Optimal. Leaf size=52 \[ \frac{\tanh (c+d x)}{b d}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{b^{3/2} d \sqrt{a+b}} \]
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Rubi [A] time = 0.0703809, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4146, 388, 208} \[ \frac{\tanh (c+d x)}{b d}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{b^{3/2} d \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\text{sech}^4(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\tanh (c+d x)}{b d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{b d}\\ &=-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b} d}+\frac{\tanh (c+d x)}{b d}\\ \end{align*}
Mathematica [B] time = 0.679913, size = 182, normalized size = 3.5 \[ \frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\sqrt{a+b} \text{sech}(c) \sinh (d x) \sqrt{b (\cosh (c)-\sinh (c))^4} \text{sech}(c+d x)+a (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left (\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )\right )}{2 b d \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4} \left (a+b \text{sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 141, normalized size = 2.7 \begin{align*} -{\frac{a}{2\,d}\ln \left ( \sqrt{a+b} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{b}+\sqrt{a+b} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+b}}}}+{\frac{a}{2\,d}\ln \left ( -\sqrt{a+b} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{b}-\sqrt{a+b} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+b}}}}+2\,{\frac{\tanh \left ( 1/2\,dx+c/2 \right ) }{bd \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27102, size = 1661, normalized size = 31.94 \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{\operatorname{sech}^{4}{\left (c + d x \right )}}{a + b \operatorname{sech}^{2}{\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 = 1.16742, size = 100, normalized size = 1.92 \begin{align*} -\frac{a \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} b d} - \frac{2}{b d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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