Optimal. Leaf size=120 \[ -\frac{a^3 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{b \left (3 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}+\frac{a^3 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.201183, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2721, 1647, 801, 635, 203, 260} \[ -\frac{a^3 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{b \left (3 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}+\frac{a^3 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 2721
Rule 1647
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a b^4}{a^2+b^2}+\frac{b^2 \left (2 a^2+b^2\right ) x}{a^2+b^2}}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{2 b^2 d}\\ &=\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\operatorname{Subst}\left (\int \left (\frac{2 a^3 b^2}{\left (a^2+b^2\right )^2 (a+x)}-\frac{b^2 \left (3 a^2 b^2+b^4+2 a^3 x\right )}{\left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{2 b^2 d}\\ &=-\frac{a^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac{\operatorname{Subst}\left (\int \frac{3 a^2 b^2+b^4+2 a^3 x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac{a^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (b^2 \left (3 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac{b \left (3 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}+\frac{a^3 \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{a^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 0.402208, size = 152, normalized size = 1.27 \[ -\frac{-a \left (a^2+b^2\right ) \text{sech}^2(c+d x)-\left (a^3-i \left (2 a^2 b+b^3\right )\right ) \log (-\sinh (c+d x)+i)-\left (a^3+i \left (2 a^2 b+b^3\right )\right ) \log (\sinh (c+d x)+i)+b \left (a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))+b \left (a^2+b^2\right ) \tanh (c+d x) \text{sech}(c+d x)+2 a^3 \log (a+b \sinh (c+d x))}{2 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 472, normalized size = 3.9 \begin{align*} -8\,{\frac{{a}^{3}\ln \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,\tanh \left ( 1/2\,dx+c/2 \right ) b-a \right ) }{d \left ( 8\,{a}^{4}+16\,{a}^{2}{b}^{2}+8\,{b}^{4} \right ) }}+{\frac{{a}^{2}b}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{{b}^{3}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}-2\,{\frac{ \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}{a}^{3}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-2\,{\frac{ \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a{b}^{2}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{{a}^{2}b}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}-{\frac{{b}^{3}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{{a}^{3}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) }+3\,{\frac{\arctan \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ){a}^{2}b}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+{\frac{{b}^{3}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }\arctan \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67531, size = 293, normalized size = 2.44 \begin{align*} -\frac{a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac{a^{3} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{{\left (3 \, a^{2} b + b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \,{\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.90572, size = 2217, normalized size = 18.48 \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{\tanh ^{3}{\left (c + d x \right )}}{a + b \sinh{\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 = 1.4486, size = 301, normalized size = 2.51 \begin{align*} \frac{\frac{a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{a^{3} \log \left ({\left | -b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a e^{\left (d x + c\right )} + b \right |}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (3 \, a^{2} b e^{c} + b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (a^{2} b e^{\left (3 \, c\right )} + b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - 2 \,{\left (a^{3} e^{\left (2 \, c\right )} + a b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} -{\left (a^{2} b e^{c} + b^{3} e^{c}\right )} e^{\left (d x\right )}}{{\left (a^{2} + b^{2}\right )}^{2}{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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