Optimal. Leaf size=61 \[ -\frac{\left (a^2-b^2\right ) \cosh (x)}{a^3}+\frac{b \left (a^2-b^2\right ) \log (a \cosh (x)+b)}{a^4}-\frac{b \cosh ^2(x)}{2 a^2}+\frac{\cosh ^3(x)}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.180163, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3872, 2837, 12, 772} \[ -\frac{\left (a^2-b^2\right ) \cosh (x)}{a^3}+\frac{b \left (a^2-b^2\right ) \log (a \cosh (x)+b)}{a^4}-\frac{b \cosh ^2(x)}{2 a^2}+\frac{\cosh ^3(x)}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2837
Rule 12
Rule 772
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{a+b \text{sech}(x)} \, dx &=-\int \frac{\cosh (x) \sinh ^3(x)}{-b-a \cosh (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{a (-b+x)} \, dx,x,-a \cosh (x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{-b+x} \, dx,x,-a \cosh (x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1-\frac{b^2}{a^2}\right )+\frac{-a^2 b+b^3}{b-x}-b x-x^2\right ) \, dx,x,-a \cosh (x)\right )}{a^4}\\ &=-\frac{\left (a^2-b^2\right ) \cosh (x)}{a^3}-\frac{b \cosh ^2(x)}{2 a^2}+\frac{\cosh ^3(x)}{3 a}+\frac{b \left (a^2-b^2\right ) \log (b+a \cosh (x))}{a^4}\\ \end{align*}
Mathematica [A] time = 0.126838, size = 66, normalized size = 1.08 \[ \frac{\left (12 a b^2-9 a^3\right ) \cosh (x)-3 a^2 b \cosh (2 x)+12 a^2 b \log (a \cosh (x)+b)+a^3 \cosh (3 x)-12 b^3 \log (a \cosh (x)+b)}{12 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.031, size = 361, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.1307, size = 173, normalized size = 2.84 \begin{align*} -\frac{{\left (3 \, a b e^{\left (-x\right )} - a^{2} + 3 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} - \frac{3 \, a b e^{\left (-2 \, x\right )} - a^{2} e^{\left (-3 \, x\right )} + 3 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} + \frac{{\left (a^{2} b - b^{3}\right )} x}{a^{4}} + \frac{{\left (a^{2} b - b^{3}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.76321, size = 1251, normalized size = 20.51 \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 \frac{\sinh ^{3}{\left (x \right )}}{a + b \operatorname{sech}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17214, size = 117, normalized size = 1.92 \begin{align*} \frac{a^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 3 \, a b{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 12 \, a^{2}{\left (e^{\left (-x\right )} + e^{x}\right )} + 12 \, b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}}{24 \, a^{3}} + \frac{{\left (a^{2} b - b^{3}\right )} \log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]