\(\int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx\) [140]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {1}{x^{2} \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {1}{x^2\,\sqrt {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]

[In]

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

[Out]

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