Optimal. Leaf size=54 \[ -\text {Int}\left (\frac {\tanh (a+b x)}{x^2},x\right )+b \cosh (2 a) \text {Chi}(2 b x)+b \sinh (2 a) \text {Shi}(2 b x)-\frac {\sinh (2 a+2 b x)}{2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, 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 = {} \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x^2} \, dx &=\int \frac {\cosh (a+b x) \sinh (a+b x)}{x^2} \, dx-\int \frac {\tanh (a+b x)}{x^2} \, dx\\ &=\int \frac {\sinh (2 a+2 b x)}{2 x^2} \, dx-\int \frac {\tanh (a+b x)}{x^2} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x^2} \, dx-\int \frac {\tanh (a+b x)}{x^2} \, dx\\ &=-\frac {\sinh (2 a+2 b x)}{2 x}+b \int \frac {\cosh (2 a+2 b x)}{x} \, dx-\int \frac {\tanh (a+b x)}{x^2} \, dx\\ &=-\frac {\sinh (2 a+2 b x)}{2 x}+(b \cosh (2 a)) \int \frac {\cosh (2 b x)}{x} \, dx+(b \sinh (2 a)) \int \frac {\sinh (2 b x)}{x} \, dx-\int \frac {\tanh (a+b x)}{x^2} \, dx\\ &=b \cosh (2 a) \text {Chi}(2 b x)-\frac {\sinh (2 a+2 b x)}{2 x}+b \sinh (2 a) \text {Shi}(2 b x)-\int \frac {\tanh (a+b x)}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 20.20, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) + \frac {1}{2} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) + \frac {1}{x} + 2 \, \int \frac {1}{x^{2} e^{\left (2 \, b x + 2 \, a\right )} + x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^2\,\mathrm {cosh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________