3.144 \(\int \frac {1}{x (a+i a \sinh (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {1}{x (a+i a \sinh (e+f x))^{3/2}},x\right ) \]

[Out]

Unintegrable(1/x/(a+I*a*sinh(f*x+e))^(3/2),x)

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Rubi [A]  time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x (a+i a \sinh (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + I*a*Sinh[e + f*x])^(3/2)),x]

[Out]

Defer[Int][1/(x*(a + I*a*Sinh[e + f*x])^(3/2)), x]

Rubi steps

\begin {align*} \int \frac {1}{x (a+i a \sinh (e+f x))^{3/2}} \, dx &=\int \frac {1}{x (a+i a \sinh (e+f x))^{3/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 21.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{x (a+i a \sinh (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(a + I*a*Sinh[e + f*x])^(3/2)),x]

[Out]

Integrate[1/(x*(a + I*a*Sinh[e + f*x])^(3/2)), x]

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fricas [A]  time = 0.60, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a^{2} f^{2} x^{2} e^{\left (f x + e\right )} - a^{2} f^{2} x^{2}\right )} {\rm integral}\left (\frac {{\left (-i \, f^{2} x^{2} + 8 i\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}} e^{\left (f x + e\right )}}{2 \, a^{2} f^{2} x^{3} e^{\left (f x + e\right )} - 2 i \, a^{2} f^{2} x^{3}}, x\right ) + {\left ({\left (-i \, f x + 2 i\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (f x + 2\right )} e^{\left (f x + e\right )}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}}}{a^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a^{2} f^{2} x^{2} e^{\left (f x + e\right )} - a^{2} f^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

((a^2*f^2*x^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*x^2*e^(f*x + e) - a^2*f^2*x^2)*integral((-I*f^2*x^2 + 8*I)*sqrt(1/
2*I*a*e^(-f*x - e))*e^(f*x + e)/(2*a^2*f^2*x^3*e^(f*x + e) - 2*I*a^2*f^2*x^3), 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*x^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*x^2*e^(f
*x + e) - a^2*f^2*x^2)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((I*a*sinh(f*x + e) + a)^(3/2)*x), x)

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maple [A]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((I*a*sinh(f*x + e) + a)^(3/2)*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+I*a*sinh(f*x+e))**(3/2),x)

[Out]

Integral(1/(x*(I*a*(sinh(e + f*x) - I))**(3/2)), x)

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