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 = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, 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 206
Rule 618
Rule 3124
Rule 3137
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 = 0.21, 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]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 455, normalized size = 4.38 \[ \left [-\frac {\sqrt {a^{2} - b^{2} + c^{2}} a c \log \left (\frac {{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{2} + 2 \, a^{2} - b^{2} + c^{2} + 2 \, {\left (a b + a c\right )} \cosh \relax (x) + 2 \, {\left (a b + a c + {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} - b^{2} + c^{2}} {\left ({\left (b + c\right )} \cosh \relax (x) + {\left (b + c\right )} \sinh \relax (x) + a\right )}}{{\left (b + c\right )} \cosh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left ({\left (b + c\right )} \cosh \relax (x) + a\right )} \sinh \relax (x) + b - c}\right ) + {\left (a^{2} b - b^{3} + b c^{2} + c^{3} + {\left (a^{2} - b^{2}\right )} c\right )} x - {\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + c \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b^{2} - b^{4} - c^{4} - {\left (a^{2} - 2 \, b^{2}\right )} c^{2}}, \frac {2 \, \sqrt {-a^{2} + b^{2} - c^{2}} a c \arctan \left (\frac {\sqrt {-a^{2} + b^{2} - c^{2}} {\left ({\left (b + c\right )} \cosh \relax (x) + {\left (b + c\right )} \sinh \relax (x) + a\right )}}{a^{2} - b^{2} + c^{2}}\right ) - {\left (a^{2} b - b^{3} + b c^{2} + c^{3} + {\left (a^{2} - b^{2}\right )} c\right )} x + {\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + c \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b^{2} - b^{4} - c^{4} - {\left (a^{2} - 2 \, b^{2}\right )} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 106, normalized size = 1.02 \[ \frac {2 \, a c \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{\sqrt {-a^{2} + b^{2} - c^{2}} {\left (b^{2} - c^{2}\right )}} + \frac {b \log \left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}{b^{2} - c^{2}} - \frac {x}{b - c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.20, size = 429, normalized size = 4.12 \[ -\frac {4 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4 b +4 c}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -2 c \tanh \left (\frac {x}{2}\right )-a -b \right ) a b}{\left (b -c \right ) \left (b +c \right ) \left (a -b \right )}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -2 c \tanh \left (\frac {x}{2}\right )-a -b \right ) b^{2}}{\left (b -c \right ) \left (b +c \right ) \left (a -b \right )}-\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) a c}{\left (b -c \right ) \left (b +c \right ) \sqrt {-a^{2}+b^{2}-c^{2}}}-\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) c b}{\left (b -c \right ) \left (b +c \right ) \sqrt {-a^{2}+b^{2}-c^{2}}}+\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) c a b}{\left (b -c \right ) \left (b +c \right ) \sqrt {-a^{2}+b^{2}-c^{2}}\, \left (a -b \right )}-\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) c \,b^{2}}{\left (b -c \right ) \left (b +c \right ) \sqrt {-a^{2}+b^{2}-c^{2}}\, \left (a -b \right )}-\frac {4 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4 b -4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.16, size = 325, normalized size = 3.12 \[ -\frac {x}{b-c}-\frac {\ln \left (\frac {2\,\left (b+a\,{\mathrm {e}}^x\right )}{{\left (b+c\right )}^2}-\frac {2\,\left (b-c+a\,{\mathrm {e}}^x\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (b+c\right )\,\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{-a^2\,b^2+a^2\,c^2+b^4-2\,b^2\,c^2+c^4}-\frac {\ln \left (\frac {2\,\left (b+a\,{\mathrm {e}}^x\right )}{{\left (b+c\right )}^2}-\frac {2\,\left (b-c+a\,{\mathrm {e}}^x\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (b+c\right )\,\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{-a^2\,b^2+a^2\,c^2+b^4-2\,b^2\,c^2+c^4} \]
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]
________________________________________________________________________________________