (12729*a^9*b^3)/64 - (1137*a^10*b^2)/8)/b^8 - (3*((tan(c
+ d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b^7 -
 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6) + (3*(((9873*a*b
^13)/16 + 8*b^14 + (198963*a^2*b^12)/16 - (13467*a^3*b^11)/8 - (240165*a^4*b^10)/8 + (68805*a^5*b^9)/16 + (307
047*a^6*b^8)/16 - 1929*a^7*b^7 - 2766*a^8*b^6 - 152*a^9*b^5)/b^8 + (3*((3*((64*a*b^15 + 1216*a^2*b^14 - 2064*a
^3*b^13 - 480*a^4*b^12 + 1968*a^5*b^11 - 704*a^6*b^10)/b^8 - (3*tan(c + d*x)*(a*24i + b*35i)*(49152*a^2*b^13 -
 49152*a^3*b^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(2048*b^8))*(a*24i + b*35i))/(32*b^2) - (tan(c + d*x)*(617
264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 +
232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*(a*24i + b*35i))/(32*b^2))*(a*24i + b*35i))/(32*b^2))*(a*24i + b*35i
))/(32*b^2) + (3*((tan(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a
^3*b^8 + 780960*a^4*b^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/
(64*b^6) - (3*(((9873*a*b^13)/16 + 8*b^14 + (198963*a^2*b^12)/16 - (13467*a^3*b^11)/8 - (240165*a^4*b^10)/8 +
(68805*a^5*b^9)/16 + (307047*a^6*b^8)/16 - 1929*a^7*b^7 - 2766*a^8*b^6 - 152*a^9*b^5)/b^8 + (3*((3*((64*a*b^15
 + 1216*a^2*b^14 - 2064*a^3*b^13 - 480*a^4*b^12 + 1968*a^5*b^11 - 704*a^6*b^10)/b^8 + (3*tan(c + d*x)*(a*24i +
 b*35i)*(49152*a^2*b^13 - 49152*a^3*b^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(2048*b^8))*(a*24i + b*35i))/(32*
b^2) + (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^10 - 919536*a^4*b^9 - 469488*a^
5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*(a*24i + b*35i))/(32*b^2))*(a*24i + b*35i))
/(32*b^2))*(a*24i + b*35i))/(32*b^2)))*(a*24i + b*35i)*3i)/(16*b^2*d) - (atan(((((78984*a*b^13 + 1024*b^14 + 1
591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b^8 - 246912*a^7*b^7 - 354
048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^3*b^13 - 61440*a^4*b^12 +
 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) - (tan(c + d*x)*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) -
 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^
11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*a^3*b^12 - 49152*a^4*b^11
+ 49152*a^5*b^10))/(64*b^6))*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^
4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^
(1/2))/(16*a^3*b^10))^(1/2) - (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^10 - 919
536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*((9*a^4*(a^3*b^11)^(
1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b
^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2))*((9*a^4*(a^3*b^11)^(
1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b
^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (tan(c + d*x)*(1239
62*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b^7 - 444642*a^5*
b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6))*((9*a^4*(a^3*b^11)^(1/2) +
b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1
/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*1i - (((78984*a*b^13 + 1024*
b^14 + 1591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b^8 - 246912*a^7*b
^7 - 354048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^3*b^13 - 61440*a^
4*b^12 + 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) + (tan(c + d*x)*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)
^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3
*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*a^3*b^12 - 49152*a
^4*b^11 + 49152*a^5*b^10))/(64*b^6))*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8
- 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^
3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^
10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*((9*a^4*(a^3
*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^
2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2))*((9*a^4*(a^3
*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^
2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) - (tan(c + d*
x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b^7 - 444
642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6))*((9*a^4*(a^3*b^11)^
(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*
b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*1i)/((((78984*a*b^13
 + 1024*b^14 + 1591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b^8 - 2469
12*a^7*b^7 - 354048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^3*b^13 -
61440*a^4*b^12 + 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) - (tan(c + d*x)*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a
^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) +
36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*a^3*b^12 -
 49152*a^4*b^11 + 49152*a^5*b^10))/(64*b^6))*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*
a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a
^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) - (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 4651
2*a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*((9*
a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 12
6*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2))*((9*
a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 12
6*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (ta
n(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b
^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6))*((9*a^4*(a^
3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b
^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (((78984*a
*b^13 + 1024*b^14 + 1591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b^8 -
 246912*a^7*b^7 - 354048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^3*b^
13 - 61440*a^4*b^12 + 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) + (tan(c + d*x)*((9*a^4*(a^3*b^11)^(1/2) + b
^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/
2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*a^3*b
^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(64*b^6))*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9
- 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) +
 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 +
 46512*a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))
*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5
 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2))
*((9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5
 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)
- (tan(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*
a^4*b^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6))*((9*a^
