\(\int \frac {1}{(a+b \sin ^2(c+d x))^5} \, dx\) [112]

Optimal result

Integrand size = 14, antiderivative size = 279 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{128 a^{9/2} (a+b)^{9/2} d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )} \]

[Out]

1/128*(128*a^4+256*a^3*b+288*a^2*b^2+160*a*b^3+35*b^4)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))/a^(9/2)/(a+b)^(9
/2)/d+1/8*b*cos(d*x+c)*sin(d*x+c)/a/(a+b)/d/(a+b*sin(d*x+c)^2)^4+7/48*b*(2*a+b)*cos(d*x+c)*sin(d*x+c)/a^2/(a+b
)^2/d/(a+b*sin(d*x+c)^2)^3+1/192*b*(104*a^2+104*a*b+35*b^2)*cos(d*x+c)*sin(d*x+c)/a^3/(a+b)^3/d/(a+b*sin(d*x+c
)^2)^2+5/384*b*(2*a+b)*(40*a^2+40*a*b+21*b^2)*cos(d*x+c)*sin(d*x+c)/a^4/(a+b)^4/d/(a+b*sin(d*x+c)^2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3263, 3252, 12, 3260, 211} \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{48 a^2 d (a+b)^2 \left (a+b \sin ^2(c+d x)\right )^3}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{384 a^4 d (a+b)^4 \left (a+b \sin ^2(c+d x)\right )}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{192 a^3 d (a+b)^3 \left (a+b \sin ^2(c+d x)\right )^2}+\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{128 a^{9/2} d (a+b)^{9/2}}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4} \]

[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 12

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

Rule 211

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

Rule 3252

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp
[(-(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 3260

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 3263

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[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] && N
eQ[a + b, 0] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \arctan \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] (verified)

Time = 11.90 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {24 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2}}+\frac {2 \sqrt {a} b \left (24576 a^6+73728 a^5 b+97280 a^4 b^2+71680 a^3 b^3+32272 a^2 b^4+8720 a b^5+1050 b^6-b \left (27648 a^5+69120 a^4 b+73616 a^3 b^2+41304 a^2 b^3+12310 a b^4+1575 b^5\right ) \cos (2 (c+d x))+2 b^2 \left (2816 a^4+5632 a^3 b+4816 a^2 b^2+2000 a b^3+315 b^4\right ) \cos (4 (c+d x))-400 a^3 b^3 \cos (6 (c+d x))-600 a^2 b^4 \cos (6 (c+d x))-410 a b^5 \cos (6 (c+d x))-105 b^6 \cos (6 (c+d x))\right ) \sin (2 (c+d x))}{(a+b)^4 (2 a+b-b \cos (2 (c+d x)))^4}}{3072 a^{9/2} d} \]

[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)

Maple [A] (verified)

Time = 3.55 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {\frac {\frac {b \left (256 a^{3}+288 a^{2} b +160 a \,b^{2}+35 b^{3}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{128 a^{4} \left (a +b \right )}+\frac {\left (2304 a^{3}+3168 a^{2} b +1760 a \,b^{2}+385 b^{3}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{384 a^{3} \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2304 a^{3}+3744 a^{2} b +2336 a \,b^{2}+511 b^{3}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{384 a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b \left (256 a^{3}+480 a^{2} b +352 a \,b^{2}+93 b^{3}\right ) \tan \left (d x +c \right )}{128 a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{4}}+\frac {\left (128 a^{4}+256 a^{3} b +288 a^{2} b^{2}+160 a \,b^{3}+35 b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{128 a^{4} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \sqrt {a \left (a +b \right )}}}{d}\)	\(336\)
default	\(\frac {\frac {\frac {b \left (256 a^{3}+288 a^{2} b +160 a \,b^{2}+35 b^{3}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{128 a^{4} \left (a +b \right )}+\frac {\left (2304 a^{3}+3168 a^{2} b +1760 a \,b^{2}+385 b^{3}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{384 a^{3} \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2304 a^{3}+3744 a^{2} b +2336 a \,b^{2}+511 b^{3}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{384 a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b \left (256 a^{3}+480 a^{2} b +352 a \,b^{2}+93 b^{3}\right ) \tan \left (d x +c \right )}{128 a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{4}}+\frac {\left (128 a^{4}+256 a^{3} b +288 a^{2} b^{2}+160 a \,b^{3}+35 b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{128 a^{4} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \sqrt {a \left (a +b \right )}}}{d}\)	\(336\)
risch	\(\text {Expression too large to display}\)	\(1743\)

[In]

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

[Out]

