\(\int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx\) [133]

Optimal result

Integrand size = 18, antiderivative size = 332 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx=-\frac {9}{16} a f^2 \cos \left (\frac {3}{4} (2 e-\pi )\right ) \operatorname {CosIntegral}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}-\frac {3}{16} a f^2 \operatorname {CosIntegral}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {1}{4} (2 e+\pi )\right ) \sqrt {a+a \sin (e+f x)}-\frac {3 a f \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{2 x}-\frac {a \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{x^2}-\frac {3}{16} a f^2 \cos \left (\frac {1}{4} (2 e+\pi )\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)} \text {Si}\left (\frac {f x}{2}\right )+\frac {9}{16} a f^2 \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {3}{4} (2 e-\pi )\right ) \sqrt {a+a \sin (e+f x)} \text {Si}\left (\frac {3 f x}{2}\right ) \] Output:

9/16*a*f^2*cos(3/2*e+1/4*Pi)*Ci(3/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*(a+a*si 
n(f*x+e))^(1/2)-3/16*a*f^2*Ci(1/2*f*x)*csc(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e 
+1/4*Pi)*(a+a*sin(f*x+e))^(1/2)-3/2*a*f*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2* 
e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/x-a*sin(1/2*e+1/4*Pi+1/2*f*x)^2*( 
a+a*sin(f*x+e))^(1/2)/x^2-3/16*a*f^2*cos(1/2*e+1/4*Pi)*csc(1/2*e+1/4*Pi+1/ 
2*f*x)*(a+a*sin(f*x+e))^(1/2)*Si(1/2*f*x)-9/16*a*f^2*csc(1/2*e+1/4*Pi+1/2* 
f*x)*sin(3/2*e+1/4*Pi)*(a+a*sin(f*x+e))^(1/2)*Si(3/2*f*x)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.89 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx=-\frac {i \left (-i a e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2\right )^{3/2} \left (-4+12 i e^{i (e+f x)}+12 e^{2 i (e+f x)}-4 i e^{3 i (e+f x)}+6 i f x+6 e^{i (e+f x)} f x+6 i e^{2 i (e+f x)} f x+6 e^{3 i (e+f x)} f x+3 i e^{i e+\frac {3 i f x}{2}} f^2 x^2 \operatorname {ExpIntegralEi}\left (-\frac {1}{2} i f x\right )+3 e^{2 i e+\frac {3 i f x}{2}} f^2 x^2 \operatorname {ExpIntegralEi}\left (\frac {i f x}{2}\right )-9 e^{\frac {3 i f x}{2}} f^2 x^2 \operatorname {ExpIntegralEi}\left (-\frac {3}{2} i f x\right )-9 i e^{\frac {3}{2} i (2 e+f x)} f^2 x^2 \operatorname {ExpIntegralEi}\left (\frac {3 i f x}{2}\right )\right )}{16 \sqrt {2} \left (i+e^{i (e+f x)}\right )^3 x^2} \] Input:

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

Output:

((-1/16*I)*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*(-4 + 
(12*I)*E^(I*(e + f*x)) + 12*E^((2*I)*(e + f*x)) - (4*I)*E^((3*I)*(e + f*x) 
) + (6*I)*f*x + 6*E^(I*(e + f*x))*f*x + (6*I)*E^((2*I)*(e + f*x))*f*x + 6* 
E^((3*I)*(e + f*x))*f*x + (3*I)*E^(I*e + ((3*I)/2)*f*x)*f^2*x^2*ExpIntegra 
lEi[(-1/2*I)*f*x] + 3*E^((2*I)*e + ((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[(I 
/2)*f*x] - 9*E^(((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[((-3*I)/2)*f*x] - (9* 
I)*E^(((3*I)/2)*(2*e + f*x))*f^2*x^2*ExpIntegralEi[((3*I)/2)*f*x]))/(Sqrt[ 
2]*(I + E^(I*(e + f*x)))^3*x^2)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.75, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3042, 3800, 3042, 3795, 3042, 3784, 3042, 3780, 3783, 3793, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int \frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{x^3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3}{x^3}dx\)

\(\Big \downarrow \) 3795

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {9}{8} f^2 \int \frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{x}dx+\frac {3}{4} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{x}dx-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {3}{4} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{x}dx-\frac {9}{8} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3}{x}dx-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3784

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {9}{8} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3}{x}dx+\frac {3}{4} f^2 \left (\sin \left (\frac {1}{4} (2 e+\pi )\right ) \int \frac {\cos \left (\frac {f x}{2}\right )}{x}dx+\cos \left (\frac {1}{4} (2 e+\pi )\right ) \int \frac {\sin \left (\frac {f x}{2}\right )}{x}dx\right )-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {9}{8} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3}{x}dx+\frac {3}{4} f^2 \left (\sin \left (\frac {1}{4} (2 e+\pi )\right ) \int \frac {\sin \left (\frac {f x}{2}+\frac {\pi }{2}\right )}{x}dx+\cos \left (\frac {1}{4} (2 e+\pi )\right ) \int \frac {\sin \left (\frac {f x}{2}\right )}{x}dx\right )-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3780

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {3}{4} f^2 \left (\sin \left (\frac {1}{4} (2 e+\pi )\right ) \int \frac {\sin \left (\frac {f x}{2}+\frac {\pi }{2}\right )}{x}dx+\cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right )\right )-\frac {9}{8} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3}{x}dx-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3783

