Optimal. Leaf size=62 \[ \frac{2 a \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^2 \sqrt{a^2+b^2}}-\frac{\cosh (x)}{b (a+b \sinh (x))}+\frac{x}{b^2} \]
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Rubi [A] time = 0.126822, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4391, 2693, 2735, 2660, 618, 206} \[ \frac{2 a \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^2 \sqrt{a^2+b^2}}-\frac{\cosh (x)}{b (a+b \sinh (x))}+\frac{x}{b^2} \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2693
Rule 2735
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(a \text{sech}(x)+b \tanh (x))^2} \, dx &=\int \frac{\cosh ^2(x)}{(a+b \sinh (x))^2} \, dx\\ &=-\frac{\cosh (x)}{b (a+b \sinh (x))}+\frac{\int \frac{\sinh (x)}{a+b \sinh (x)} \, dx}{b}\\ &=\frac{x}{b^2}-\frac{\cosh (x)}{b (a+b \sinh (x))}-\frac{a \int \frac{1}{a+b \sinh (x)} \, dx}{b^2}\\ &=\frac{x}{b^2}-\frac{\cosh (x)}{b (a+b \sinh (x))}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^2}\\ &=\frac{x}{b^2}-\frac{\cosh (x)}{b (a+b \sinh (x))}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{b^2}\\ &=\frac{x}{b^2}+\frac{2 a \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^2 \sqrt{a^2+b^2}}-\frac{\cosh (x)}{b (a+b \sinh (x))}\\ \end{align*}
Mathematica [C] time = 3.68398, size = 659, normalized size = 10.63 \[ \frac{\cosh (x) \left (\sqrt{a+i b} \left (\sqrt{b^2} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \left (2 \sqrt [4]{-1} a \sqrt{b} (b+i a) \sin ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{a-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{b}}\right )-\sqrt{a-i b} \left (a^2+b^2\right ) \sqrt{1+i \sinh (x)} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}\right )+2 i a^2 b \sqrt{a-i b} \sqrt{1+i \sinh (x)} \tan ^{-1}\left (\frac{\sqrt{-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{i b} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}}}\right )+2 \sinh (x) \left (\sqrt [4]{-1} \sqrt{b^2} b^{3/2} (b+i a) \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sin ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{a-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{b}}\right )+i a b^2 \sqrt{a-i b} \sqrt{1+i \sinh (x)} \tan ^{-1}\left (\frac{\sqrt{-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{i b} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}}}\right )\right )\right )+2 i a \sqrt{b^2} (b+i a) \sqrt{1+i \sinh (x)} (a+b \sinh (x)) \tanh ^{-1}\left (\frac{\sqrt{a-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{a+i b} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}}}\right )\right )}{b \sqrt{b^2} (a-i b)^{3/2} (a+i b)^{3/2} \sqrt{1+i \sinh (x)} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}} (a+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 119, normalized size = 1.9 \begin{align*}{\frac{1}{{b}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) a}}+2\,{\frac{1}{b \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }}-2\,{\frac{a}{{b}^{2}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{1}{{b}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -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 = 2.40107, size = 933, normalized size = 15.05 \begin{align*} -\frac{{\left (a^{2} b + b^{3}\right )} x \cosh \left (x\right )^{2} +{\left (a^{2} b + b^{3}\right )} x \sinh \left (x\right )^{2} - 2 \, a^{2} b - 2 \, b^{3} +{\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) - a b + 2 \,{\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) -{\left (a^{2} b + b^{3}\right )} x + 2 \,{\left (a^{3} + a b^{2} +{\left (a^{3} + a b^{2}\right )} x\right )} \cosh \left (x\right ) + 2 \,{\left (a^{3} + a b^{2} +{\left (a^{2} b + b^{3}\right )} x \cosh \left (x\right ) +{\left (a^{3} + a b^{2}\right )} x\right )} \sinh \left (x\right )}{a^{2} b^{3} + b^{5} -{\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{2} -{\left (a^{2} b^{3} + b^{5}\right )} \sinh \left (x\right )^{2} - 2 \,{\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right ) - 2 \,{\left (a^{3} b^{2} + a b^{4} +{\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \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{1}{\left (a \operatorname{sech}{\left (x \right )} + b \tanh{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19518, size = 131, normalized size = 2.11 \begin{align*} -\frac{a \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} b^{2}} + \frac{x}{b^{2}} + \frac{2 \,{\left (a e^{x} - b\right )}}{{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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