Optimal. Leaf size=89 \[ -\frac {3 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^4 (a \cosh (x)+a+c \sinh (x))}+\frac {\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^5}-\frac {a \sinh (x)+c \cosh (x)}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3129, 3153, 3124, 31} \[ \frac {\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^5}-\frac {3 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^4 (a \cosh (x)+a+c \sinh (x))}-\frac {a \sinh (x)+c \cosh (x)}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3124
Rule 3129
Rule 3153
Rubi steps
\begin {align*} \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^3} \, dx &=-\frac {c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac {\int \frac {-2 a+a \cosh (x)+c \sinh (x)}{(a+a \cosh (x)+c \sinh (x))^2} \, dx}{2 c^2}\\ &=-\frac {c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac {3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}+\frac {\left (3 a^2-c^2\right ) \int \frac {1}{a+a \cosh (x)+c \sinh (x)} \, dx}{2 c^4}\\ &=-\frac {c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac {3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}+\frac {\left (3 a^2-c^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a+2 c x} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{c^4}\\ &=\frac {\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^5}-\frac {c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac {3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.55, size = 148, normalized size = 1.66 \[ \frac {4 \left (c^2-3 a^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {6 c \left (c^2-a^2\right ) \sinh \left (\frac {x}{2}\right )}{a \cosh \left (\frac {x}{2}\right )+c \sinh \left (\frac {x}{2}\right )}+4 \left (3 a^2-c^2\right ) \log \left (a \cosh \left (\frac {x}{2}\right )+c \sinh \left (\frac {x}{2}\right )\right )+\frac {c^2 (a-c) (a+c)}{\left (a \cosh \left (\frac {x}{2}\right )+c \sinh \left (\frac {x}{2}\right )\right )^2}-6 a c \tanh \left (\frac {x}{2}\right )+c^2 \left (-\text {sech}^2\left (\frac {x}{2}\right )\right )}{8 c^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 1504, normalized size = 16.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.13, size = 205, normalized size = 2.30 \[ \frac {{\left (3 \, a^{3} + 3 \, a^{2} c - a c^{2} - c^{3}\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{2 \, {\left (a c^{5} + c^{6}\right )}} - \frac {{\left (3 \, a^{2} - c^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, c^{5}} + \frac {3 \, a^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} c e^{\left (3 \, x\right )} - a c^{2} e^{\left (3 \, x\right )} - c^{3} e^{\left (3 \, x\right )} + 9 \, a^{3} e^{\left (2 \, x\right )} - 3 \, a c^{2} e^{\left (2 \, x\right )} + 9 \, a^{3} e^{x} - 9 \, a^{2} c e^{x} + a c^{2} e^{x} - c^{3} e^{x} + 3 \, a^{3} - 6 \, a^{2} c + 3 \, a c^{2}}{{\left (a e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + a - c\right )}^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.27, size = 138, normalized size = 1.55 \[ \frac {\tanh ^{2}\left (\frac {x}{2}\right )}{8 c^{3}}-\frac {3 a \tanh \left (\frac {x}{2}\right )}{4 c^{4}}-\frac {a^{4}}{8 c^{5} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {a^{2}}{4 c^{3} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {1}{8 c \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {a^{3}}{c^{5} \left (a +c \tanh \left (\frac {x}{2}\right )\right )}-\frac {a}{c^{3} \left (a +c \tanh \left (\frac {x}{2}\right )\right )}+\frac {3 \ln \left (a +c \tanh \left (\frac {x}{2}\right )\right ) a^{2}}{2 c^{5}}-\frac {\ln \left (a +c \tanh \left (\frac {x}{2}\right )\right )}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 248, normalized size = 2.79 \[ -\frac {3 \, a^{3} + 6 \, a^{2} c + 3 \, a c^{2} + {\left (9 \, a^{3} + 9 \, a^{2} c + a c^{2} + c^{3}\right )} e^{\left (-x\right )} + 3 \, {\left (3 \, a^{3} - a c^{2}\right )} e^{\left (-2 \, x\right )} + {\left (3 \, a^{3} - 3 \, a^{2} c - a c^{2} + c^{3}\right )} e^{\left (-3 \, x\right )}}{a^{2} c^{4} + 2 \, a c^{5} + c^{6} + 4 \, {\left (a^{2} c^{4} + a c^{5}\right )} e^{\left (-x\right )} + 2 \, {\left (3 \, a^{2} c^{4} - c^{6}\right )} e^{\left (-2 \, x\right )} + 4 \, {\left (a^{2} c^{4} - a c^{5}\right )} e^{\left (-3 \, x\right )} + {\left (a^{2} c^{4} - 2 \, a c^{5} + c^{6}\right )} e^{\left (-4 \, x\right )}} + \frac {{\left (3 \, a^{2} - c^{2}\right )} \log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{2 \, c^{5}} - \frac {{\left (3 \, a^{2} - c^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+a\,\mathrm {cosh}\relax (x)+c\,\mathrm {sinh}\relax (x)\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________