\(\int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx\) [131]

Optimal result

Integrand size = 18, antiderivative size = 221 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx=\frac {1}{2} a \cos \left (\frac {3}{4} (2 e-\pi )\right ) \operatorname {CosIntegral}\left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}+\frac {3}{2} a \operatorname {CosIntegral}\left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {1}{4} (2 e+\pi )\right ) \sqrt {a+a \sin (e+f x)}+\frac {3}{2} a \cos \left (\frac {1}{4} (2 e+\pi )\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)} \text {Si}\left (\frac {f x}{2}\right )-\frac {1}{2} a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {3}{4} (2 e-\pi )\right ) \sqrt {a+a \sin (e+f x)} \text {Si}\left (\frac {3 f x}{2}\right ) \] Output:

-1/2*a*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*sin(f* 
x+e))^(1/2)+3/2*a*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*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)+1/2*a*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 [A] (verified)

Time = 2.37 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.57 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx=\frac {(a (1+\sin (e+f x)))^{3/2} \left (3 \operatorname {CosIntegral}\left (\frac {f x}{2}\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )+\operatorname {CosIntegral}\left (\frac {3 f x}{2}\right ) \left (-\cos \left (\frac {3 e}{2}\right )+\sin \left (\frac {3 e}{2}\right )\right )+\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (3 \text {Si}\left (\frac {f x}{2}\right )+(1+2 \sin (e)) \text {Si}\left (\frac {3 f x}{2}\right )\right )\right )}{2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:

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

Output:

((a*(1 + Sin[e + f*x]))^(3/2)*(3*CosIntegral[(f*x)/2]*(Cos[e/2] + Sin[e/2] 
) + CosIntegral[(3*f*x)/2]*(-Cos[(3*e)/2] + Sin[(3*e)/2]) + (Cos[e/2] - Si 
n[e/2])*(3*SinIntegral[(f*x)/2] + (1 + 2*Sin[e])*SinIntegral[(3*f*x)/2]))) 
/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.56, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3042, 3800, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*a*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*((Cos[(3*(2*e - Pi) 
)/4]*CosIntegral[(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)
 

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 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 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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.59 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx=\frac {\sqrt {2} {\left (a f \cos \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) \operatorname {Ci}\left (\frac {3}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a f \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) \operatorname {Ci}\left (\frac {1}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + a f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) \operatorname {Si}\left (\frac {3}{2} \, f x\right ) + 3 \, a f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) \operatorname {Si}\left (\frac {1}{2} \, f x\right )\right )} \sqrt {a}}{2 \, f} \] Input:

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

Output:

1/2*sqrt(2)*(a*f*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)) + 3*a*f*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)) + a*f*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) + 3*a*f*sgn(cos(-1/4*pi + 1/2*f 
*x + 1/2*e))*sin(1/4*pi - 1/2*e)*sin_integral(1/2*f*x))*sqrt(a)/f
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1)/x,x) + int((sqrt(sin(e + f*x) + 1)*s 
in(e + f*x))/x,x))