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

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\text {Int}\left (\frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 23.84 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81

\[\int \frac {1}{x^{2} \left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 9.95 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 52.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {1}{x^{2} \left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 (a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

[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)