Optimal. Leaf size=121 \[ \frac{a^2 b \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{a \left (a^2-b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}-\frac{a^2 b \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac{\text{sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.229112, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2837, 12, 1647, 801, 635, 203, 260} \[ \frac{a^2 b \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{a \left (a^2-b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}-\frac{a^2 b \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac{\text{sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 1647
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\text{sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{x^2}{b^2 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{x^2}{(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) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^2}{a^2+b^2}-\frac{a b^2 x}{a^2+b^2}}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{2 b d}\\ &=-\frac{\text{sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)}-\frac{a b^2 \left (a^2-b^2-2 a 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 d}\\ &=\frac{a^2 b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\text{sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac{(a b) \operatorname{Subst}\left (\int \frac{a^2-b^2-2 a x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac{a^2 b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\text{sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{\left (a^2 b\right ) \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 (a b \left (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{a \left (a^2-b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}-\frac{a^2 b \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{a^2 b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\text{sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 0.327329, size = 130, normalized size = 1.07 \[ -\frac{b \left (a^2+b^2\right ) \text{sech}^2(c+d x)+a \left (a^2+b^2\right ) \tanh (c+d x) \text{sech}(c+d x)+a \left (\left (a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))+a ((b+i a) \log (-\sinh (c+d x)+i)+(b-i a) \log (\sinh (c+d x)+i)-2 b \log (a+b \sinh (c+d x)))\right )}{2 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 475, normalized size = 3.9 \begin{align*} 4\,{\frac{{a}^{2}b\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 ( 4\,{a}^{4}+8\,{a}^{2}{b}^{2}+4\,{b}^{4} \right ) }}+{\frac{{a}^{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}}+{\frac{a{b}^{2}}{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}^{2}b}{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}{b}^{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}}}-{\frac{{a}^{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{b}^{2}}{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 ) }\arctan \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{a{b}^{2}}{d \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }\arctan \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{{a}^{2}b}{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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77111, size = 296, normalized size = 2.45 \begin{align*} \frac{a^{2} b \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^{2} b \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 (a^{3} - a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a 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 = 2.41718, size = 2241, normalized size = 18.52 \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 ^{2}{\left (c + d x \right )} \operatorname{sech}{\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 = 1.26063, size = 305, normalized size = 2.52 \begin{align*} -\frac{\frac{a^{2} b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{a^{2} b \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 (a^{3} e^{c} - a b^{2} 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^{3} e^{\left (3 \, c\right )} + a b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 2 \,{\left (a^{2} b e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} -{\left (a^{3} e^{c} + a b^{2} 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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