Optimal. Leaf size=85 \[ \frac{\left (a^2+b^2\right ) \sinh (c+d x)}{a^3 d}-\frac{b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b)}{a^4 d}-\frac{b \sinh ^2(c+d x)}{2 a^2 d}+\frac{\sinh ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.182335, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2837, 12, 772} \[ \frac{\left (a^2+b^2\right ) \sinh (c+d x)}{a^3 d}-\frac{b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b)}{a^4 d}-\frac{b \sinh ^2(c+d x)}{2 a^2 d}+\frac{\sinh ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2837
Rule 12
Rule 772
Rubi steps
\begin{align*} \int \frac{\cosh ^3(c+d x)}{a+b \text{csch}(c+d x)} \, dx &=i \int \frac{\cosh ^3(c+d x) \sinh (c+d x)}{i b+i a \sinh (c+d 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 (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{i b+x} \, dx,x,i a \sinh (c+d x)\right )}{a^4 d}\\ &=-\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 (c+d x)\right )}{a^4 d}\\ &=-\frac{b \left (a^2+b^2\right ) \log (b+a \sinh (c+d x))}{a^4 d}+\frac{\left (a^2+b^2\right ) \sinh (c+d x)}{a^3 d}-\frac{b \sinh ^2(c+d x)}{2 a^2 d}+\frac{\sinh ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.223873, size = 75, normalized size = 0.88 \[ \frac{6 a \left (a^2+b^2\right ) \sinh (c+d x)-6 b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b)-3 a^2 b \sinh ^2(c+d x)+2 a^3 \sinh ^3(c+d x)}{6 a^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 428, normalized size = 5. \begin{align*} -{\frac{1}{3\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{b}{2\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{1}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{b}{2\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{{b}^{2}}{d{a}^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{b}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{{b}^{3}}{d{a}^{4}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{b}{d{a}^{2}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( 1/2\,dx+c/2 \right ) -b \right ) }-{\frac{{b}^{3}}{d{a}^{4}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( 1/2\,dx+c/2 \right ) -b \right ) }-{\frac{1}{3\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{b}{2\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{1}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{b}{2\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{b}^{2}}{d{a}^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{b}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{{b}^{3}}{d{a}^{4}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21209, size = 247, normalized size = 2.91 \begin{align*} -\frac{{\left (3 \, a b e^{\left (-d x - c\right )} - a^{2} - 3 \,{\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, a^{3} d} - \frac{{\left (a^{2} b + b^{3}\right )}{\left (d x + c\right )}}{a^{4} d} - \frac{3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + a^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \,{\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, a^{3} d} - \frac{{\left (a^{2} b + b^{3}\right )} \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82615, size = 1613, normalized size = 18.98 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname{csch}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18495, size = 215, normalized size = 2.53 \begin{align*} -\frac{{\left (a^{2} b + b^{3}\right )} \log \left ({\left | a{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, b \right |}\right )}{a^{4} d} + \frac{a^{2} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, b^{2} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{24 \, a^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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