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*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)

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Rubi [A]  time = 0.439522, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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 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 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 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]

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*}

Mathematica [A]  time = 0.91874, 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]

-(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])/S
qrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - (96*(Sqrt[a] - Sqrt[b])^5*Sqrt[b]*ArcTanh[((S
qrt[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)])/(192*b^3*d)

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Maple [B]  time = 0.132, size = 880, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

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]

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/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/(((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))-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))-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*arc
tan(tan(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 13.9979, size = 6926, normalized size = 27.48 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError