Optimal. Leaf size=52 \[ \frac{\sinh (c+d x)}{a d}-\frac{b \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{a^{3/2} d \sqrt{a+b}} \]
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Rubi [A] time = 0.061165, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4147, 388, 205} \[ \frac{\sinh (c+d x)}{a d}-\frac{b \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{a^{3/2} d \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 4147
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\sinh (c+d x)}{a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{a d}\\ &=-\frac{b \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{a^{3/2} \sqrt{a+b} d}+\frac{\sinh (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.12126, size = 52, normalized size = 1. \[ \frac{\sqrt{a} \sinh (c+d x)-\frac{b \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a+b}}}{a^{3/2} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 128, normalized size = 2.5 \begin{align*} -{\frac{b}{d}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+b}}}}-{\frac{b}{d}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}-2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+b}}}}-{\frac{1}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}} \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*} \frac{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}}{2 \, a d} - \frac{1}{2} \, \int \frac{4 \,{\left (b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}\right )}}{a^{2} e^{\left (4 \, d x + 4 \, c\right )} + a^{2} + 2 \,{\left (a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37508, size = 1891, normalized size = 36.37 \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{\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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