\(\int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx\) [742]

Optimal result

Integrand size = 27, antiderivative size = 329 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {4 (b c-a d) \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 (b c-a d) \left (2 a c d+b \left (c^2-3 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \left (2 a b c d-a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^2 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}} \] Output:

2/3*(-a*d+b*c)^2*cos(f*x+e)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(3/2)-4/3*(-a*d 
+b*c)*(2*a*c*d+b*(c^2-3*d^2))*cos(f*x+e)/d/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^ 
(1/2)+4/3*(-a*d+b*c)*(2*a*c*d+b*(c^2-3*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)/d^2/(c^2-d^2)^2/f/( 
(c+d*sin(f*x+e))/(c+d))^(1/2)+2/3*(2*a*b*c*d-a^2*d^2+b^2*(2*c^2-3*d^2))*In 
verseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+ 
e))/(c+d))^(1/2)/d^2/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (d^2 \left (-8 a b c d+a^2 \left (3 c^2+d^2\right )+b^2 \left (c^2+3 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )-2 \left (-2 a^2 c d^2+a b d \left (c^2+3 d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) (-c-d \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d)^2 (c+d)^2}-\frac {d (-b c+a d) \cos (e+f x) \left (-b c^3-5 a c^2 d+5 b c d^2+a d^3-2 d \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^2}\right )}{3 d^2 f (c+d \sin (e+f x))^{3/2}} \] Input:

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

Output:

(2*(((d^2*(-8*a*b*c*d + a^2*(3*c^2 + d^2) + b^2*(c^2 + 3*d^2))*EllipticF[( 
-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - 2*(-2*a^2*c*d^2 + a*b*d*(c^2 + 3*d^ 
2) + b^2*(c^3 - 3*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/ 
(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*(-c - d*Sin 
[e + f*x])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)^2*(c + d)^2) - (d* 
(-(b*c) + a*d)*Cos[e + f*x]*(-(b*c^3) - 5*a*c^2*d + 5*b*c*d^2 + a*d^3 - 2* 
d*(2*a*c*d + b*(c^2 - 3*d^2))*Sin[e + f*x]))/(c^2 - d^2)^2))/(3*d^2*f*(c + 
 d*Sin[e + f*x])^(3/2))
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3269, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3269

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {2 \int \frac {d \left (-\left (\left (3 c^2+d^2\right ) a^2\right )+8 b c d a-b^2 \left (c^2+3 d^2\right )\right )+2 (b c-a d) \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {4 (b c-a d) \left (2 a c d+b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {d \left (-\left (\left (3 c^2+d^2\right ) a^2\right )+8 b c d a-b^2 \left (c^2+3 d^2\right )\right )+2 (b c-a d) \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {4 (b c-a d) \left (2 a c d+b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {d \left (-\left (\left (3 c^2+d^2\right ) a^2\right )+8 b c d a-b^2 \left (c^2+3 d^2\right )\right )+2 (b c-a d) \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {4 (b c-a d) \left (2 a c d+b c^2-3 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

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

Output:

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

Defintions of rubi rules used

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 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*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]
 

rule 3233

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

rule 3269

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1037\) vs. \(2(314)=628\).

Time = 4.33 (sec) , antiderivative size = 1038, normalized size of antiderivative = 3.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1038\)
parts	\(\text {Expression too large to display}\)	\(2388\)

Input:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 1356, normalized size of antiderivative = 4.12 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/9*((4*b^2*c^6 + 4*a*b*c^5*d - 8*a*b*c^3*d^3 + 4*a^2*c^2*d^4 - 12*a*b*c* 
d^5 + (a^2 - 5*b^2)*c^4*d^2 + 3*(a^2 + 3*b^2)*d^6 - (4*b^2*c^4*d^2 + 4*a*b 
*c^3*d^3 - 12*a*b*c*d^5 + (a^2 - 9*b^2)*c^2*d^4 + 3*(a^2 + 3*b^2)*d^6)*cos 
(f*x + e)^2 + 2*(4*b^2*c^5*d + 4*a*b*c^4*d^2 - 12*a*b*c^2*d^4 + (a^2 - 9*b 
^2)*c^3*d^3 + 3*(a^2 + 3*b^2)*c*d^5)*sin(f*x + e))*sqrt(1/2*I*d)*weierstra 
ssPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3* 
(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (4*b^2*c^6 + 4*a*b*c^ 
5*d - 8*a*b*c^3*d^3 + 4*a^2*c^2*d^4 - 12*a*b*c*d^5 + (a^2 - 5*b^2)*c^4*d^2 
 + 3*(a^2 + 3*b^2)*d^6 - (4*b^2*c^4*d^2 + 4*a*b*c^3*d^3 - 12*a*b*c*d^5 + ( 
a^2 - 9*b^2)*c^2*d^4 + 3*(a^2 + 3*b^2)*d^6)*cos(f*x + e)^2 + 2*(4*b^2*c^5* 
d + 4*a*b*c^4*d^2 - 12*a*b*c^2*d^4 + (a^2 - 9*b^2)*c^3*d^3 + 3*(a^2 + 3*b^ 
2)*c*d^5)*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3 
*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d 
*sin(f*x + e) + 2*I*c)/d) - 6*(-I*b^2*c^5*d - I*a*b*c^4*d^2 - 4*I*a*b*c^2* 
d^4 - 3*I*a*b*d^6 + 2*I*(a^2 + b^2)*c^3*d^3 + I*(2*a^2 + 3*b^2)*c*d^5 + (I 
*b^2*c^3*d^3 + I*a*b*c^2*d^4 + 3*I*a*b*d^6 - I*(2*a^2 + 3*b^2)*c*d^5)*cos( 
f*x + e)^2 + 2*(-I*b^2*c^4*d^2 - I*a*b*c^3*d^3 - 3*I*a*b*c*d^5 + I*(2*a^2 
+ 3*b^2)*c^2*d^4)*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 
- 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4 
*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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