\(\int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx\) [757]

Optimal result

Integrand size = 27, antiderivative size = 399 \[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (7 b c^2-4 a c d-3 b d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 d \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 (b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) (b c-a d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

-2/3*d^2*cos(f*x+e)/(-a*d+b*c)/(c^2-d^2)/f/(c+d*sin(f*x+e))^(3/2)-2/3*d^2* 
(-4*a*c*d+7*b*c^2-3*b*d^2)*cos(f*x+e)/(-a*d+b*c)^2/(c^2-d^2)^2/f/(c+d*sin( 
f*x+e))^(1/2)+2/3*d*(-4*a*c*d+7*b*c^2-3*b*d^2)*EllipticE(cos(1/2*e+1/4*Pi+ 
1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/(-a*d+b*c)^2/(c^2 
-d^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2/3*d*InverseJacobiAM(1/2*e-1/4*P 
i+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(-a*d+b* 
c)/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)-2*b^2*EllipticPi(cos(1/2*e+1/4*Pi+1/ 
2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/( 
a+b)/(-a*d+b*c)^2/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.96 (sec) , antiderivative size = 1079, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]
 

Output:

(Sqrt[c + d*Sin[e + f*x]]*((-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2 
)*(c + d*Sin[e + f*x])^2) + (2*(-7*b*c^2*d^2*Cos[e + f*x] + 4*a*c*d^3*Cos[ 
e + f*x] + 3*b*d^4*Cos[e + f*x]))/(3*(b*c - a*d)^2*(c^2 - d^2)^2*(c + d*Si 
n[e + f*x]))))/f + ((-2*(6*b^2*c^4 - 12*a*b*c^3*d + 6*a^2*c^2*d^2 - 19*b^2 
*c^2*d^2 + 8*a*b*c*d^3 + 2*a^2*d^4 + 9*b^2*d^4)*EllipticPi[(2*b)/(a + b), 
(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(( 
a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(-12*b^2*c^3*d - 8*a*b*c^2*d^2 + 
 8*a^2*c*d^3 + 4*b^2*c*d^3 + 8*a*b*d^4)*Cos[e + f*x]*((b*c - a*d)*Elliptic 
F[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d) 
] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]* 
Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c 
+ d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e 
 + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqr 
t[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + 
d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(7*b^2*c^2*d^2 - 4*a*b*c*d^3 - 3*b^2*d^ 
4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcS 
inh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(- 
2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + 
d*Sin[e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d) 
)/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]],...
 

Rubi [A] (verified)

Time = 3.89 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.08, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3281, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 3281

\(\displaystyle \frac {2 \int -\frac {-b d^2 \sin ^2(e+f x)+d (3 b c-a d) \sin (e+f x)+3 \left (a c d-b \left (c^2-d^2\right )\right )}{2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-b d^2 \sin ^2(e+f x)+d (3 b c-a d) \sin (e+f x)+3 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {-b d^2 \sin (e+f x)^2+d (3 b c-a d) \sin (e+f x)+3 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3534

\(\displaystyle -\frac {\frac {2 \int -\frac {-b \left (7 b c^2-4 a d c-3 b d^2\right ) \sin ^2(e+f x) d^2+a^2 \left (3 c^2+d^2\right ) d^2+2 \left (-\left (\left (3 c^3-c d^2\right ) b^2\right )-2 a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x) d+3 b^2 \left (c^2-d^2\right )^2-a b \left (6 c^3 d-2 c d^3\right )}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-b \left (7 b c^2-4 a d c-3 b d^2\right ) \sin ^2(e+f x) d^2+a^2 \left (3 c^2+d^2\right ) d^2-2 a b c \left (3 c^2-d^2\right ) d+2 \left (-\left (\left (3 c^3-c d^2\right ) b^2\right )-2 a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x) d+3 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {-b \left (7 b c^2-4 a d c-3 b d^2\right ) \sin (e+f x)^2 d^2+a^2 \left (3 c^2+d^2\right ) d^2-2 a b c \left (3 c^2-d^2\right ) d+2 \left (-\left (\left (3 c^3-c d^2\right ) b^2\right )-2 a d \left (c^2-d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x) d+3 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\int -\frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-d \left (-4 a c d+7 b c^2-3 b d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-d \left (-4 a c d+7 b c^2-3 b d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-d \left (-4 a c d+7 b c^2-3 b d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\int \frac {b^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x) d^2+b \left (c^2-d^2\right ) \left (3 \left (c^2-d^2\right ) b^2+a c d b-a^2 d^2\right ) d}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+b d^2 \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+b d^2 \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {3 b^3 d \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {\frac {6 b^3 d \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}+\frac {2 b d^2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{\left (c^2-d^2\right ) (b c-a d)}}{3 \left (c^2-d^2\right ) (b c-a d)}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}\)

Input:

Int[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]
 

Output:

(-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3 
/2)) - ((2*d^2*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*Cos[e + f*x])/((b*c - a*d)*(c 
^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - ((-2*d*(7*b*c^2 - 4*a*c*d - 3*b*d^ 
2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/ 
(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((2*b*d^2*(b*c - a*d)*(c^2 - d^2) 
*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c 
 + d)])/(f*Sqrt[c + d*Sin[e + f*x]]) + (6*b^3*d*(c^2 - d^2)^2*EllipticPi[( 
2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x]) 
/(c + d)])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]]))/(b*d))/((b*c - a*d)*(c^2 
- d^2)))/(3*(b*c - a*d)*(c^2 - d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3281

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 
))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n 
 + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2* 
n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))
 

rule 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && GtQ[c + d, 0]
 

rule 3286

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt 
[c + d*Sin[e + f*x]]   Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + 
 d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]
 

rule 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1066\) vs. \(2(383)=766\).

Time = 2.58 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.67

method	result	size
default	\(\text {Expression too large to display}\)	\(1067\)

Input:

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

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(2*b/(a*d-b*c)^2*(c/d-1)*((c+d*sin 
(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c 
-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(( 
(c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+d/( 
a*d-b*c)*(2/3/(c^2-d^2)/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x 
+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e) 
^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/ 
(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2 
)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d 
))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e) 
)/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1 
/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin( 
f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d 
))^(1/2),((c-d)/(c+d))^(1/2))))-b*d/(a*d-b*c)^2*(2*d*cos(f*x+e)^2/(c^2-d^2 
)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2/(c^2-d^2)*c*(c/d-1)*((c+d*sin( 
f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c- 
d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+ 
e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+ 
e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^ 
(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} b \,d^{3}+\sin \left (f x +e \right )^{3} a \,d^{3}+3 \sin \left (f x +e \right )^{3} b c \,d^{2}+3 \sin \left (f x +e \right )^{2} a c \,d^{2}+3 \sin \left (f x +e \right )^{2} b \,c^{2} d +3 \sin \left (f x +e \right ) a \,c^{2} d +\sin \left (f x +e \right ) b \,c^{3}+a \,c^{3}}d x \] Input:

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

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**4*b*d**3 + sin(e + f*x)**3*a*d 
**3 + 3*sin(e + f*x)**3*b*c*d**2 + 3*sin(e + f*x)**2*a*c*d**2 + 3*sin(e + 
f*x)**2*b*c**2*d + 3*sin(e + f*x)*a*c**2*d + sin(e + f*x)*b*c**3 + a*c**3) 
,x)