\(\int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx\) [741]

Optimal result

Integrand size = 27, antiderivative size = 228 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (2 b^2 c^2-2 a b c d+\left (a^2-b^2\right ) d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 b (b c-a d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

2*(-a*d+b*c)^2*cos(f*x+e)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)-2*(2*b^2*c^ 
2-2*a*b*c*d+(a^2-b^2)*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)/f/((c+d*sin(f*x+e))/(c+ 
d))^(1/2)-4*b*(-a*d+b*c)*InverseJacobiAM(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/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \left (\frac {(b c-a d)^2 \cos (e+f x)}{c^2-d^2}+\frac {\left (\left (2 a b c d-a^2 d^2+b^2 \left (-2 c^2+d^2\right )\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+2 b (c-d) (b c-a d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d) d}\right )}{d f \sqrt {c+d \sin (e+f x)}} \] Input:

Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3269, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3269

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2),x]
 

Output:

(2*(b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) 
+ ((2*(2*b^2*c^2 - 2*a*b*c*d + (a^2 - b^2)*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)]) - (4*b*(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)])/(d*f*Sqrt[c + d*Sin[e + 
f*x]]))/(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 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. \(882\) vs. \(2(223)=446\).

Time = 3.98 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.87

method	result	size
default	\(\frac {\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (\frac {b \left (\frac {2 b d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {4 a d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}-\frac {2 b c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {2 d \cos \left (f x +e \right )^{2}}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{d^{2}}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(883\)
parts	\(\text {Expression too large to display}\)	\(1647\)

Input:

int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.35 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

2/3*(3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*sqrt(d*sin(f*x + e) + c)*cos( 
f*x + e) - (4*b^2*c^4 - 4*a*b*c^3*d + 6*a*b*c*d^3 - (a^2 + 5*b^2)*c^2*d^2 
+ (4*b^2*c^3*d - 4*a*b*c^2*d^2 + 6*a*b*d^4 - (a^2 + 5*b^2)*c*d^3)*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) - (4*b^2*c^4 - 4*a*b*c^3*d + 6*a*b*c*d^3 - (a^2 + 5*b^2)*c^2*d^2 + ( 
4*b^2*c^3*d - 4*a*b*c^2*d^2 + 6*a*b*d^4 - (a^2 + 5*b^2)*c*d^3)*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) + 3*(-2*I*b^2*c^3*d + 2*I*a*b*c^2*d^2 - I*(a^2 - b^2)*c*d^3 + (-2*I*b 
^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(a^2 - b^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 + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(2*I*b 
^2*c^3*d - 2*I*a*b*c^2*d^2 + I*(a^2 - b^2)*c*d^3 + (2*I*b^2*c^2*d^2 - 2*I* 
a*b*c*d^3 + I*(a^2 - b^2)*d^4)*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassZet 
a(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassP 
Inverse(-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)))/((c^2*d^4 - d^6)*f*sin( 
f*x + e) + (c^3*d^3 - c*d^5)*f)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sin(f*x + e) + a)^2/(d*sin(f*x + e) + c)^(3/2), x)
 

Giac [F]

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

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sin(f*x + e) + a)^2/(d*sin(f*x + e) + c)^(3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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