4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*
a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) - (9276
9*a*b^11 - 312*a^11*b + 576*a^12 - 11865*b^12 - 305868*a^2*b^10 + 535356*a^3*b^9 - 487854*a^4*b^8 + 117486*a^5
*b^7 + 203868*a^6*b^6 - 208908*a^7*b^5 + 61119*a^8*b^4 + 12729*a^9*b^3 - 9096*a^10*b^2)/(64*b^8)))*((9*a^4*(a^
3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) - 9*a^2*b^9 - 84*a^3*b^8 - 126*a^4*b^7 - 36*a^5*b^6 - a^6*b^5 + 126*a^2*b
^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*2i)/d - ((ta
n(c + d*x)*(8*a + 55*b))/(16*b^2) + (tan(c + d*x)^3*(6*a + 35*b))/(6*b^2) + (tan(c + d*x)^5*(8*a + 41*b))/(16*
b^2))/(d*(3*tan(c + d*x)^2 + 3*tan(c + d*x)^4 + tan(c + d*x)^6 + 1)) - (atan(((((78984*a*b^13 + 1024*b^14 + 15
91704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b^8 - 246912*a^7*b^7 - 3540
48*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^3*b^13 - 61440*a^4*b^12 +
251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) - (tan(c + d*x)*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) +
 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^
11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*a^3*b^12 - 49152*a^4*b^11
+ 49152*a^5*b^10))/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a
^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)
^(1/2))/(16*a^3*b^10))^(1/2) - (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*a^3*b^10 - 91
9536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*(-(9*a^4*(a^3*b^11)
^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3
*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2))*(-(9*a^4*(a^3*b^11
)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^
3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (tan(c + d*x)*(1
23962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b^7 - 444642*a
^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2
) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11
)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*1i - (((78984*a*b^13 + 1
024*b^14 + 1591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b^8 - 246912*a
^7*b^7 - 354048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^3*b^13 - 6144
0*a^4*b^12 + 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) + (tan(c + d*x)*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*
b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*
a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*a^3*b^12 - 49
152*a^4*b^11 + 49152*a^5*b^10))/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^
3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3
*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*b^12 + 46512*
a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64*b^6))*(-(9*a
^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126
*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2))*(-(9*
a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 12
6*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) - (ta
n(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 + 780960*a^4*b
^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6))*(-(9*a^4*(a
^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*
b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*1i)/((((789
84*a*b^13 + 1024*b^14 + 1591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 2456376*a^6*b
^8 - 246912*a^7*b^7 - 354048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 - 264192*a^
3*b^13 - 61440*a^4*b^12 + 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) - (tan(c + d*x)*(-(9*a^4*(a^3*b^11)^(1/2
) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11
)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^13 - 49152*
a^3*b^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^
2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(
1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) - (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13 - 10240*a*
b^12 + 46512*a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8*b^5))/(64
*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 +
a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))
^(1/2))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 +
 a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10)
)^(1/2) + (tan(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370*a^3*b^8 +
 780960*a^4*b^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2))/(64*b^6)
)*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b
^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2
) + (((78984*a*b^13 + 1024*b^14 + 1591704*a^2*b^12 - 215472*a^3*b^11 - 3842640*a^4*b^10 + 550440*a^5*b^9 + 245
6376*a^6*b^8 - 246912*a^7*b^7 - 354048*a^8*b^6 - 19456*a^9*b^5)/(128*b^8) + (((8192*a*b^15 + 155648*a^2*b^14 -
 264192*a^3*b^13 - 61440*a^4*b^12 + 251904*a^5*b^11 - 90112*a^6*b^10)/(128*b^8) + (tan(c + d*x)*(-(9*a^4*(a^3*
b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2
*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2)*(49152*a^2*b^1
3 - 49152*a^3*b^12 - 49152*a^4*b^11 + 49152*a^5*b^10))/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1
/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a
^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10))^(1/2) + (tan(c + d*x)*(617264*a^2*b^11 - 1024*b^13
- 10240*a*b^12 + 46512*a^3*b^10 - 919536*a^4*b^9 - 469488*a^5*b^8 + 498944*a^6*b^7 + 232448*a^7*b^6 + 5120*a^8
*b^5))/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*
a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*
a^3*b^10))^(1/2))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36
*a^5*b^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16
*a^3*b^10))^(1/2) - (tan(c + d*x)*(123962*a*b^10 - 3776*a^10*b - 128*a^11 + 11153*b^11 - 387826*a^2*b^9 + 2370
*a^3*b^8 + 780960*a^4*b^7 - 444642*a^5*b^6 - 387534*a^6*b^5 + 261366*a^7*b^4 + 118095*a^8*b^3 - 74000*a^9*b^2)
)/(64*b^6))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b
^6 + a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b
^10))^(1/2) - (92769*a*b^11 - 312*a^11*b + 576*a^12 - 11865*b^12 - 305868*a^2*b^10 + 535356*a^3*b^9 - 487854*a
^4*b^8 + 117486*a^5*b^7 + 203868*a^6*b^6 - 208908*a^7*b^5 + 61119*a^8*b^4 + 12729*a^9*b^3 - 9096*a^10*b^2)/(64
*b^8)))*(-(9*a^4*(a^3*b^11)^(1/2) + b^4*(a^3*b^11)^(1/2) + 9*a^2*b^9 + 84*a^3*b^8 + 126*a^4*b^7 + 36*a^5*b^6 +
 a^6*b^5 + 126*a^2*b^2*(a^3*b^11)^(1/2) + 36*a*b^3*(a^3*b^11)^(1/2) + 84*a^3*b*(a^3*b^11)^(1/2))/(16*a^3*b^10)
)^(1/2)*2i)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**10/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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