3.112 \(\int \frac{1}{(a+b \sin ^2(c+d x))^5} \, dx\)

Optimal. Leaf size=279 \[ \frac{\left (288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{128 a^{9/2} d (a+b)^{9/2}}+\frac{5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{384 a^4 d (a+b)^4 \left (a+b \sin ^2(c+d x)\right )}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{192 a^3 d (a+b)^3 \left (a+b \sin ^2(c+d x)\right )^2}+\frac{7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{48 a^2 d (a+b)^2 \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4} \]

[Out]

((128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(128*a^(
9/2)*(a + b)^(9/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^4) + (7*b*(2*a + b
)*Cos[c + d*x]*Sin[c + d*x])/(48*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2)^3) + (b*(104*a^2 + 104*a*b + 35*b^2)*C
os[c + d*x]*Sin[c + d*x])/(192*a^3*(a + b)^3*d*(a + b*Sin[c + d*x]^2)^2) + (5*b*(2*a + b)*(40*a^2 + 40*a*b + 2
1*b^2)*Cos[c + d*x]*Sin[c + d*x])/(384*a^4*(a + b)^4*d*(a + b*Sin[c + d*x]^2))

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Rubi [A]  time = 0.530857, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3184, 3173, 12, 3181, 205} \[ \frac{\left (288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{128 a^{9/2} d (a+b)^{9/2}}+\frac{5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{384 a^4 d (a+b)^4 \left (a+b \sin ^2(c+d x)\right )}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{192 a^3 d (a+b)^3 \left (a+b \sin ^2(c+d x)\right )^2}+\frac{7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{48 a^2 d (a+b)^2 \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x]^2)^(-5),x]

[Out]

((128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(128*a^(
9/2)*(a + b)^(9/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^4) + (7*b*(2*a + b
)*Cos[c + d*x]*Sin[c + d*x])/(48*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2)^3) + (b*(104*a^2 + 104*a*b + 35*b^2)*C
os[c + d*x]*Sin[c + d*x])/(192*a^3*(a + b)^3*d*(a + b*Sin[c + d*x]^2)^2) + (5*b*(2*a + b)*(40*a^2 + 40*a*b + 2
1*b^2)*Cos[c + d*x]*Sin[c + d*x])/(384*a^4*(a + b)^4*d*(a + b*Sin[c + d*x]^2))

Rule 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
e + f*x]^2)^(p + 1))/(2*a*f*(p + 1)*(a + b)), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^(p
+ 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ
[a + b, 0] && LtQ[p, -1]

Rule 3173

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Sim
p[((A*b - a*B)*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1))/(2*a*f*(a + b)*(p + 1)), x] - Dist[1/
(2*a*(a + b)*(p + 1)), Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*(p + 1) + b*(2*p + 3)) + 2*(A*b -
a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

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{1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx &=\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}-\frac{\int \frac{-8 a-7 b+6 b \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx}{8 a (a+b)}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac{7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}-\frac{\int \frac{-48 a^2-76 a b-35 b^2+28 b (2 a+b) \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx}{48 a^2 (a+b)^2}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac{7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}-\frac{\int \frac{-192 a^3-392 a^2 b-340 a b^2-105 b^3+2 b \left (104 a^2+104 a b+35 b^2\right ) \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx}{192 a^3 (a+b)^3}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac{7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac{5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}-\frac{\int -\frac{3 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right )}{a+b \sin ^2(c+d x)} \, dx}{384 a^4 (a+b)^4}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac{7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac{5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}+\frac{\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \int \frac{1}{a+b \sin ^2(c+d x)} \, dx}{128 a^4 (a+b)^4}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac{7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac{5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}+\frac{\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{128 a^4 (a+b)^4 d}\\ &=\frac{\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{128 a^{9/2} (a+b)^{9/2} d}+\frac{b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac{7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac{b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac{5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.89236, size = 312, normalized size = 1.12 \[ \frac{\frac{24 \left (288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{(a+b)^{9/2}}+\frac{2 \sqrt{a} b \sin (2 (c+d x)) \left (-400 a^3 b^3 \cos (6 (c+d x))-600 a^2 b^4 \cos (6 (c+d x))-b \left (73616 a^3 b^2+41304 a^2 b^3+69120 a^4 b+27648 a^5+12310 a b^4+1575 b^5\right ) \cos (2 (c+d x))+2 b^2 \left (4816 a^2 b^2+5632 a^3 b+2816 a^4+2000 a b^3+315 b^4\right ) \cos (4 (c+d x))+97280 a^4 b^2+71680 a^3 b^3+32272 a^2 b^4+73728 a^5 b+24576 a^6-410 a b^5 \cos (6 (c+d x))+8720 a b^5-105 b^6 \cos (6 (c+d x))+1050 b^6\right )}{(a+b)^4 (2 a-b \cos (2 (c+d x))+b)^4}}{3072 a^{9/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x]^2)^(-5),x]

