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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0776774, 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 = {} \[ \int \frac{1}{x \sqrt{a+i a \sinh (e+f x)}} \, dx \]

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 = 3.84339, size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a+i a \sinh (e+f x)}} \, dx \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt{a+ia\sinh \left ( fx+e \right ) }}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{i \, a \sinh \left (f x + e\right ) + a} 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))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

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)*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*x*e^(2*f*x +
2*e) - 2*I*a*x*e^(f*x + e) - a*x), x)

________________________________________________________________________________________

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

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(a*(I*sinh(e + f*x) + 1))), x)

________________________________________________________________________________________

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

[Out]

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