3.2.29 \(\int \frac {(a+i a \sinh (e+f x))^{3/2}}{x^2} \, dx\) [129]

Optimal. Leaf size=302 \[ -\frac {2 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{x}-\frac {3}{4} a f \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (6 e-i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{4} a f \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{4} a f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right )-\frac {3}{4} a f \cosh \left (\frac {1}{4} (6 e-i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {3 f x}{2}\right ) \]

[Out]

-2*a*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^2*(a+I*a*sinh(f*x+e))^(1/2)/x+3/4*a*f*cosh(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)+3/4*I*a*f*sinh(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)+3/4*I*a*f*Chi(3/2*f*x)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*cosh(3/2*e+1/4*I*P
i)*(a+I*a*sinh(f*x+e))^(1/2)+3/4*a*f*Chi(1/2*f*x)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*sinh(1/2*e+1/4*I*Pi)*(a+I*a*sin
h(f*x+e))^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 302, 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 = {3400, 3394, 3384, 3379, 3382} \begin {gather*} -\frac {3}{4} a f \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}{4} a f \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 {3}{4} a f \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 {3}{4} a f \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)}-\frac {2 a \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/x - (3*a*f*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]])/4 + (3*a*f*CoshIntegral[(f*x)/2]*Sec
h[e/2 + (I/4)*Pi + (f*x)/2]*Sinh[(2*e + I*Pi)/4]*Sqrt[a + I*a*Sinh[e + f*x]])/4 + (3*a*f*Cosh[(2*e + I*Pi)/4]*
Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2])/4 - (3*a*f*Cosh[(6*e - I*Pi)
/4]*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(3*f*x)/2])/4

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 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && 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*} \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x^2} \, 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^2} \, dx\right )\\ &=-\frac {2 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{x}+\left (3 a f \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 {\cosh \left (\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}\right )}{4 x}+\frac {\cosh \left (\frac {1}{4} (6 e+i \pi )+\frac {3 f x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac {2 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{x}+\frac {1}{4} \left (3 a f \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 {\cosh \left (\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}\right )}{x} \, dx+\frac {1}{4} \left (3 a f \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 {\cosh \left (\frac {1}{4} (6 e+i \pi )+\frac {3 f x}{2}\right )}{x} \, dx\\ &=-\frac {2 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{x}+\frac {1}{4} \left (3 i a f \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}{4} \left (3 i a f \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}{4} \left (3 i a f \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 {1}{4} \left (3 i a f \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 {2 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{x}-\frac {3}{4} a f \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}{4} a f \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 {3}{4} a f \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 {3}{4} a f \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 = 0.43, size = 243, normalized size = 0.80 \begin {gather*} \frac {a (-i+\sinh (e+f x)) \sqrt {a+i a \sinh (e+f x)} \left (-6 i \cosh \left (\frac {1}{2} (e+f x)\right )+2 i \cosh \left (\frac {3}{2} (e+f x)\right )-3 f x \text {Chi}\left (\frac {f x}{2}\right ) \left (\cosh \left (\frac {e}{2}\right )-i \sinh \left (\frac {e}{2}\right )\right )-3 f x \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 )+6 \sinh \left (\frac {1}{2} (e+f x)\right )+2 \sinh \left (\frac {3}{2} (e+f x)\right )+3 i f x \cosh \left (\frac {e}{2}\right ) \text {Shi}\left (\frac {f x}{2}\right )-3 f x \sinh \left (\frac {e}{2}\right ) \text {Shi}\left (\frac {f x}{2}\right )-3 i f x \cosh \left (\frac {3 e}{2}\right ) \text {Shi}\left (\frac {3 f x}{2}\right )-3 f x \sinh \left (\frac {3 e}{2}\right ) \text {Shi}\left (\frac {3 f x}{2}\right )\right )}{4 x \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-I + Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]]*((-6*I)*Cosh[(e + f*x)/2] + (2*I)*Cosh[(3*(e + f*x))/2] -
3*f*x*CoshIntegral[(f*x)/2]*(Cosh[e/2] - I*Sinh[e/2]) - 3*f*x*CoshIntegral[(3*f*x)/2]*(Cosh[(3*e)/2] + I*Sinh[
(3*e)/2]) + 6*Sinh[(e + f*x)/2] + 2*Sinh[(3*(e + f*x))/2] + (3*I)*f*x*Cosh[e/2]*SinhIntegral[(f*x)/2] - 3*f*x*
Sinh[e/2]*SinhIntegral[(f*x)/2] - (3*I)*f*x*Cosh[(3*e)/2]*SinhIntegral[(3*f*x)/2] - 3*f*x*Sinh[(3*e)/2]*SinhIn
tegral[(3*f*x)/2]))/(4*x*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3)

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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