1/d*((1/128*b*(256*a^3+288*a^2*b+160*a*b^2+35*b^3)/a^4/(a+b)*tan(d*x+c)^7+1/384*(2304*a^3+3168*a^2*b+1760*a*b^
2+385*b^3)/a^3*b/(a^2+2*a*b+b^2)*tan(d*x+c)^5+1/384*(2304*a^3+3744*a^2*b+2336*a*b^2+511*b^3)/a^2*b/(a^3+3*a^2*
b+3*a*b^2+b^3)*tan(d*x+c)^3+1/128*b*(256*a^3+480*a^2*b+352*a*b^2+93*b^3)/a/(a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)
*tan(d*x+c))/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+a)^4+1/128*(128*a^4+256*a^3*b+288*a^2*b^2+160*a*b^3+35*b^4)/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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (261) = 522\).

Time = 0.38 (sec) , antiderivative size = 2017, normalized size of antiderivative = 7.23 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (261) = 522\).

Time = 0.40 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {3 \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (256 \, a^{6} b + 1056 \, a^{5} b^{2} + 1792 \, a^{4} b^{3} + 1635 \, a^{3} b^{4} + 873 \, a^{2} b^{5} + 265 \, a b^{6} + 35 \, b^{7}\right )} \tan \left (d x + c\right )^{7} + {\left (2304 \, a^{6} b + 7776 \, a^{5} b^{2} + 10400 \, a^{4} b^{3} + 7073 \, a^{3} b^{4} + 2530 \, a^{2} b^{5} + 385 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + {\left (2304 \, a^{6} b + 6048 \, a^{5} b^{2} + 6080 \, a^{4} b^{3} + 2847 \, a^{3} b^{4} + 511 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (256 \, a^{6} b + 480 \, a^{5} b^{2} + 352 \, a^{4} b^{3} + 93 \, a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{12} + 4 \, a^{11} b + 6 \, a^{10} b^{2} + 4 \, a^{9} b^{3} + a^{8} b^{4} + {\left (a^{12} + 8 \, a^{11} b + 28 \, a^{10} b^{2} + 56 \, a^{9} b^{3} + 70 \, a^{8} b^{4} + 56 \, a^{7} b^{5} + 28 \, a^{6} b^{6} + 8 \, a^{5} b^{7} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{8} + 4 \, {\left (a^{12} + 7 \, a^{11} b + 21 \, a^{10} b^{2} + 35 \, a^{9} b^{3} + 35 \, a^{8} b^{4} + 21 \, a^{7} b^{5} + 7 \, a^{6} b^{6} + a^{5} b^{7}\right )} \tan \left (d x + c\right )^{6} + 6 \, {\left (a^{12} + 6 \, a^{11} b + 15 \, a^{10} b^{2} + 20 \, a^{9} b^{3} + 15 \, a^{8} b^{4} + 6 \, a^{7} b^{5} + a^{6} b^{6}\right )} \tan \left (d x + c\right )^{4} + 4 \, {\left (a^{12} + 5 \, a^{11} b + 10 \, a^{10} b^{2} + 10 \, a^{9} b^{3} + 5 \, a^{8} b^{4} + a^{7} b^{5}\right )} \tan \left (d x + c\right )^{2}}}{384 \, d} \]

[In]

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

[Out]

1/384*(3*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))
/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4)*sqrt((a + b)*a)) + (3*(256*a^6*b + 1056*a^5*b^2 + 1792*a^4
*b^3 + 1635*a^3*b^4 + 873*a^2*b^5 + 265*a*b^6 + 35*b^7)*tan(d*x + c)^7 + (2304*a^6*b + 7776*a^5*b^2 + 10400*a^
4*b^3 + 7073*a^3*b^4 + 2530*a^2*b^5 + 385*a*b^6)*tan(d*x + c)^5 + (2304*a^6*b + 6048*a^5*b^2 + 6080*a^4*b^3 +
2847*a^3*b^4 + 511*a^2*b^5)*tan(d*x + c)^3 + 3*(256*a^6*b + 480*a^5*b^2 + 352*a^4*b^3 + 93*a^3*b^4)*tan(d*x +
c))/(a^12 + 4*a^11*b + 6*a^10*b^2 + 4*a^9*b^3 + a^8*b^4 + (a^12 + 8*a^11*b + 28*a^10*b^2 + 56*a^9*b^3 + 70*a^8
*b^4 + 56*a^7*b^5 + 28*a^6*b^6 + 8*a^5*b^7 + a^4*b^8)*tan(d*x + c)^8 + 4*(a^12 + 7*a^11*b + 21*a^10*b^2 + 35*a
^9*b^3 + 35*a^8*b^4 + 21*a^7*b^5 + 7*a^6*b^6 + a^5*b^7)*tan(d*x + c)^6 + 6*(a^12 + 6*a^11*b + 15*a^10*b^2 + 20
*a^9*b^3 + 15*a^8*b^4 + 6*a^7*b^5 + a^6*b^6)*tan(d*x + c)^4 + 4*(a^12 + 5*a^11*b + 10*a^10*b^2 + 10*a^9*b^3 +
5*a^8*b^4 + a^7*b^5)*tan(d*x + c)^2))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (261) = 522\).

