\(\int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx\) [138]

Optimal result

Integrand size = 19, antiderivative size = 207 \[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\frac {4 i x \text {arctanh}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {4 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}} \]

[Out]

-4*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)/f/(a+I*a*sinh(f*x+e))^(1/2)+4*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))/f^2/(a+I*a*sinh(f*x+e))^(1/2)-4*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))/f^2/(a+I*a*sinh(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3400, 4267, 2317, 2438} \[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}} \]

[In]

Int[x/Sqrt[a + I*a*Sinh[e + f*x]],x]

[Out]

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \int x \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{\sqrt {a+i a \sinh (e+f x)}} \\ & = \frac {4 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 )}{f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (2 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \log \left (1-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (2 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \log \left (1+e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{f \sqrt {a+i a \sinh (e+f x)}} \\ & = \frac {4 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 )}{f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (4 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (4 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}} \\ & = \frac {4 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 )}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {4 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 )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 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 )}{f^2 \sqrt {a+i a \sinh (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\frac {(2-2 i) (-1)^{3/4} \left (i e \arctan \left (\sqrt [4]{-1} e^{\frac {1}{2} (e+f x)}\right )+\frac {1}{2} (e+f x) \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-\frac {1}{2} (e+f x) \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-\operatorname {PolyLog}\left (2,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+\operatorname {PolyLog}\left (2,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}} \]

[In]

Integrate[x/Sqrt[a + I*a*Sinh[e + f*x]],x]

[Out]

((2 - 2*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]))/(f^2*Sqrt[a + I*a*Sinh[e + f*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {x}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate(x/(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {x}{\sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}}\, dx \]

[In]

integrate(x/(a+I*a*sinh(f*x+e))**(1/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {x}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate(x/(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {x}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate(x/(a+I*a*sinh(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {x}{\sqrt {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]

[In]

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

[Out]

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