3.413 \(\int \frac {\cos ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=155 \[ -\frac {\left (\sqrt {a}-\sqrt {b}\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/2} d}-\frac {\sin (c+d x) \cos (c+d x)}{2 b d}-\frac {5 x}{2 b} \]

[Out]

-5/2*x/b-1/2*cos(d*x+c)*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)
)^(5/2)/a^(3/4)/b^(3/2)/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))^(5/2)/a^(3/
4)/b^(3/2)/d

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Rubi [A]  time = 0.29, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {\left (\sqrt {a}-\sqrt {b}\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/2} d}-\frac {\sin (c+d x) \cos (c+d x)}{2 b d}-\frac {5 x}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

(-5*x)/(2*b) - ((Sqrt[a] - Sqrt[b])^(5/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b
^(3/2)*d) + ((Sqrt[a] + Sqrt[b])^(5/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3
/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*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 ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2 \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 )^2}-\frac {2}{b \left (1+x^2\right )}+\frac {3 a+b+2 (a-b) x^2}{b \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 {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {\operatorname {Subst}\left (\int \frac {3 a+b+2 (a-b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d}\\ &=-\frac {2 x}{b}-\frac {\cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 b d}+\frac {\left (2 a-2 b+\frac {(a-b) (a+b)}{\sqrt {a} \sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b d}+\frac {\left (2 a-2 b-\frac {a^2-b^2}{\sqrt {a} \sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=-\frac {5 x}{2 b}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/2} d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/2} d}-\frac {\cos (c+d x) \sin (c+d x)}{2 b d}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 194, normalized size = 1.25 \[ \frac {\frac {2 \sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^3 \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {2 \sqrt {b} \left (\sqrt {a}-\sqrt {b}\right )^3 \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}-a}}-10 b (c+d x)-b \sin (2 (c+d x))}{4 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

(-10*b*(c + d*x) + (2*(Sqrt[a] + Sqrt[b])^3*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]]) + (2*(Sqrt[a] - Sqrt[b])^3*Sqrt[b]*ArcTanh[((Sqrt[a] - Sqrt[b]
)*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - b*Sin[2*(c + d*x)])/(4*b^2
*d)

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fricas [B]  time = 0.92, size = 1751, normalized size = 11.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/8*(b*d*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*
b - 5*b^2)/(a*b^3*d^2))*log(5/4*a^4 - 7/2*a^2*b^2 + 2*a*b^3 + 1/4*b^4 - 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^
4)*cos(d*x + c)^2 + 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))
*cos(d*x + c)*sin(d*x + c) + (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*
b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3
*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt
((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) - b*d*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*
a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3*d^2))*log(5/4*a^4 - 7/2*a^
2*b^2 + 2*a*b^3 + 1/4*b^4 - 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2 - 1/2*(2*a^3*b^4*d^3*sqrt(
(25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin(d*x + c) + (5*a^4*b + 15*a
^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b
^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b
^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^
3 + b^4)/(a^3*b^5*d^4))) + b*d*sqrt(-(a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*
b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*d^2))*log(-5/4*a^4 + 7/2*a^2*b^2 - 2*a*b^3 - 1/4*b^4 + 1/4*(5*a^4 - 1
4*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2 + 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b
^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin(d*x + c) - (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c
)*sin(d*x + c))*sqrt(-(a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2
+ 10*a*b + 5*b^2)/(a*b^3*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3
*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) - b*d*sqrt(-(a*b
^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*
d^2))*log(-5/4*a^4 + 7/2*a^2*b^2 - 2*a*b^3 - 1/4*b^4 + 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2
- 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin
(d*x + c) - (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^3*d^2*sqrt((25
*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*d^2)) + 1/4*(2*
(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a
^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) - 20*d*x - 4*cos(d*x + c)*sin(d*x + c))/(b*d)

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giac [B]  time = 1.00, size = 995, normalized size = 6.42 \[ -\frac {\frac {5 \, {\left (d x + c\right )}}{b} + \frac {{\left (2 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} b^{2} {\left | -a + b \right |} - {\left (9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b - 15 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left | -a + b \right |} {\left | b \right |} + {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b - 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 7 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left | -a + b \right |}\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 + \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right )} {\left | b \right |}} - \frac {{\left (2 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} b^{2} {\left | -a + b \right |} + {\left (9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b - 15 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left | -a + b \right |} {\left | b \right |} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b - 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 7 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left | -a + b \right |}\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 - \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right )} {\left | b \right |}} + \frac {\tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} b}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/2*(5*(d*x + c)/b + (2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(
a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (9*sqrt(a^2 - a*b
+ sqrt(a*b)*(a - b))*a^3*b - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*a^3*b - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 7*sqrt(a^2 - a*b + sqrt(a*b)
*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c
)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 12
*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)) - (2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 -
6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*
abs(-a + b) + (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 -
9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) + (3
*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2
- 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*
abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b - sqrt(a^2*b^2 - (a*b - b^2)*a*b))/
(a*b - b^2))))/((3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)) + tan(d*x + c)/((tan(d*x + c
)^2 + 1)*b))/d

