Optimal. Leaf size=75 \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{a^2 d \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d} \]
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Rubi [A] time = 0.115501, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4146, 414, 522, 206, 208} \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{a^2 d \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 414
Rule 522
Rule 206
Rule 208
Rubi steps
\begin{align*} \int \frac{\cosh ^2(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\operatorname{Subst}\left (\int \frac{a-b-b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{(a-2 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a^2 d}\\ &=\frac{(a-2 b) x}{2 a^2}+\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{a^2 \sqrt{a+b} d}+\frac{\cosh (c+d x) \sinh (c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.227966, size = 67, normalized size = 0.89 \[ \frac{\frac{4 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a+b}}+2 (a-2 b) (c+d x)+a \sinh (2 (c+d x))}{4 a^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 278, normalized size = 3.7 \begin{align*} -{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{b}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{2\,d{a}^{2}}{b}^{{\frac{3}{2}}}\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 ){\frac{1}{\sqrt{a+b}}}}-{\frac{1}{2\,d{a}^{2}}{b}^{{\frac{3}{2}}}\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 ){\frac{1}{\sqrt{a+b}}}}+{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{b}{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 [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 = 2.4233, size = 2188, normalized size = 29.17 \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{\cosh ^{2}{\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 [B] time = 1.18639, size = 178, normalized size = 2.37 \begin{align*} \frac{b^{2} \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}} a^{2} d} + \frac{{\left (d x + c\right )}{\left (a - 2 \, b\right )}}{2 \, a^{2} d} + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} - \frac{{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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