3.139 \(\int \frac {1}{x \sqrt {a+i a \sinh (e+f x)}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {1}{x \sqrt {a+i a \sinh (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 3.80, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {a+i a \sinh (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}} e^{\left (f x + e\right )}}{a x e^{\left (f x + e\right )} - i \, a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(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*e^(f*x + e) - I*a*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {a +i a \sinh \left (f x +e \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________