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

Optimal. Leaf size=261 \[ \frac {3}{2} i a \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{2} i a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Shi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \]

[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)

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Rubi [A]  time = 0.27, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3319, 3312, 3303, 3298, 3301} \[ \frac {3}{2} i a \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{2} i a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Shi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \]

Antiderivative was successfully verified.

[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 3298

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 3301

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 3303

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 3312

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 3319

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

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Mathematica [A]  time = 0.74, size = 146, normalized size = 0.56 \[ \frac {a \sqrt {a+i a \sinh (e+f x)} \left (3 \text {Chi}\left (\frac {f x}{2}\right ) \left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right )-\text {Chi}\left (\frac {3 f x}{2}\right ) \left (\cosh \left (\frac {3 e}{2}\right )-i \sinh \left (\frac {3 e}{2}\right )\right )+\left (\sinh \left (\frac {e}{2}\right )+i \cosh \left (\frac {e}{2}\right )\right ) \left (3 \text {Shi}\left (\frac {f x}{2}\right )+(1+2 i \sinh (e)) \text {Shi}\left (\frac {3 f x}{2}\right )\right )\right )}{2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[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]))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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