Optimal. Leaf size=665 \[ \frac{2 \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{a d \sqrt{a+b}}-\frac{(3 a-b) \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}}+\frac{2 b^2 \tanh (c+d x)}{a d \left (a^2-b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}-\frac{4 a b^2 \tanh (c+d x)}{d \left (a^2-b^2\right )^2 \sqrt{a+b \text{sech}(c+d x)}}-\frac{b^2 \tanh (c+d x)}{d \left (a^2-b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}+\frac{2 \sqrt{a+b} \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^2 d}-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}-\frac{2 \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d \sqrt{a+b}}+\frac{4 a \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}} \]
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Rubi [A] time = 0.98349, antiderivative size = 665, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3896, 3785, 4058, 3921, 3784, 3832, 4004, 3875, 3833, 4003, 4005} \[ \frac{2 b^2 \tanh (c+d x)}{a d \left (a^2-b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}-\frac{4 a b^2 \tanh (c+d x)}{d \left (a^2-b^2\right )^2 \sqrt{a+b \text{sech}(c+d x)}}-\frac{b^2 \tanh (c+d x)}{d \left (a^2-b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}+\frac{2 \sqrt{a+b} \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^2 d}-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}+\frac{2 \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d \sqrt{a+b}}-\frac{(3 a-b) \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}}-\frac{2 \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d \sqrt{a+b}}+\frac{4 a \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3896
Rule 3785
Rule 4058
Rule 3921
Rule 3784
Rule 3832
Rule 4004
Rule 3875
Rule 3833
Rule 4003
Rule 4005
Rubi steps
\begin{align*} \int \frac{\coth ^2(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}} \, dx &=-\int \left (-\frac{1}{(a+b \text{sech}(c+d x))^{3/2}}-\frac{\text{csch}^2(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}}\right ) \, dx\\ &=\int \frac{1}{(a+b \text{sech}(c+d x))^{3/2}} \, dx+\int \frac{\text{csch}^2(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}} \, dx\\ &=-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}+\frac{2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \text{sech}(c+d x)}}+\frac{1}{2} (3 b) \int \frac{\text{sech}(c+d x)}{(a+b \text{sech}(c+d x))^{5/2}} \, dx-\frac{2 \int \frac{\frac{1}{2} \left (-a^2+b^2\right )+\frac{1}{2} a b \text{sech}(c+d x)+\frac{1}{2} b^2 \text{sech}^2(c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}-\frac{b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d (a+b \text{sech}(c+d x))^{3/2}}+\frac{2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \text{sech}(c+d x)}}-\frac{2 \int \frac{\frac{1}{2} \left (-a^2+b^2\right )+\left (\frac{a b}{2}-\frac{b^2}{2}\right ) \text{sech}(c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}-\frac{b \int \frac{\text{sech}(c+d x) \left (-\frac{3 a}{2}+\frac{1}{2} b \text{sech}(c+d x)\right )}{(a+b \text{sech}(c+d x))^{3/2}} \, dx}{a^2-b^2}-\frac{b^2 \int \frac{\text{sech}(c+d x) (1+\text{sech}(c+d x))}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a \sqrt{a+b} d}-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}-\frac{b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d (a+b \text{sech}(c+d x))^{3/2}}-\frac{4 a b^2 \tanh (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \text{sech}(c+d x)}}+\frac{2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \text{sech}(c+d x)}}+\frac{\int \frac{1}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{a}-\frac{b \int \frac{\text{sech}(c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{a (a+b)}+\frac{(2 b) \int \frac{\text{sech}(c+d x) \left (\frac{1}{4} \left (3 a^2+b^2\right )+a b \text{sech}(c+d x)\right )}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=-\frac{2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a \sqrt{a+b} d}+\frac{2 \coth (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a \sqrt{a+b} d}+\frac{2 \sqrt{a+b} \coth (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a^2 d}-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}-\frac{b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d (a+b \text{sech}(c+d x))^{3/2}}-\frac{4 a b^2 \tanh (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \text{sech}(c+d x)}}+\frac{2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \text{sech}(c+d x)}}+\frac{((3 a-b) b) \int \frac{\text{sech}(c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{2 (a-b) (a+b)^2}+\frac{\left (2 a b^2\right ) \int \frac{\text{sech}(c+d x) (1+\text{sech}(c+d x))}{\sqrt{a+b \text{sech}(c+d x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{4 a \coth (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{(a-b) (a+b)^{3/2} d}-\frac{2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a \sqrt{a+b} d}-\frac{(3 a-b) \coth (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{(a-b) (a+b)^{3/2} d}+\frac{2 \coth (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a \sqrt{a+b} d}+\frac{2 \sqrt{a+b} \coth (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (1+\text{sech}(c+d x))}{a-b}}}{a^2 d}-\frac{\coth (c+d x)}{d (a+b \text{sech}(c+d x))^{3/2}}-\frac{b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d (a+b \text{sech}(c+d x))^{3/2}}-\frac{4 a b^2 \tanh (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \text{sech}(c+d x)}}+\frac{2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \text{sech}(c+d x)}}\\ \end{align*}
Mathematica [F] time = 104.747, size = 0, normalized size = 0. \[ \int \frac{\coth ^2(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.166, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{2} \left ( a+b{\rm sech} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (d x + c\right )^{2}}{{\left (b \operatorname{sech}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname{sech}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (d x + c\right )^{2}}{{\left (b \operatorname{sech}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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