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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{1}{x (a+i a \sinh (e+f x))^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0876549, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 21.6249, size = 0, normalized size = 0. \[ \int \frac{1}{x (a+i a \sinh (e+f x))^{3/2}} \, dx \]

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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Maple [A]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+ia\sinh \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{\frac{1}{2}}{\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{i \, a e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a e^{\left (f x + e\right )} - i \, a} e^{\left (-\frac{1}{2} \, f x - \frac{1}{2} \, e\right )} +{\left (a^{2} f^{2} x^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{2} x^{2} e^{\left (f x + e\right )} + i \, a^{2} f^{2} x^{2}\right )}{\rm integral}\left (\frac{\sqrt{\frac{1}{2}}{\left (-i \, f^{2} x^{2} + 8 i\right )} \sqrt{i \, a e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a e^{\left (f x + e\right )} - i \, a} e^{\left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}}{2 \, a^{2} f^{2} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - 4 i \, a^{2} f^{2} x^{3} e^{\left (f x + e\right )} - 2 \, a^{2} f^{2} x^{3}}, x\right )}{a^{2} f^{2} x^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{2} x^{2} e^{\left (f x + e\right )} + i \, a^{2} f^{2} x^{2}} \end{align*}

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]

(sqrt(1/2)*((-I*f*x + 2*I)*e^(2*f*x + 2*e) + (f*x + 2)*e^(f*x + e))*sqrt(I*a*e^(2*f*x + 2*e) + 2*a*e^(f*x + e)
- I*a)*e^(-1/2*f*x - 1/2*e) + (a^2*f^2*x^2*e^(3*f*x + 3*e) - 3*I*a^2*f^2*x^2*e^(2*f*x + 2*e) - 3*a^2*f^2*x^2*
e^(f*x + e) + I*a^2*f^2*x^2)*integral(sqrt(1/2)*(-I*f^2*x^2 + 8*I)*sqrt(I*a*e^(2*f*x + 2*e) + 2*a*e^(f*x + e)
- I*a)*e^(1/2*f*x + 1/2*e)/(2*a^2*f^2*x^3*e^(2*f*x + 2*e) - 4*I*a^2*f^2*x^3*e^(f*x + e) - 2*a^2*f^2*x^3), x))/
(a^2*f^2*x^2*e^(3*f*x + 3*e) - 3*I*a^2*f^2*x^2*e^(2*f*x + 2*e) - 3*a^2*f^2*x^2*e^(f*x + e) + I*a^2*f^2*x^2)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a \left (i \sinh{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

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)