Time = 0.38 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {3 \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt {a^{2} + a b}} + \frac {768 \, a^{6} b \tan \left (d x + c\right )^{7} + 3168 \, a^{5} b^{2} \tan \left (d x + c\right )^{7} + 5376 \, a^{4} b^{3} \tan \left (d x + c\right )^{7} + 4905 \, a^{3} b^{4} \tan \left (d x + c\right )^{7} + 2619 \, a^{2} b^{5} \tan \left (d x + c\right )^{7} + 795 \, a b^{6} \tan \left (d x + c\right )^{7} + 105 \, b^{7} \tan \left (d x + c\right )^{7} + 2304 \, a^{6} b \tan \left (d x + c\right )^{5} + 7776 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} + 10400 \, a^{4} b^{3} \tan \left (d x + c\right )^{5} + 7073 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 2530 \, a^{2} b^{5} \tan \left (d x + c\right )^{5} + 385 \, a b^{6} \tan \left (d x + c\right )^{5} + 2304 \, a^{6} b \tan \left (d x + c\right )^{3} + 6048 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 6080 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} + 2847 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 511 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 768 \, a^{6} b \tan \left (d x + c\right ) + 1440 \, a^{5} b^{2} \tan \left (d x + c\right ) + 1056 \, a^{4} b^{3} \tan \left (d x + c\right ) + 279 \, a^{3} b^{4} \tan \left (d x + c\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{4}}}{384 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 16.12 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (256\,a^3\,b+480\,a^2\,b^2+352\,a\,b^3+93\,b^4\right )}{128\,a\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2304\,a^3\,b+3744\,a^2\,b^2+2336\,a\,b^3+511\,b^4\right )}{384\,a^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (2304\,a^3\,b+3168\,a^2\,b^2+1760\,a\,b^3+385\,b^4\right )}{384\,a^3\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (256\,a^3\,b+288\,a^2\,b^2+160\,a\,b^3+35\,b^4\right )}{128\,a^4\,\left (a+b\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (6\,a^4+12\,a^3\,b+6\,a^2\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (4\,a^4+4\,b\,a^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (4\,a^4+12\,a^3\,b+12\,a^2\,b^2+4\,a\,b^3\right )+a^4+{\mathrm {tan}\left (c+d\,x\right )}^8\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )\right )}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{9/2}}\right )\,\left (128\,a^4+256\,a^3\,b+288\,a^2\,b^2+160\,a\,b^3+35\,b^4\right )}{128\,a^{9/2}\,d\,{\left (a+b\right )}^{9/2}} \]

[In]

int(1/(a + b*sin(c + d*x)^2)^5,x)

[Out]

((tan(c + d*x)*(352*a*b^3 + 256*a^3*b + 93*b^4 + 480*a^2*b^2))/(128*a*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b
^2)) + (tan(c + d*x)^3*(2336*a*b^3 + 2304*a^3*b + 511*b^4 + 3744*a^2*b^2))/(384*a^2*(3*a*b^2 + 3*a^2*b + a^3 +
b^3)) + (tan(c + d*x)^5*(1760*a*b^3 + 2304*a^3*b + 385*b^4 + 3168*a^2*b^2))/(384*a^3*(2*a*b + a^2 + b^2)) + (
tan(c + d*x)^7*(160*a*b^3 + 256*a^3*b + 35*b^4 + 288*a^2*b^2))/(128*a^4*(a + b)))/(d*(tan(c + d*x)^4*(12*a^3*b
+ 6*a^4 + 6*a^2*b^2) + tan(c + d*x)^2*(4*a^3*b + 4*a^4) + tan(c + d*x)^6*(4*a*b^3 + 12*a^3*b + 4*a^4 + 12*a^2
*b^2) + a^4 + tan(c + d*x)^8*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2))) + (atan((tan(c + d*x)*(2*a + 2*b)*(
4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2))/(2*a^(1/2)*(a + b)^(9/2)))*(160*a*b^3 + 256*a^3*b + 128*a^4 + 35*b
^4 + 288*a^2*b^2))/(128*a^(9/2)*d*(a + b)^(9/2))