Optimal. Leaf size=83 \[ \frac{b^2}{2 a^2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )}+\frac{b (2 a+b) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d (a+b)^2}+\frac{\log (\sinh (c+d x))}{d (a+b)^2} \]
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Rubi [A] time = 0.128867, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4138, 446, 88} \[ \frac{b^2}{2 a^2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )}+\frac{b (2 a+b) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d (a+b)^2}+\frac{\log (\sinh (c+d x))}{d (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4138
Rule 446
Rule 88
Rubi steps
\begin{align*} \int \frac{\coth (c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^5}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{(1-x) (b+a x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{(a+b)^2 (-1+x)}+\frac{b^2}{a (a+b) (b+a x)^2}-\frac{b (2 a+b)}{a (a+b)^2 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{b^2}{2 a^2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}+\frac{b (2 a+b) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 (a+b)^2 d}+\frac{\log (\sinh (c+d x))}{(a+b)^2 d}\\ \end{align*}
Mathematica [A] time = 0.287691, size = 115, normalized size = 1.39 \[ \frac{a \sinh ^2(c+d x) \left (2 a^2 \log (\sinh (c+d x))+b (2 a+b) \log \left (a \sinh ^2(c+d x)+a+b\right )\right )+(a+b) \left (2 a^2 \log (\sinh (c+d x))+b \left ((2 a+b) \log \left (a \sinh ^2(c+d x)+a+b\right )+b\right )\right )}{a^2 d (a+b)^2 (a \cosh (2 (c+d x))+a+2 b)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.084, size = 292, normalized size = 3.5 \begin{align*} -{\frac{1}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-2\,{\frac{{b}^{2} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d \left ( a+b \right ) ^{2} \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+b \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) a}}+{\frac{b}{da \left ( a+b \right ) ^{2}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }+{\frac{{b}^{2}}{2\,d{a}^{2} \left ( a+b \right ) ^{2}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }+{\frac{1}{d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{d{a}^{2}}\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 = 1.37354, size = 282, normalized size = 3.4 \begin{align*} \frac{2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{4} + a^{3} b + 2 \,{\left (a^{4} + 3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{4} + a^{3} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac{{\left (2 \, a b + b^{2}\right )} \log \left (2 \,{\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \,{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac{d x + c}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.30547, size = 2457, normalized size = 29.6 \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{\coth{\left (c + d x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.69404, size = 332, normalized size = 4. \begin{align*} \frac{\frac{{\left (2 \, a b + b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{4} + 2 \, a^{3} b + a^{2} b^{2}} + \frac{2 \, e^{\left (2 \, c\right )} \log \left ({\left | -e^{\left (2 \, d x + 2 \, c\right )} + 1 \right |}\right )}{a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}} - \frac{2 \, d x}{a^{2}} - \frac{2 \, a b e^{\left (4 \, d x + 4 \, c\right )} + b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b + b^{2}}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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