Optimal. Leaf size=90 \[ -\frac{2 a \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac{b \sinh (x)+c \cosh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))} \]
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Rubi [A] time = 0.0875916, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3129, 12, 3124, 618, 206} \[ -\frac{2 a \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac{b \sinh (x)+c \cosh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))} \]
Antiderivative was successfully verified.
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Rule 3129
Rule 12
Rule 3124
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cosh (x)+c \sinh (x))^2} \, dx &=-\frac{c \cosh (x)+b \sinh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac{\int \frac{a}{a+b \cosh (x)+c \sinh (x)} \, dx}{a^2-b^2+c^2}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac{a \int \frac{1}{a+b \cosh (x)+c \sinh (x)} \, dx}{a^2-b^2+c^2}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^2-b^2+c^2}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac{(4 a) \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 )}{a^2-b^2+c^2}\\ &=-\frac{2 a \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac{c \cosh (x)+b \sinh (x)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.267378, size = 105, normalized size = 1.17 \[ \frac{\left (b^2-c^2\right ) \sinh (x)-a c}{b \left (-a^2+b^2-c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac{2 a \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )+c}{\sqrt{-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 191, normalized size = 2.1 \begin{align*} -2\,{\frac{1}{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-2\,c\tanh \left ( x/2 \right ) -a-b} \left ( -{\frac{ \left ( ab-{b}^{2}+{c}^{2} \right ) \tanh \left ( x/2 \right ) }{{a}^{3}-{a}^{2}b-a{b}^{2}+a{c}^{2}+{b}^{3}-b{c}^{2}}}-{\frac{ac}{{a}^{3}-{a}^{2}b-a{b}^{2}+a{c}^{2}+{b}^{3}-b{c}^{2}}} \right ) }-2\,{\frac{a}{ \left ({a}^{2}-{b}^{2}+{c}^{2} \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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 2.19876, size = 2912, normalized size = 32.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.13329, size = 150, normalized size = 1.67 \begin{align*} \frac{2 \, a \arctan \left (\frac{b e^{x} + c e^{x} + a}{\sqrt{-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{2} - b^{2} + c^{2}\right )} \sqrt{-a^{2} + b^{2} - c^{2}}} + \frac{2 \,{\left (a e^{x} + b - c\right )}}{{\left (a^{2} - b^{2} + c^{2}\right )}{\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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