\(\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx\) [491]

Optimal result

Integrand size = 25, antiderivative size = 231 \[ \int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 f}-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a (3 c+5 d) \left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d f \sqrt {c+d \sin (e+f x)}} \] Output:

-2/15*a*(3*c+5*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f-2/5*a*cos(f*x+e)*(c+ 
d*sin(f*x+e))^(3/2)/f-2/15*a*(3*c^2+20*c*d+9*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/f/((c+d*sin( 
f*x+e))/(c+d))^(1/2)-2/15*a*(3*c+5*d)*(c^2-d^2)*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/f/(c+ 
d*sin(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.62 (sec) , antiderivative size = 2625, normalized size of antiderivative = 11.36 \[ \int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

a*((c^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((C 
sc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt 
[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d 
*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^ 
2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot 
[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan 
[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*S 
qrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqr 
t[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + 
 Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[ 
1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - A 
rcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*S 
qrt[1 + Cot[e]^2]*Sin[e]]))/(5*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]) 
^2) + (4*c*d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, 
 -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d 
*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*( 
c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Co 
t[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTa 
n[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - A 
rcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*S...
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3232, 27, 3042, 3232, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

\(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {c+d \sin (e+f x)} (a (5 c+3 d)+a (3 c+5 d) \sin (e+f x))dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \sqrt {c+d \sin (e+f x)} (a (5 c+3 d)+a (3 c+5 d) \sin (e+f x))dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \int \sqrt {c+d \sin (e+f x)} (a (5 c+3 d)+a (3 c+5 d) \sin (e+f x))dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 a (3 c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\)

Input:

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

Output:

(-2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + ((-2*a*(3*c + 5*d)* 
Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((2*a*(3*c^2 + 20*c*d + 9*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*a*(3*c + 5*d)*(c^2 - d^2)*E 
llipticF[(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]]))/3)/5
 

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 3232

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1033\) vs. \(2(216)=432\).

Time = 2.82 (sec) , antiderivative size = 1034, normalized size of antiderivative = 4.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1034\)
parts	\(\text {Expression too large to display}\)	\(1449\)

Input:

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

Output:

2/15*a*(18*((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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2), 
((c-d)/(c+d))^(1/2))*c^3*d+14*c^2*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+s 
in(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*si 
n(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^2-18*c*((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) 
*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^3-14*((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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^( 
1/2))*d^4-3*((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)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2) 
,((c-d)/(c+d))^(1/2))*c^4-20*((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)*EllipticE(((c+d*sin(f*x 
+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d-6*((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)*Ellip 
ticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^2+20*((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)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/ 
2))*c*d^3+9*((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)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.15 \[ \int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

-2/45*((6*a*c^3 - 5*a*c^2*d - 18*a*c*d^2 - 15*a*d^3)*sqrt(1/2*I*d)*weierst 
rassPInverse(-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*a*c^3 - 5*a*c^2* 
d - 18*a*c*d^2 - 15*a*d^3)*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*(3*I*a*c^2*d + 20*I*a*c*d^2 + 9*I*a*d^3)* 
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*(-3*I*a*c^2*d - 20*I*a*c*d^2 - 9*I*a*d^3)*sqrt(-1/2*I*d)*weierstrass 
Zeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 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)) + 3*(3*a*d^3*cos(f*x 
+ e)*sin(f*x + e) + (6*a*c*d^2 + 5*a*d^3)*cos(f*x + e))*sqrt(d*sin(f*x + e 
) + c))/(d^2*f)
 

Sympy [F]

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

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

Output:

a*(Integral(c*sqrt(c + d*sin(e + f*x)), x) + Integral(c*sqrt(c + d*sin(e + 
 f*x))*sin(e + f*x), x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x) 
, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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