Optimal. Leaf size=74 \[ -\frac{2 (a A+b B) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{\cosh (x) (A b-a B)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0828692, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2754, 12, 2660, 618, 206} \[ -\frac{2 (a A+b B) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{\cosh (x) (A b-a B)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2754
Rule 12
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \sinh (x)}{(a+b \sinh (x))^2} \, dx &=-\frac{(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\int \frac{-a A-b B}{a+b \sinh (x)} \, dx}{a^2+b^2}\\ &=-\frac{(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{(a A+b B) \int \frac{1}{a+b \sinh (x)} \, dx}{a^2+b^2}\\ &=-\frac{(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{(2 (a A+b B)) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^2+b^2}\\ &=-\frac{(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{(4 (a A+b B)) \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 )}{a^2+b^2}\\ &=-\frac{2 (a A+b B) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{(A b-a B) \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.152572, size = 82, normalized size = 1.11 \[ \frac{\frac{2 (a A+b B) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+\frac{\cosh (x) (a B-A b)}{a+b \sinh (x)}}{a^2+b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 113, normalized size = 1.5 \begin{align*} -2\,{\frac{1}{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a} \left ( -{\frac{b \left ( Ab-aB \right ) \tanh \left ( x/2 \right ) }{a \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{Ab-aB}{{a}^{2}+{b}^{2}}} \right ) }+2\,{\frac{Aa+bB}{ \left ({a}^{2}+{b}^{2} \right ) ^{3/2}}{\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 = 1.88599, size = 1065, normalized size = 14.39 \begin{align*} -\frac{2 \, B a^{3} b - 2 \, A a^{2} b^{2} + 2 \, B a b^{3} - 2 \, A b^{4} -{\left (A a b^{2} + B b^{3} -{\left (A a b^{2} + B b^{3}\right )} \cosh \left (x\right )^{2} -{\left (A a b^{2} + B b^{3}\right )} \sinh \left (x\right )^{2} - 2 \,{\left (A a^{2} b + B a b^{2}\right )} \cosh \left (x\right ) - 2 \,{\left (A a^{2} b + B a b^{2} +{\left (A a b^{2} + B b^{3}\right )} \cosh \left (x\right )\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 ) - 2 \,{\left (B a^{4} - A a^{3} b + B a^{2} b^{2} - A a b^{3}\right )} \cosh \left (x\right ) - 2 \,{\left (B a^{4} - A a^{3} b + B a^{2} b^{2} - A a b^{3}\right )} \sinh \left (x\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} -{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{2} -{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sinh \left (x\right )^{2} - 2 \,{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cosh \left (x\right ) - 2 \,{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} +{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.36898, size = 161, normalized size = 2.18 \begin{align*} \frac{{\left (A a + B b\right )} \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 )}{{\left (a^{2} + b^{2}\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (B a^{2} e^{x} - A a b e^{x} - B a b + A b^{2}\right )}}{{\left (a^{2} b + b^{3}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]