3.2.45 \(\int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx\) [145]

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

[Out]

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

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Rubi [A]
time = 0.06, 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 = {} \begin {gather*} \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 17.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^2/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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^2), x)

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

[Out]

((a^2*f^2*x^3*e^(2*f*x + 2*e) - 2*I*a^2*f^2*x^3*e^(f*x + e) - a^2*f^2*x^3)*integral(1/2*(-I*f^2*x^2 + 24*I)*sq
rt(1/2*I*a*e^(-f*x - e))*e^(f*x + e)/(a^2*f^2*x^4*e^(f*x + e) - I*a^2*f^2*x^4), x) + ((-I*f*x + 4*I)*e^(2*f*x
+ 2*e) + (f*x + 4)*e^(f*x + e))*sqrt(1/2*I*a*e^(-f*x - e)))/(a^2*f^2*x^3*e^(2*f*x + 2*e) - 2*I*a^2*f^2*x^3*e^(
f*x + e) - a^2*f^2*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
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(1/x^2/(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^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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