[Out]

((24*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a +
b)^(9/2) + (2*Sqrt[a]*b*(24576*a^6 + 73728*a^5*b + 97280*a^4*b^2 + 71680*a^3*b^3 + 32272*a^2*b^4 + 8720*a*b^5
+ 1050*b^6 - b*(27648*a^5 + 69120*a^4*b + 73616*a^3*b^2 + 41304*a^2*b^3 + 12310*a*b^4 + 1575*b^5)*Cos[2*(c +
d*x)] + 2*b^2*(2816*a^4 + 5632*a^3*b + 4816*a^2*b^2 + 2000*a*b^3 + 315*b^4)*Cos[4*(c + d*x)] - 400*a^3*b^3*Cos
[6*(c + d*x)] - 600*a^2*b^4*Cos[6*(c + d*x)] - 410*a*b^5*Cos[6*(c + d*x)] - 105*b^6*Cos[6*(c + d*x)])*Sin[2*(c
+ d*x)])/((a + b)^4*(2*a + b - b*Cos[2*(c + d*x)])^4))/(3072*a^(9/2)*d)

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Maple [B]  time = 0.098, size = 1249, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+sin(d*x+c)^2*b)^5,x)

[Out]

2/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b/a/(a+b)*tan(d*x+c)^7+9/4/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b^2/a
^2/(a+b)*tan(d*x+c)^7+5/4/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b^3/a^3/(a+b)*tan(d*x+c)^7+35/128/d/(a*tan(d*x
+c)^2+tan(d*x+c)^2*b+a)^4*b^4/a^4/(a+b)*tan(d*x+c)^7+6/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b/(a^2+2*a*b+b^2)
*tan(d*x+c)^5+33/4/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4/a*b^2/(a^2+2*a*b+b^2)*tan(d*x+c)^5+55/12/d/(a*tan(d*x
+c)^2+tan(d*x+c)^2*b+a)^4/a^2*b^3/(a^2+2*a*b+b^2)*tan(d*x+c)^5+385/384/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4/a
^3*b^4/(a^2+2*a*b+b^2)*tan(d*x+c)^5+6/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*a*b/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(
d*x+c)^3+39/4/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b^2/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)^3+73/12/d/(a*tan(
d*x+c)^2+tan(d*x+c)^2*b+a)^4/a*b^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)^3+511/384/d/(a*tan(d*x+c)^2+tan(d*x+c)
^2*b+a)^4/a^2*b^4/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)^3+2/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b*a^2/(a^4+4*
a^3*b+6*a^2*b^2+4*a*b^3+b^4)*tan(d*x+c)+15/4/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b^2*a/(a^4+4*a^3*b+6*a^2*b^
2+4*a*b^3+b^4)*tan(d*x+c)+11/4/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b^3/(a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)*t
an(d*x+c)+93/128/d/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4*b^4/a/(a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)*tan(d*x+c)+1/
d/(a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))+2/d/a/(a^4+4*a^
3*b+6*a^2*b^2+4*a*b^3+b^4)/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b+9/4/d/a^2/(a^4+4*a^3*b+6
*a^2*b^2+4*a*b^3+b^4)/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b^2+5/4/d/a^3/(a^4+4*a^3*b+6*a^
2*b^2+4*a*b^3+b^4)/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b^3+35/128/d/a^4/(a^4+4*a^3*b+6*a^
2*b^2+4*a*b^3+b^4)/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))*b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x+c)^2)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.9847, size = 4771, normalized size = 17.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x+c)^2)^5,x, algorithm="fricas")

[Out]

