Optimal. Leaf size=104 \[ -\frac{2 a c \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt{a^2-b^2+c^2}}+\frac{b \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\frac{c x}{b^2-c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.114731, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3137, 3124, 618, 206} \[ -\frac{2 a c \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt{a^2-b^2+c^2}}+\frac{b \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\frac{c x}{b^2-c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3137
Rule 3124
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{a+b \cosh (x)+c \sinh (x)} \, dx &=-\frac{c x}{b^2-c^2}+\frac{b \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac{(a c) \int \frac{1}{a+b \cosh (x)+c \sinh (x)} \, dx}{b^2-c^2}\\ &=-\frac{c x}{b^2-c^2}+\frac{b \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac{(2 a c) \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^2-c^2}\\ &=-\frac{c x}{b^2-c^2}+\frac{b \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\frac{(4 a c) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2-b^2+c^2\right )-x^2} \, dx,x,2 c+2 (-a+b) \tanh \left (\frac{x}{2}\right )\right )}{b^2-c^2}\\ &=-\frac{c x}{b^2-c^2}-\frac{2 a c \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt{a^2-b^2+c^2}}+\frac{b \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}\\ \end{align*}
Mathematica [A] time = 0.186858, size = 86, normalized size = 0.83 \[ \frac{\frac{2 a c \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )+c}{\sqrt{-a^2+b^2-c^2}}\right )}{\sqrt{-a^2+b^2-c^2}}+b \log (a+b \cosh (x)+c \sinh (x))-c x}{b^2-c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.036, size = 429, normalized size = 4.1 \begin{align*} -4\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) }{4\,b-4\,c}}-4\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) -1 \right ) }{4\,b+4\,c}}+{\frac{ab}{ \left ( b-c \right ) \left ( b+c \right ) \left ( a-b \right ) }\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,c\tanh \left ( x/2 \right ) -a-b \right ) }-{\frac{{b}^{2}}{ \left ( b-c \right ) \left ( b+c \right ) \left ( a-b \right ) }\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,c\tanh \left ( x/2 \right ) -a-b \right ) }-2\,{\frac{ac}{ \left ( b-c \right ) \left ( b+c \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{cb}{ \left ( b-c \right ) \left ( b+c \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) }+2\,{\frac{acb}{ \left ( b-c \right ) \left ( b+c \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}} \left ( a-b \right ) }\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{c{b}^{2}}{ \left ( b-c \right ) \left ( b+c \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}} \left ( a-b \right ) }\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{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 [A] time = 2.64441, size = 1110, normalized size = 10.67 \begin{align*} \left [-\frac{\sqrt{a^{2} - b^{2} + c^{2}} a c \log \left (\frac{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} +{\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} + 2 \, a^{2} - b^{2} + c^{2} + 2 \,{\left (a b + a c\right )} \cosh \left (x\right ) + 2 \,{\left (a b + a c +{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} - b^{2} + c^{2}}{\left ({\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right ) + a\right )}}{{\left (b + c\right )} \cosh \left (x\right )^{2} +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left ({\left (b + c\right )} \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b - c}\right ) +{\left (a^{2} b - b^{3} + b c^{2} + c^{3} +{\left (a^{2} - b^{2}\right )} c\right )} x -{\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b^{2} - b^{4} - c^{4} -{\left (a^{2} - 2 \, b^{2}\right )} c^{2}}, \frac{2 \, \sqrt{-a^{2} + b^{2} - c^{2}} a c \arctan \left (\frac{\sqrt{-a^{2} + b^{2} - c^{2}}{\left ({\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2} + c^{2}}\right ) -{\left (a^{2} b - b^{3} + b c^{2} + c^{3} +{\left (a^{2} - b^{2}\right )} c\right )} x +{\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b^{2} - b^{4} - c^{4} -{\left (a^{2} - 2 \, b^{2}\right )} c^{2}}\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 = 1.14693, size = 143, normalized size = 1.38 \begin{align*} \frac{2 \, a c \arctan \left (\frac{b e^{x} + c e^{x} + a}{\sqrt{-a^{2} + b^{2} - c^{2}}}\right )}{\sqrt{-a^{2} + b^{2} - c^{2}}{\left (b^{2} - c^{2}\right )}} + \frac{b \log \left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}{b^{2} - c^{2}} - \frac{x}{b - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]