Optimal. Leaf size=121 \[ \frac{2 \left (a b B-A \left (b^2-c^2\right )\right ) \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt{a^2-b^2+c^2}}-\frac{B c \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac{b B x}{b^2-c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.1179, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3138, 3124, 618, 206} \[ \frac{2 \left (a b B-A \left (b^2-c^2\right )\right ) \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt{a^2-b^2+c^2}}-\frac{B c \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac{b B x}{b^2-c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3138
Rule 3124
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{a+b \cosh (x)+c \sinh (x)} \, dx &=\frac{b B x}{b^2-c^2}-\frac{B c \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\left (A-\frac{a b B}{b^2-c^2}\right ) \int \frac{1}{a+b \cosh (x)+c \sinh (x)} \, dx\\ &=\frac{b B x}{b^2-c^2}-\frac{B c \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\left (2 \left (A-\frac{a b B}{b^2-c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{b B x}{b^2-c^2}-\frac{B c \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\left (4 \left (A-\frac{a b B}{b^2-c^2}\right )\right ) \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 )\\ &=\frac{b B x}{b^2-c^2}-\frac{2 \left (A-\frac{a b B}{b^2-c^2}\right ) \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\sqrt{a^2-b^2+c^2}}-\frac{B c \log (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}\\ \end{align*}
Mathematica [A] time = 0.179444, size = 104, normalized size = 0.86 \[ \frac{B (b x-c \log (a+b \cosh (x)+c \sinh (x)))-\frac{2 \left (a b B+A \left (c^2-b^2\right )\right ) \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )+c}{\sqrt{-a^2+b^2-c^2}}\right )}{\sqrt{-a^2+b^2-c^2}}}{(b-c) (b+c)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.051, size = 574, normalized size = 4.7 \begin{align*} \text{result too large to display} \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 = 2.38994, size = 1214, normalized size = 10.03 \begin{align*} \left [-\frac{{\left (B a b - A b^{2} + A c^{2}\right )} \sqrt{a^{2} - b^{2} + c^{2}} \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 (B a^{2} b - B b^{3} + B b c^{2} + B c^{3} +{\left (B a^{2} - B b^{2}\right )} c\right )} x +{\left (B c^{3} +{\left (B a^{2} - B b^{2}\right )} c\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 \,{\left (B a b - A b^{2} + A c^{2}\right )} \sqrt{-a^{2} + b^{2} - c^{2}} \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 (B a^{2} b - B b^{3} + B b c^{2} + B c^{3} +{\left (B a^{2} - B b^{2}\right )} c\right )} x +{\left (B c^{3} +{\left (B a^{2} - B b^{2}\right )} c\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 = 1.14437, size = 165, normalized size = 1.36 \begin{align*} -\frac{B c \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{B x}{b - c} - \frac{2 \,{\left (B a b - A b^{2} + A c^{2}\right )} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]