[-1/1536*(3*((128*a^4*b^4 + 256*a^3*b^5 + 288*a^2*b^6 + 160*a*b^7 + 35*b^8)*cos(d*x + c)^8 + 128*a^8 + 768*a^7
*b + 2080*a^6*b^2 + 3360*a^5*b^3 + 3555*a^4*b^4 + 2508*a^3*b^5 + 1138*a^2*b^6 + 300*a*b^7 + 35*b^8 - 4*(128*a^
5*b^3 + 384*a^4*b^4 + 544*a^3*b^5 + 448*a^2*b^6 + 195*a*b^7 + 35*b^8)*cos(d*x + c)^6 + 6*(128*a^6*b^2 + 512*a^
5*b^3 + 928*a^4*b^4 + 992*a^3*b^5 + 643*a^2*b^6 + 230*a*b^7 + 35*b^8)*cos(d*x + c)^4 - 4*(128*a^7*b + 640*a^6*
b^2 + 1440*a^5*b^3 + 1920*a^4*b^4 + 1635*a^3*b^5 + 873*a^2*b^6 + 265*a*b^7 + 35*b^8)*cos(d*x + c)^2)*sqrt(-a^2
- a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(
d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*
(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*(5*(80*a^5*b^4 + 200*a^4*b^5 + 202*a^3*b^6 + 103*a^2*b^7
+ 21*a*b^8)*cos(d*x + c)^7 - (1408*a^6*b^3 + 4824*a^5*b^4 + 6724*a^4*b^5 + 4923*a^3*b^6 + 1930*a^2*b^7 + 315*a
*b^8)*cos(d*x + c)^5 + (1728*a^7*b^2 + 7456*a^6*b^3 + 13370*a^5*b^4 + 12969*a^4*b^5 + 7327*a^3*b^6 + 2315*a^2*
b^7 + 315*a*b^8)*cos(d*x + c)^3 - 3*(256*a^8*b + 1312*a^7*b^2 + 2848*a^6*b^3 + 3427*a^5*b^4 + 2508*a^4*b^5 + 1
138*a^3*b^6 + 300*a^2*b^7 + 35*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^10*b^4 + 5*a^9*b^5 + 10*a^8*b^6 + 10*a^7
*b^7 + 5*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^8 - 4*(a^11*b^3 + 6*a^10*b^4 + 15*a^9*b^5 + 20*a^8*b^6 + 15*a^7*b^7
+ 6*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^6 + 6*(a^12*b^2 + 7*a^11*b^3 + 21*a^10*b^4 + 35*a^9*b^5 + 35*a^8*b^6 +
21*a^7*b^7 + 7*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^4 - 4*(a^13*b + 8*a^12*b^2 + 28*a^11*b^3 + 56*a^10*b^4 + 70*a
^9*b^5 + 56*a^8*b^6 + 28*a^7*b^7 + 8*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^2 + (a^14 + 9*a^13*b + 36*a^12*b^2 + 84
*a^11*b^3 + 126*a^10*b^4 + 126*a^9*b^5 + 84*a^8*b^6 + 36*a^7*b^7 + 9*a^6*b^8 + a^5*b^9)*d), -1/768*(3*((128*a^
4*b^4 + 256*a^3*b^5 + 288*a^2*b^6 + 160*a*b^7 + 35*b^8)*cos(d*x + c)^8 + 128*a^8 + 768*a^7*b + 2080*a^6*b^2 +
3360*a^5*b^3 + 3555*a^4*b^4 + 2508*a^3*b^5 + 1138*a^2*b^6 + 300*a*b^7 + 35*b^8 - 4*(128*a^5*b^3 + 384*a^4*b^4
+ 544*a^3*b^5 + 448*a^2*b^6 + 195*a*b^7 + 35*b^8)*cos(d*x + c)^6 + 6*(128*a^6*b^2 + 512*a^5*b^3 + 928*a^4*b^4
+ 992*a^3*b^5 + 643*a^2*b^6 + 230*a*b^7 + 35*b^8)*cos(d*x + c)^4 - 4*(128*a^7*b + 640*a^6*b^2 + 1440*a^5*b^3 +
1920*a^4*b^4 + 1635*a^3*b^5 + 873*a^2*b^6 + 265*a*b^7 + 35*b^8)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((
2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))) + 2*(5*(80*a^5*b^4 + 200*a^4*b^5
+ 202*a^3*b^6 + 103*a^2*b^7 + 21*a*b^8)*cos(d*x + c)^7 - (1408*a^6*b^3 + 4824*a^5*b^4 + 6724*a^4*b^5 + 4923*a
^3*b^6 + 1930*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^5 + (1728*a^7*b^2 + 7456*a^6*b^3 + 13370*a^5*b^4 + 12969*a^4*b
^5 + 7327*a^3*b^6 + 2315*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^3 - 3*(256*a^8*b + 1312*a^7*b^2 + 2848*a^6*b^3 + 34
27*a^5*b^4 + 2508*a^4*b^5 + 1138*a^3*b^6 + 300*a^2*b^7 + 35*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^10*b^4 + 5*
a^9*b^5 + 10*a^8*b^6 + 10*a^7*b^7 + 5*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^8 - 4*(a^11*b^3 + 6*a^10*b^4 + 15*a^9*
b^5 + 20*a^8*b^6 + 15*a^7*b^7 + 6*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^6 + 6*(a^12*b^2 + 7*a^11*b^3 + 21*a^10*b^4
+ 35*a^9*b^5 + 35*a^8*b^6 + 21*a^7*b^7 + 7*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^4 - 4*(a^13*b + 8*a^12*b^2 + 28*
a^11*b^3 + 56*a^10*b^4 + 70*a^9*b^5 + 56*a^8*b^6 + 28*a^7*b^7 + 8*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^2 + (a^14
+ 9*a^13*b + 36*a^12*b^2 + 84*a^11*b^3 + 126*a^10*b^4 + 126*a^9*b^5 + 84*a^8*b^6 + 36*a^7*b^7 + 9*a^6*b^8 + a^
5*b^9)*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(1/(a+b*sin(d*x+c)**2)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.16196, size = 707, normalized size = 2.53 \begin{align*} \frac{\frac{3 \,{\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )}{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac{a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt{a^{2} + a b}}\right )\right )}}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt{a^{2} + a b}} + \frac{768 \, a^{6} b \tan \left (d x + c\right )^{7} + 3168 \, a^{5} b^{2} \tan \left (d x + c\right )^{7} + 5376 \, a^{4} b^{3} \tan \left (d x + c\right )^{7} + 4905 \, a^{3} b^{4} \tan \left (d x + c\right )^{7} + 2619 \, a^{2} b^{5} \tan \left (d x + c\right )^{7} + 795 \, a b^{6} \tan \left (d x + c\right )^{7} + 105 \, b^{7} \tan \left (d x + c\right )^{7} + 2304 \, a^{6} b \tan \left (d x + c\right )^{5} + 7776 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} + 10400 \, a^{4} b^{3} \tan \left (d x + c\right )^{5} + 7073 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 2530 \, a^{2} b^{5} \tan \left (d x + c\right )^{5} + 385 \, a b^{6} \tan \left (d x + c\right )^{5} + 2304 \, a^{6} b \tan \left (d x + c\right )^{3} + 6048 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 6080 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} + 2847 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 511 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 768 \, a^{6} b \tan \left (d x + c\right ) + 1440 \, a^{5} b^{2} \tan \left (d x + c\right ) + 1056 \, a^{4} b^{3} \tan \left (d x + c\right ) + 279 \, a^{3} b^{4} \tan \left (d x + c\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )}{\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{4}}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x+c)^2)^5,x, algorithm="giac")

