Optimal. Leaf size=60 \[ -\frac{2 A \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}-\frac{B \log (a+b \sinh (x))}{a}+\frac{B \log (\sinh (x))}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.150008, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {4401, 2660, 618, 206, 2721, 36, 29, 31} \[ -\frac{2 A \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}-\frac{B \log (a+b \sinh (x))}{a}+\frac{B \log (\sinh (x))}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4401
Rule 2660
Rule 618
Rule 206
Rule 2721
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \coth (x)}{a+b \sinh (x)} \, dx &=\int \left (\frac{A}{a+b \sinh (x)}+\frac{B \coth (x)}{a+b \sinh (x)}\right ) \, dx\\ &=A \int \frac{1}{a+b \sinh (x)} \, dx+B \int \frac{\coth (x)}{a+b \sinh (x)} \, dx\\ &=(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 \operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,b \sinh (x)\right )\\ &=-\left ((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 )\right )+\frac{B \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,b \sinh (x)\right )}{a}-\frac{B \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sinh (x)\right )}{a}\\ &=-\frac{2 A \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}+\frac{B \log (\sinh (x))}{a}-\frac{B \log (a+b \sinh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.15923, size = 65, normalized size = 1.08 \[ \frac{2 A \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+\frac{B (\log (\sinh (x))-\log (a+b \sinh (x)))}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 73, normalized size = 1.2 \begin{align*}{\frac{B}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{B}{a}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }+2\,{\frac{A}{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.23613, size = 502, normalized size = 8.37 \begin{align*} \frac{\sqrt{a^{2} + b^{2}} A a \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 (B a^{2} + B b^{2}\right )} \log \left (\frac{2 \,{\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\left (B a^{2} + B b^{2}\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{3} + a b^{2}} \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{A + B \coth{\left (x \right )}}{a + b \sinh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17266, size = 138, normalized size = 2.3 \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}}} + \frac{B \log \left (e^{x} + 1\right )}{a} - \frac{B \log \left ({\left | b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b \right |}\right )}{a} + \frac{B \log \left ({\left | e^{x} - 1 \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]