Optimal. Leaf size=47 \[ \frac{\cosh (c+d x)}{a d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{a^{3/2} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0484901, antiderivative size = 47, 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 = {4133, 321, 205} \[ \frac{\cosh (c+d x)}{a d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{a^{3/2} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4133
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{\sinh (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\cosh (c+d x)}{a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{a^{3/2} d}+\frac{\cosh (c+d x)}{a d}\\ \end{align*}
Mathematica [C] time = 1.04528, size = 328, normalized size = 6.98 \[ \frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{a \left (\tan ^{-1}\left (\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sqrt{a}+i \sqrt{a+b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b}}\right )\right )}{\sqrt{b}}-\frac{(a+4 b) \left (\tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )\right )}{\sqrt{b}}+4 \sqrt{a} \cosh (c+d x)\right )}{8 a^{3/2} d \left (a+b \text{sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{da{\rm sech} \left (dx+c\right )}}+{\frac{b}{da}\arctan \left ({b{\rm sech} \left (dx+c\right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.78275, size = 1643, normalized size = 34.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]