Optimal. Leaf size=107 \[ \frac{b x \left (a^2-2 b^2\right )}{2 a^4}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{2 b^4 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}-\frac{b \sinh (x) \cosh (x)}{2 a^2}+\frac{\sinh ^2(x) \cosh (x)}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.4582, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3853, 4104, 3919, 3831, 2660, 618, 206} \[ \frac{b x \left (a^2-2 b^2\right )}{2 a^4}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{2 b^4 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}-\frac{b \sinh (x) \cosh (x)}{2 a^2}+\frac{\sinh ^2(x) \cosh (x)}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3853
Rule 4104
Rule 3919
Rule 3831
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{a+b \text{csch}(x)} \, dx &=\frac{\cosh (x) \sinh ^2(x)}{3 a}-\frac{i \int \frac{\left (-3 i b-2 i a \text{csch}(x)-2 i b \text{csch}^2(x)\right ) \sinh ^2(x)}{a+b \text{csch}(x)} \, dx}{3 a}\\ &=-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}+\frac{\int \frac{\left (-2 \left (2 a^2-3 b^2\right )-a b \text{csch}(x)+3 b^2 \text{csch}^2(x)\right ) \sinh (x)}{a+b \text{csch}(x)} \, dx}{6 a^2}\\ &=-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}+\frac{i \int \frac{-3 i b \left (a^2-2 b^2\right )-3 i a b^2 \text{csch}(x)}{a+b \text{csch}(x)} \, dx}{6 a^3}\\ &=\frac{b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}+\frac{b^4 \int \frac{\text{csch}(x)}{a+b \text{csch}(x)} \, dx}{a^4}\\ &=\frac{b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}+\frac{b^3 \int \frac{1}{1+\frac{a \sinh (x)}{b}} \, dx}{a^4}\\ &=\frac{b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^4}\\ &=\frac{b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (1+\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}-2 \tanh \left (\frac{x}{2}\right )\right )}{a^4}\\ &=\frac{b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac{2 b^4 \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a^2+b^2}}\right )}{a^4 \sqrt{a^2+b^2}}-\frac{\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac{b \cosh (x) \sinh (x)}{2 a^2}+\frac{\cosh (x) \sinh ^2(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.412106, size = 104, normalized size = 0.97 \[ \frac{\left (12 a b^2-9 a^3\right ) \cosh (x)+3 b \left (\frac{8 b^3 \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+2 a^2 x-a^2 \sinh (2 x)-4 b^2 x\right )+a^3 \cosh (3 x)}{12 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.039, size = 262, normalized size = 2.5 \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}{2\,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}{2\,{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 ) }+2\,{\frac{{b}^{4}}{{a}^{4}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \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}{2\,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}{2\,{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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.0559, size = 1947, normalized size = 18.2 \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{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20164, size = 209, normalized size = 1.95 \begin{align*} \frac{b^{4} \log \left (\frac{{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a^{4}} + \frac{a^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} - 9 \, a^{2} e^{x} + 12 \, b^{2} e^{x}}{24 \, a^{3}} + \frac{{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} + \frac{{\left (3 \, a^{2} b e^{x} + a^{3} - 3 \,{\left (3 \, a^{3} - 4 \, a b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]