Optimal. Leaf size=57 \[ \frac{\left (a^2+b^2\right ) \sinh (x)}{a^3}-\frac{b \left (a^2+b^2\right ) \log (a \sinh (x)+b)}{a^4}-\frac{b \sinh ^2(x)}{2 a^2}+\frac{\sinh ^3(x)}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15953, antiderivative size = 57, 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} \[ \frac{\left (a^2+b^2\right ) \sinh (x)}{a^3}-\frac{b \left (a^2+b^2\right ) \log (a \sinh (x)+b)}{a^4}-\frac{b \sinh ^2(x)}{2 a^2}+\frac{\sinh ^3(x)}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2837
Rule 12
Rule 772
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{a+b \text{csch}(x)} \, dx &=i \int \frac{\cosh ^3(x) \sinh (x)}{i b+i a \sinh (x)} \, dx\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{a (i b+x)} \, dx,x,i a \sinh (x)\right )}{a^3}\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{i b+x} \, dx,x,i a \sinh (x)\right )}{a^4}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (a^2 \left (1+\frac{b^2}{a^2}\right )-\frac{b \left (a^2+b^2\right )}{b-i x}+i b x-x^2\right ) \, dx,x,i a \sinh (x)\right )}{a^4}\\ &=-\frac{b \left (a^2+b^2\right ) \log (b+a \sinh (x))}{a^4}+\frac{\left (a^2+b^2\right ) \sinh (x)}{a^3}-\frac{b \sinh ^2(x)}{2 a^2}+\frac{\sinh ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.107951, size = 56, normalized size = 0.98 \[ \frac{6 a \left (a^2+b^2\right ) \sinh (x)-6 b \left (a^2+b^2\right ) \log (a \sinh (x)+b)-3 a^2 b \sinh ^2(x)+2 a^3 \sinh ^3(x)}{6 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.035, size = 274, normalized size = 4.8 \begin{align*} -{\frac{1}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{b}{2\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{b}{2\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{{b}^{2}}{{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{b}{{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{{b}^{3}}{{a}^{4}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{b}{{a}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \right ) }-{\frac{{b}^{3}}{{a}^{4}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \right ) }-{\frac{1}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{b}{2\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{b}{2\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{b}^{2}}{{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{b}{{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{{b}^{3}}{{a}^{4}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.00699, size = 171, normalized size = 3. \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 = 1.61395, size = 1251, normalized size = 21.95 \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{\cosh ^{3}{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15823, size = 131, normalized size = 2.3 \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]