\(\int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx\) [128]

Optimal result

Integrand size = 21, antiderivative size = 261 \[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\frac {3}{2} i a \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{2} i a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right )+\frac {1}{2} i a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {3 f x}{2}\right ) \]

[Out]

3/2*a*sinh(1/2*e+1/4*I*Pi)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*Shi(1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)+1/2*I*a*cosh(3/
2*e+1/4*I*Pi)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*Shi(3/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)+3/2*a*Chi(1/2*f*x)*sech(1/2*
e+1/4*I*Pi+1/2*f*x)*cosh(1/2*e+1/4*I*Pi)*(a+I*a*sinh(f*x+e))^(1/2)+1/2*I*a*Chi(3/2*f*x)*sech(1/2*e+1/4*I*Pi+1/
2*f*x)*sinh(3/2*e+1/4*I*Pi)*(a+I*a*sinh(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3400, 3393, 3384, 3379, 3382} \[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\frac {3}{2} i a \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{2} i a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Shi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \]

[In]

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

[Out]

((3*I)/2)*a*CoshIntegral[(f*x)/2]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(2*e - I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*
x]] + (I/2)*a*CoshIntegral[(3*f*x)/2]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(6*e + I*Pi)/4]*Sqrt[a + I*a*Sinh[e
+ f*x]] + ((3*I)/2)*a*Cosh[(2*e - I*Pi)/4]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhInte
gral[(f*x)/2] + (I/2)*a*Cosh[(6*e + I*Pi)/4]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIn
tegral[(3*f*x)/2]

Rule 3379

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

Rule 3382

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

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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 3393

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

Rubi steps \begin{align*} \text {integral}& = -\left (\left (2 a \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh ^3\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )}{x} \, dx\right ) \\ & = -\left (\left (2 i a \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \left (\frac {3 i \sinh \left (\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}\right )}{4 x}+\frac {i \sinh \left (\frac {1}{4} (6 e+i \pi )+\frac {3 f x}{2}\right )}{4 x}\right ) \, dx\right ) \\ & = \frac {1}{2} \left (a \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh \left (\frac {1}{4} (6 e+i \pi )+\frac {3 f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (3 a \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh \left (\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}\right )}{x} \, dx \\ & = \frac {1}{2} \left (3 a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh \left (\frac {f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh \left (\frac {3 f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (3 a \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\cosh \left (\frac {f x}{2}\right )}{x} \, dx+\frac {1}{2} \left (a \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\cosh \left (\frac {3 f x}{2}\right )}{x} \, dx \\ & = \frac {3}{2} i a \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{2} i a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right )+\frac {1}{2} i a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {3 f x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.72 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.56 \[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\frac {a \sqrt {a+i a \sinh (e+f x)} \left (3 \text {Chi}\left (\frac {f x}{2}\right ) \left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right )-\text {Chi}\left (\frac {3 f x}{2}\right ) \left (\cosh \left (\frac {3 e}{2}\right )-i \sinh \left (\frac {3 e}{2}\right )\right )+\left (i \cosh \left (\frac {e}{2}\right )+\sinh \left (\frac {e}{2}\right )\right ) \left (3 \text {Shi}\left (\frac {f x}{2}\right )+(1+2 i \sinh (e)) \text {Shi}\left (\frac {3 f x}{2}\right )\right )\right )}{2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

(a*Sqrt[a + I*a*Sinh[e + f*x]]*(3*CoshIntegral[(f*x)/2]*(Cosh[e/2] + I*Sinh[e/2]) - CoshIntegral[(3*f*x)/2]*(C
osh[(3*e)/2] - I*Sinh[(3*e)/2]) + (I*Cosh[e/2] + Sinh[e/2])*(3*SinhIntegral[(f*x)/2] + (1 + (2*I)*Sinh[e])*Sin
hIntegral[(3*f*x)/2])))/(2*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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