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maple [B]  time = 0.66, size = 483, normalized size = 3.12 \[ -\frac {\tan \left (d x +c \right )}{2 d b \left (\tan ^{2}\left (d x +c \right )+1\right )}-\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{2 d b}+\frac {\arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right ) a^{2}}{2 d b \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{d b \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {\arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right ) a^{2}}{2 d b \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{d b \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {\arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{d \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {b \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {\arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{d \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {b \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a-b*sin(d*x+c)^4),x)

[Out]

-1/2/d/b*tan(d*x+c)/(tan(d*x+c)^2+1)-5/2/d/b*arctan(tan(d*x+c))+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))*a^2+1/d/b*a/(((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)*arctan((a-
b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^2+1/d/b*a/(((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/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))-1/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)^6/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/4*(4*b*d*integrate(-4*(4*(a*b + 3*b^2)*cos(6*d*x + 6*c)^2 + 4*(40*a^2 - 23*a*b + 3*b^2)*cos(4*d*x + 4*c)^2
+ 4*(a*b + 3*b^2)*cos(2*d*x + 2*c)^2 + 4*(a*b + 3*b^2)*sin(6*d*x + 6*c)^2 + 4*(40*a^2 - 23*a*b + 3*b^2)*sin(4*
d*x + 4*c)^2 + 2*(8*a^2 + 41*a*b - 13*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a*b + 3*b^2)*sin(2*d*x + 2*c
)^2 - ((a*b + 3*b^2)*cos(6*d*x + 6*c) + 2*(5*a*b - b^2)*cos(4*d*x + 4*c) + (a*b + 3*b^2)*cos(2*d*x + 2*c))*cos
(8*d*x + 8*c) - (a*b + 3*b^2 - 2*(8*a^2 + 41*a*b - 13*b^2)*cos(4*d*x + 4*c) - 8*(a*b + 3*b^2)*cos(2*d*x + 2*c)
)*cos(6*d*x + 6*c) - 2*(5*a*b - b^2 - (8*a^2 + 41*a*b - 13*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b + 3*
b^2)*cos(2*d*x + 2*c) - ((a*b + 3*b^2)*sin(6*d*x + 6*c) + 2*(5*a*b - b^2)*sin(4*d*x + 4*c) + (a*b + 3*b^2)*sin
(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((8*a^2 + 41*a*b - 13*b^2)*sin(4*d*x + 4*c) + 4*(a*b + 3*b^2)*sin(2*d*x +
2*c))*sin(6*d*x + 6*c))/(b^3*cos(8*d*x + 8*c)^2 + 16*b^3*cos(6*d*x + 6*c)^2 + 16*b^3*cos(2*d*x + 2*c)^2 + b^3*
sin(8*d*x + 8*c)^2 + 16*b^3*sin(6*d*x + 6*c)^2 + 16*b^3*sin(2*d*x + 2*c)^2 - 8*b^3*cos(2*d*x + 2*c) + b^3 + 4*
(64*a^2*b - 48*a*b^2 + 9*b^3)*cos(4*d*x + 4*c)^2 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*sin(4*d*x + 4*c)^2 + 16*(8*
a*b^2 - 3*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*b^3*cos(6*d*x + 6*c) + 4*b^3*cos(2*d*x + 2*c) - b^3 +
2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)
*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a*b^2 - 3*b^3 - 4*(8*a*b^2 - 3*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4
*c) - 4*(2*b^3*sin(6*d*x + 6*c) + 2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c
) + 16*(2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + 10*d*x + sin(2*d*
x + 2*c))/(b*d)

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mupad [B]  time = 18.04, size = 3088, normalized size = 19.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(a - b*sin(c + d*x)^4),x)

[Out]

