Optimal. Leaf size=89 \[ \frac{\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{2 c^5}-\frac{3 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^4 (a \cosh (x)+a+c \sinh (x))}-\frac{a \sinh (x)+c \cosh (x)}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2} \]
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Rubi [A] time = 0.0936418, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3129, 3153, 3124, 31} \[ \frac{\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{2 c^5}-\frac{3 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^4 (a \cosh (x)+a+c \sinh (x))}-\frac{a \sinh (x)+c \cosh (x)}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2} \]
Antiderivative was successfully verified.
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Rule 3129
Rule 3153
Rule 3124
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(a+a \cosh (x)+c \sinh (x))^3} \, dx &=-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{\int \frac{-2 a+a \cosh (x)+c \sinh (x)}{(a+a \cosh (x)+c \sinh (x))^2} \, dx}{2 c^2}\\ &=-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}+\frac{\left (3 a^2-c^2\right ) \int \frac{1}{a+a \cosh (x)+c \sinh (x)} \, dx}{2 c^4}\\ &=-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}+\frac{\left (3 a^2-c^2\right ) \operatorname{Subst}\left (\int \frac{1}{2 a+2 c x} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{c^4}\\ &=\frac{\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{2 c^5}-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.494929, size = 148, normalized size = 1.66 \[ \frac{4 \left (c^2-3 a^2\right ) \log \left (\cosh \left (\frac{x}{2}\right )\right )+\frac{6 c \left (c^2-a^2\right ) \sinh \left (\frac{x}{2}\right )}{a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )}+4 \left (3 a^2-c^2\right ) \log \left (a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )\right )+\frac{c^2 (a-c) (a+c)}{\left (a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )\right )^2}-6 a c \tanh \left (\frac{x}{2}\right )+c^2 \left (-\text{sech}^2\left (\frac{x}{2}\right )\right )}{8 c^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 138, normalized size = 1.6 \begin{align*}{\frac{1}{8\,{c}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{3\,a}{4\,{c}^{4}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{{a}^{4}}{8\,{c}^{5}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{2}}{4\,{c}^{3}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{8\,c} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{3}}{{c}^{5}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{a}{{c}^{3}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{3\,{a}^{2}}{2\,{c}^{5}}\ln \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{2\,{c}^{3}}\ln \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12245, size = 335, normalized size = 3.76 \begin{align*} -\frac{3 \, a^{3} + 6 \, a^{2} c + 3 \, a c^{2} +{\left (9 \, a^{3} + 9 \, a^{2} c + a c^{2} + c^{3}\right )} e^{\left (-x\right )} + 3 \,{\left (3 \, a^{3} - a c^{2}\right )} e^{\left (-2 \, x\right )} +{\left (3 \, a^{3} - 3 \, a^{2} c - a c^{2} + c^{3}\right )} e^{\left (-3 \, x\right )}}{a^{2} c^{4} + 2 \, a c^{5} + c^{6} + 4 \,{\left (a^{2} c^{4} + a c^{5}\right )} e^{\left (-x\right )} + 2 \,{\left (3 \, a^{2} c^{4} - c^{6}\right )} e^{\left (-2 \, x\right )} + 4 \,{\left (a^{2} c^{4} - a c^{5}\right )} e^{\left (-3 \, x\right )} +{\left (a^{2} c^{4} - 2 \, a c^{5} + c^{6}\right )} e^{\left (-4 \, x\right )}} + \frac{{\left (3 \, a^{2} - c^{2}\right )} \log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{2 \, c^{5}} - \frac{{\left (3 \, a^{2} - c^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.82751, size = 3359, normalized size = 37.74 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17161, size = 277, normalized size = 3.11 \begin{align*} \frac{{\left (3 \, a^{3} + 3 \, a^{2} c - a c^{2} - c^{3}\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{2 \,{\left (a c^{5} + c^{6}\right )}} - \frac{{\left (3 \, a^{2} - c^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, c^{5}} + \frac{3 \, a^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} c e^{\left (3 \, x\right )} - a c^{2} e^{\left (3 \, x\right )} - c^{3} e^{\left (3 \, x\right )} + 9 \, a^{3} e^{\left (2 \, x\right )} - 3 \, a c^{2} e^{\left (2 \, x\right )} + 9 \, a^{3} e^{x} - 9 \, a^{2} c e^{x} + a c^{2} e^{x} - c^{3} e^{x} + 3 \, a^{3} - 6 \, a^{2} c + 3 \, a c^{2}}{{\left (a e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + a - c\right )}^{2} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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