Optimal. Leaf size=71 \[ \frac{B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac{\sinh (x) (b B-c C)}{b \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0687751, antiderivative size = 71, 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 = {3156, 12, 3075} \[ \frac{B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac{\sinh (x) (b B-c C)}{b \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3156
Rule 12
Rule 3075
Rubi steps
\begin{align*} \int \frac{B \cosh (x)+C \sinh (x)}{(b \cosh (x)+c \sinh (x))^3} \, dx &=\frac{B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac{\int \frac{2 (b B-c C)}{(b \cosh (x)+c \sinh (x))^2} \, dx}{2 \left (b^2-c^2\right )}\\ &=\frac{B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac{(b B-c C) \int \frac{1}{(b \cosh (x)+c \sinh (x))^2} \, dx}{b^2-c^2}\\ &=\frac{B c-b C}{2 \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))^2}+\frac{(b B-c C) \sinh (x)}{b \left (b^2-c^2\right ) (b \cosh (x)+c \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.158388, size = 70, normalized size = 0.99 \[ \frac{C \left (c^2-b^2\right )+b \sinh (2 x) (b B-c C)+c \cosh (2 x) (b B-c C)}{2 b (b-c) (b+c) (b \cosh (x)+c \sinh (x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.083, size = 63, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+2\,c\tanh \left ( x/2 \right ) +b \right ) ^{2}} \left ( -{\frac{B \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{b}}-{\frac{ \left ( Bc+bC \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{{b}^{2}}}-{\frac{B\tanh \left ( x/2 \right ) }{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.11157, size = 455, normalized size = 6.41 \begin{align*} 2 \, B{\left (\frac{{\left (b - c\right )} e^{\left (-2 \, x\right )}}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \,{\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} +{\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}} + \frac{b}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \,{\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} +{\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}}\right )} - 2 \, C{\left (\frac{{\left (b - c\right )} e^{\left (-2 \, x\right )}}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \,{\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} +{\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}} + \frac{c}{b^{4} - 2 \, b^{2} c^{2} + c^{4} + 2 \,{\left (b^{4} - 2 \, b^{3} c + 2 \, b c^{3} - c^{4}\right )} e^{\left (-2 \, x\right )} +{\left (b^{4} - 4 \, b^{3} c + 6 \, b^{2} c^{2} - 4 \, b c^{3} + c^{4}\right )} e^{\left (-4 \, x\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.98492, size = 556, normalized size = 7.83 \begin{align*} -\frac{2 \,{\left ({\left ({\left (2 \, B + C\right )} b + B c\right )} \cosh \left (x\right ) +{\left (C b +{\left (B + 2 \, C\right )} c\right )} \sinh \left (x\right )\right )}}{{\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{3} + 3 \,{\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} +{\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \sinh \left (x\right )^{3} +{\left (3 \, b^{4} + 4 \, b^{3} c - 2 \, b^{2} c^{2} - 4 \, b c^{3} - c^{4}\right )} \cosh \left (x\right ) +{\left (b^{4} + 4 \, b^{3} c + 2 \, b^{2} c^{2} - 4 \, b c^{3} - 3 \, c^{4} + 3 \,{\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{2}\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 = 1.20253, size = 95, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (B b e^{\left (2 \, x\right )} + C b e^{\left (2 \, x\right )} + B c e^{\left (2 \, x\right )} + C c e^{\left (2 \, x\right )} + B b - C c\right )}}{{\left (b^{2} + 2 \, b c + c^{2}\right )}{\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]