(atan((a^4*b^8*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3
+ 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*240i - a^3*b^9*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*
b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*108i - a^5*b^7*sin
(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)
^(1/2))/(16*a^3*b^6))^(1/2)*80i - a^6*b^6*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*
b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*120i + a^7*b^5*sin(c + d*x)*(-(5*a^2*
(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6
))^(1/2)*60i + a^8*b^4*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 +
a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*8i - a^3*b^11*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^
2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*64i + a^4*b
^10*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a
^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*128i + a^5*b^9*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2)
+ 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*6080i + a^6*b^8*sin(c + d*x)*
(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(1
6*a^3*b^6))^(3/2)*4032i + a^7*b^7*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10
*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*320i + a^5*b^11*sin(c + d*x)*(-(5*a^2*(a^3*b^
7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(5/2
)*3072i + a^6*b^10*sin(c + d*x)*(-(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*
b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(5/2)*3072i)/(55*a^2*b^9*cos(c + d*x) + 540*a^3*b^8*cos(c + d*x) +
1035*a^4*b^7*cos(c + d*x) + 45*a^5*b^6*cos(c + d*x) + a^6*b^5*cos(c + d*x) + 110*a^7*b^4*cos(c + d*x) + 5*a^8
*b^3*cos(c + d*x) + 50*a^6*cos(c + d*x)*(a^3*b^7)^(1/2) + 10*b^6*cos(c + d*x)*(a^3*b^7)^(1/2) + a*b^10*cos(c +
d*x) + 195*a*b^5*cos(c + d*x)*(a^3*b^7)^(1/2) + 75*a^5*b*cos(c + d*x)*(a^3*b^7)^(1/2) + 1002*a^2*b^4*cos(c +
d*x)*(a^3*b^7)^(1/2) + 490*a^3*b^3*cos(c + d*x)*(a^3*b^7)^(1/2) - 30*a^4*b^2*cos(c + d*x)*(a^3*b^7)^(1/2)))*(-
(5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) + 5*a^2*b^5 + 10*a^3*b^4 + a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*
a^3*b^6))^(1/2)*2i)/d + (atan((a^4*b^8*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5
- 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*240i - a^3*b^9*sin(c + d*x)*((5*a^2*(a^3*
b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1
/2)*108i - a^5*b^7*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b
^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*80i - a^6*b^6*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3
*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*120i + a^7*b^5*si
n(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)
^(1/2))/(16*a^3*b^6))^(1/2)*60i + a^8*b^4*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b
^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(1/2)*8i - a^3*b^11*sin(c + d*x)*((5*a^2*(a^
3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^
(3/2)*64i + a^4*b^10*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4
*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*128i + a^5*b^9*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(
a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*6080i + a^6*b^
8*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*
b^7)^(1/2))/(16*a^3*b^6))^(3/2)*4032i + a^7*b^7*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5
*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(3/2)*320i + a^5*b^11*sin(c + d*x)*((5
*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^
3*b^6))^(5/2)*3072i + a^6*b^10*sin(c + d*x)*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3
*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(1/2))/(16*a^3*b^6))^(5/2)*3072i)/(55*a^2*b^9*cos(c + d*x) + 540*a^3*b^8*cos
(c + d*x) + 1035*a^4*b^7*cos(c + d*x) + 45*a^5*b^6*cos(c + d*x) + a^6*b^5*cos(c + d*x) + 110*a^7*b^4*cos(c + d
*x) + 5*a^8*b^3*cos(c + d*x) - 50*a^6*cos(c + d*x)*(a^3*b^7)^(1/2) - 10*b^6*cos(c + d*x)*(a^3*b^7)^(1/2) + a*b
^10*cos(c + d*x) - 195*a*b^5*cos(c + d*x)*(a^3*b^7)^(1/2) - 75*a^5*b*cos(c + d*x)*(a^3*b^7)^(1/2) - 1002*a^2*b
^4*cos(c + d*x)*(a^3*b^7)^(1/2) - 490*a^3*b^3*cos(c + d*x)*(a^3*b^7)^(1/2) + 30*a^4*b^2*cos(c + d*x)*(a^3*b^7)
^(1/2)))*((5*a^2*(a^3*b^7)^(1/2) + b^2*(a^3*b^7)^(1/2) - 5*a^2*b^5 - 10*a^3*b^4 - a^4*b^3 + 10*a*b*(a^3*b^7)^(
1/2))/(16*a^3*b^6))^(1/2)*2i)/d - sin(2*c + 2*d*x)/(4*b*d) - (5*atan(sin(c + d*x)/cos(c + d*x)))/(2*b*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)**6/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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