\(\int \frac {x}{(a+i a \sinh (e+f x))^{3/2}} \, dx\) [143]

Optimal result

Integrand size = 19, antiderivative size = 288 \[ \int \frac {x}{(a+i a \sinh (e+f x))^{3/2}} \, dx=\frac {1}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x \text {arctanh}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}+\frac {i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}} \] Output:

1/a/f^2/(a+I*a*sinh(f*x+e))^(1/2)-I*x*arctanh(exp(1/2*e+3/4*I*Pi+1/2*f*x)) 
*cosh(1/2*e+1/4*I*Pi+1/2*f*x)/a/f/(a+I*a*sinh(f*x+e))^(1/2)+I*cosh(1/2*e+1 
/4*I*Pi+1/2*f*x)*polylog(2,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^2/(a+I*a*sinh( 
f*x+e))^(1/2)-I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,-exp(1/2*e+3/4*I*Pi 
+1/2*f*x))/a/f^2/(a+I*a*sinh(f*x+e))^(1/2)+1/2*x*tanh(1/2*e+1/4*I*Pi+1/2*f 
*x)/a/f/(a+I*a*sinh(f*x+e))^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.73 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.90 \[ \int \frac {x}{(a+i a \sinh (e+f x))^{3/2}} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \left ((2+i f x) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )+(1-i) (-1)^{3/4} \left (i e \arctan \left (\sqrt [4]{-1} e^{\frac {1}{2} (e+f x)}\right )+\frac {1}{2} (e+f x) \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-\frac {1}{2} (e+f x) \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-\operatorname {PolyLog}\left (2,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+\operatorname {PolyLog}\left (2,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2+2 f x \sinh \left (\frac {1}{2} (e+f x)\right )\right )}{2 f^2 (a+i a \sinh (e+f x))^{3/2}} \] Input:

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

Output:

((Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])*((2 + I*f*x)*(Cosh[(e + f*x)/2] 
 + I*Sinh[(e + f*x)/2]) + (1 - I)*(-1)^(3/4)*(I*e*ArcTan[(-1)^(1/4)*E^((e 
+ f*x)/2)] + ((e + f*x)*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)])/2 - ((e + f*x 
)*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)])/2 - PolyLog[2, -((-1)^(3/4)*E^((e + 
 f*x)/2))] + PolyLog[2, (-1)^(3/4)*E^((e + f*x)/2)])*(Cosh[(e + f*x)/2] + 
I*Sinh[(e + f*x)/2])^2 + 2*f*x*Sinh[(e + f*x)/2]))/(2*f^2*(a + I*a*Sinh[e 
+ f*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3800, 3042, 4673, 3042, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \int x \text {sech}^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4673

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \int x \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx+\frac {2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \int x \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx+\frac {2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (\frac {2 i \int \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {2 i \int \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}+\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )+\frac {2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (\frac {4 i \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}-\frac {4 i \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}+\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )+\frac {2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )+\frac {2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

Input:

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

Output:

(Cosh[e/2 + (I/4)*Pi + (f*x)/2]*((((4*I)*x*ArcTanh[E^((2*e - I*Pi)/4 + (f* 
x)/2)])/f + ((4*I)*PolyLog[2, -E^((2*e - I*Pi)/4 + (f*x)/2)])/f^2 - ((4*I) 
*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/f^2)/2 + (2*Sech[e/2 + (I/4)*Pi 
 + (f*x)/2])/f^2 + (x*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Tanh[e/2 + (I/4)*Pi + 
 (f*x)/2])/f))/(2*a*Sqrt[a + I*a*Sinh[e + f*x]])
 

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

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])
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

((a^2*f^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*e^(f*x + e) - a^2*f^2)*integral(-1 
/2*I*sqrt(1/2*I*a*e^(-f*x - e))*x*e^(f*x + e)/(a^2*e^(f*x + e) - I*a^2), x 
) + ((-I*f*x - 2*I)*e^(2*f*x + 2*e) + (f*x - 2)*e^(f*x + e))*sqrt(1/2*I*a* 
e^(-f*x - e)))/(a^2*f^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*e^(f*x + e) - a^2*f^ 
2)
 

Sympy [F]

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

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

Output:

Integral(x/(I*a*(sinh(e + f*x) - I))**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {x}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:

int(x/(a + a*sinh(e + f*x)*1i)^(3/2),x)
 

Output:

int(x/(a + a*sinh(e + f*x)*1i)^(3/2), x)
 

Reduce [F]

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

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

Output:

(sqrt(a)*( - int((sqrt(sinh(e + f*x)*i + 1)*sinh(e + f*x)*x)/(sinh(e + f*x 
)**3*i + sinh(e + f*x)**2 + sinh(e + f*x)*i + 1),x)*i + int((sqrt(sinh(e + 
 f*x)*i + 1)*x)/(sinh(e + f*x)**3*i + sinh(e + f*x)**2 + sinh(e + f*x)*i + 
 1),x)))/a**2