Optimal. Leaf size=38 \[ \frac{\text{sech}^3(a+b x)}{3 b}+\frac{\text{sech}(a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0292979, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2622, 302, 207} \[ \frac{\text{sech}^3(a+b x)}{3 b}+\frac{\text{sech}(a+b x)}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2622
Rule 302
Rule 207
Rubi steps
\begin{align*} \int \text{csch}(a+b x) \text{sech}^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=\frac{\text{sech}(a+b x)}{b}+\frac{\text{sech}^3(a+b x)}{3 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}+\frac{\text{sech}(a+b x)}{b}+\frac{\text{sech}^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0272027, size = 41, normalized size = 1.08 \[ \frac{\text{sech}^3(a+b x)}{3 b}+\frac{\text{sech}(a+b x)}{b}+\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 33, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{1}{3\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}}+ \left ( \cosh \left ( bx+a \right ) \right ) ^{-1}-2\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.0422, size = 146, normalized size = 3.84 \begin{align*} -\frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac{2 \,{\left (3 \, e^{\left (-b x - a\right )} + 10 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}\right )}}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.0867, size = 1958, normalized size = 51.53 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}{\left (a + b x \right )} \operatorname{sech}^{4}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18892, size = 124, normalized size = 3.26 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right )}{2 \, b} + \frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{2 \, b} + \frac{2 \,{\left (3 \,{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}}{3 \, b{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]