Optimal. Leaf size=30 \[ -i \text{Unintegrable}\left (\frac{i \sinh (c+d x)}{a+b \tanh ^3(c+d x)},x\right ) \]
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Rubi [A] time = 0.0276825, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=-\left (i \int \frac{i \sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx\right )\\ \end{align*}
Mathematica [A] time = 0.226157, size = 409, normalized size = 13.63 \[ \frac{b \text{RootSum}\left [\text{$\#$1}^6 a+3 \text{$\#$1}^4 a+3 \text{$\#$1}^2 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b+a-b\& ,\frac{4 \text{$\#$1}^4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 \text{$\#$1}^4 a c+2 \text{$\#$1}^4 a d x-2 \text{$\#$1}^4 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\text{$\#$1}^4 b c-\text{$\#$1}^4 b d x+4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 a c+2 a d x+b c+b d x}{\text{$\#$1}^5 a+2 \text{$\#$1}^3 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} a+\text{$\#$1} b}\& \right ]+6 a \cosh (c+d x)-6 b \sinh (c+d x)}{6 d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.108, size = 164, normalized size = 5.5 \begin{align*} -4\,{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+{\frac{b}{3\,d \left ( a-b \right ) \left ( a+b \right ) }\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}a-2\,{{\it \_R}}^{3}b+6\,{{\it \_R}}^{2}a-2\,{\it \_R}\,b+a}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }}+4\,{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a + b\right )} e^{\left (-d x\right )}}{2 \,{\left (a^{2} d e^{c} - b^{2} d e^{c}\right )}} + \frac{1}{2} \, \int \frac{4 \,{\left ({\left (2 \, a b e^{\left (5 \, c\right )} - b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} +{\left (2 \, a b e^{c} + b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{3} - a^{2} b - a b^{2} + b^{3} +{\left (a^{3} e^{\left (6 \, c\right )} + a^{2} b e^{\left (6 \, c\right )} - a b^{2} e^{\left (6 \, c\right )} - b^{3} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \,{\left (a^{3} e^{\left (4 \, c\right )} - a^{2} b e^{\left (4 \, c\right )} - a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \,{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [A] time = 1.62107, size = 263, normalized size = 8.77 \begin{align*} \frac{\frac{e^{\left (d x + 8 \, c\right )}}{a e^{\left (7 \, c\right )} + b e^{\left (7 \, c\right )}} + \frac{e^{\left (-d x\right )}}{a e^{c} - b e^{c}}}{2 \, d} + \frac{\frac{6 \,{\left (2 \, a b e^{c} + b^{2} e^{c}\right )} d x}{a d - b d} - \frac{{\left (2 \, a b e^{c} + b^{2} e^{c}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a d - b d}}{3 \,{\left (a^{2} - b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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