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {9}{8} f^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3}{x}dx+\frac {3}{4} f^2 \left (\sin \left (\frac {1}{4} (2 e+\pi )\right ) \operatorname {CosIntegral}\left (\frac {f x}{2}\right )+\cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right )\right )-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {9}{8} f^2 \int \left (\frac {3 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}+\frac {\sin \left (\frac {3 e}{2}+\frac {3 f x}{2}-\frac {\pi }{4}\right )}{4 x}\right )dx+\frac {3}{4} f^2 \left (\sin \left (\frac {1}{4} (2 e+\pi )\right ) \operatorname {CosIntegral}\left (\frac {f x}{2}\right )+\cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right )\right )-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {3}{4} f^2 \left (\sin \left (\frac {1}{4} (2 e+\pi )\right ) \operatorname {CosIntegral}\left (\frac {f x}{2}\right )+\cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right )\right )-\frac {9}{8} f^2 \left (\frac {3}{4} \sin \left (\frac {1}{4} (2 e+\pi )\right ) \operatorname {CosIntegral}\left (\frac {f x}{2}\right )+\frac {1}{4} \cos \left (\frac {3}{4} (2 e-\pi )\right ) \operatorname {CosIntegral}\left (\frac {3 f x}{2}\right )-\frac {1}{4} \sin \left (\frac {3}{4} (2 e-\pi )\right ) \text {Si}\left (\frac {3 f x}{2}\right )+\frac {3}{4} \cos \left (\frac {1}{4} (2 e+\pi )\right ) \text {Si}\left (\frac {f x}{2}\right )\right )-\frac {\sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 x^2}-\frac {3 f \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{4 x}\right )\)

Input:

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

Output:

2*a*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*((-3*f*Cos[e/2 + Pi 
/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]^2)/(4*x) - Sin[e/2 + Pi/4 + (f*x)/ 
2]^3/(2*x^2) + (3*f^2*(CosIntegral[(f*x)/2]*Sin[(2*e + Pi)/4] + Cos[(2*e + 
 Pi)/4]*SinIntegral[(f*x)/2]))/4 - (9*f^2*((Cos[(3*(2*e - Pi))/4]*CosInteg 
ral[(3*f*x)/2])/4 + (3*CosIntegral[(f*x)/2]*Sin[(2*e + Pi)/4])/4 + (3*Cos[ 
(2*e + Pi)/4]*SinIntegral[(f*x)/2])/4 - (Sin[(3*(2*e - Pi))/4]*SinIntegral 
[(3*f*x)/2])/4))/8)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 3795

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

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{x^{3}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

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

Output:

Integral((a*(sin(e + f*x) + 1))**(3/2)/x**3, x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1256 vs. \(2 (248) = 496\).

Time = 0.48 (sec) , antiderivative size = 1256, normalized size of antiderivative = 3.78 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/16*sqrt(2)*(9*pi^2*a*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn( 
cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 18*pi*(pi - 2*f*x - 2*e)*a*f^3*cos(3/4*p 
i - 3/2*e)*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 9*( 
pi - 2*f*x - 2*e)^2*a*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn(co 
s(-1/4*pi + 1/2*f*x + 1/2*e)) - 36*pi*a*e*f^3*cos(3/4*pi - 3/2*e)*cos_inte 
gral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*(pi - 2*f*x - 2*e)* 
a*e*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn(cos(-1/4*pi + 1/2*f* 
x + 1/2*e)) + 36*a*e^2*f^3*cos(3/4*pi - 3/2*e)*cos_integral(3/2*f*x)*sgn(c 
os(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*pi^2*a*f^3*cos(1/4*pi - 1/2*e)*cos_inte 
gral(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 6*pi*(pi - 2*f*x - 2*e 
)*a*f^3*cos(1/4*pi - 1/2*e)*cos_integral(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f* 
x + 1/2*e)) + 3*(pi - 2*f*x - 2*e)^2*a*f^3*cos(1/4*pi - 1/2*e)*cos_integra 
l(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 12*pi*a*e*f^3*cos(1/4*pi 
- 1/2*e)*cos_integral(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*(p 
i - 2*f*x - 2*e)*a*e*f^3*cos(1/4*pi - 1/2*e)*cos_integral(1/2*f*x)*sgn(cos 
(-1/4*pi + 1/2*f*x + 1/2*e)) + 12*a*e^2*f^3*cos(1/4*pi - 1/2*e)*cos_integr 
al(1/2*f*x)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 9*pi^2*a*f^3*sgn(cos(-1/ 
4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 3/2*e)*sin_integral(3/2*f*x) - 18*pi 
*(pi - 2*f*x - 2*e)*a*f^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(3/4*pi - 
 3/2*e)*sin_integral(3/2*f*x) + 9*(pi - 2*f*x - 2*e)^2*a*f^3*sgn(cos(-1...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a)*a*( - 2*sqrt(sin(e + f*x) + 1) + int((sqrt(sin(e + f*x) + 1)*cos( 
e + f*x))/(sin(e + f*x)*x**2 + x**2),x)*f*x**2 + 4*int((sqrt(sin(e + f*x) 
+ 1)*sin(e + f*x))/x**3,x)*x**2))/(4*x**2)