[Out]

1/384*(3*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b)
+ arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^
4)*sqrt(a^2 + a*b)) + (768*a^6*b*tan(d*x + c)^7 + 3168*a^5*b^2*tan(d*x + c)^7 + 5376*a^4*b^3*tan(d*x + c)^7 +
4905*a^3*b^4*tan(d*x + c)^7 + 2619*a^2*b^5*tan(d*x + c)^7 + 795*a*b^6*tan(d*x + c)^7 + 105*b^7*tan(d*x + c)^7
+ 2304*a^6*b*tan(d*x + c)^5 + 7776*a^5*b^2*tan(d*x + c)^5 + 10400*a^4*b^3*tan(d*x + c)^5 + 7073*a^3*b^4*tan(d*
x + c)^5 + 2530*a^2*b^5*tan(d*x + c)^5 + 385*a*b^6*tan(d*x + c)^5 + 2304*a^6*b*tan(d*x + c)^3 + 6048*a^5*b^2*t
an(d*x + c)^3 + 6080*a^4*b^3*tan(d*x + c)^3 + 2847*a^3*b^4*tan(d*x + c)^3 + 511*a^2*b^5*tan(d*x + c)^3 + 768*a
^6*b*tan(d*x + c) + 1440*a^5*b^2*tan(d*x + c) + 1056*a^4*b^3*tan(d*x + c) + 279*a^3*b^4*tan(d*x + c))/((a^8 +
4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^4))/d