Optimal. Leaf size=38 \[ \frac {\sinh (x)}{b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}} \]
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Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3186, 388, 205} \[ \frac {\sinh (x)}{b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3186
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{a+b+b x^2} \, dx,x,\sinh (x)\right )\\ &=\frac {\sinh (x)}{b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{b}\\ &=-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}}+\frac {\sinh (x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.00 \[ \frac {\sinh (x)}{b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 498, normalized size = 13.11 \[ \left [\frac {{\left (a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a b + b^{2}\right )} \sinh \relax (x)^{2} - \sqrt {-a b - b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )} \log \left (\frac {b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} - 2 \, {\left (2 \, a + 3 \, b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} - 2 \, a - 3 \, b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} - {\left (2 \, a + 3 \, b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)\right )} \sqrt {-a b - b^{2}} + b}{b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} + 2 \, a + b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} + {\left (2 \, a + b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + b}\right ) - a b - b^{2}}{2 \, {\left ({\left (a b^{2} + b^{3}\right )} \cosh \relax (x) + {\left (a b^{2} + b^{3}\right )} \sinh \relax (x)\right )}}, \frac {{\left (a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a b + b^{2}\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {a b + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )} \arctan \left (\frac {b \cosh \relax (x)^{3} + 3 \, b \cosh \relax (x) \sinh \relax (x)^{2} + b \sinh \relax (x)^{3} + {\left (4 \, a + 3 \, b\right )} \cosh \relax (x) + {\left (3 \, b \cosh \relax (x)^{2} + 4 \, a + 3 \, b\right )} \sinh \relax (x)}{2 \, \sqrt {a b + b^{2}}}\right ) - 2 \, \sqrt {a b + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )} \arctan \left (\frac {\sqrt {a b + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{2 \, {\left (a + b\right )}}\right ) - a b - b^{2}}{2 \, {\left ({\left (a b^{2} + b^{3}\right )} \cosh \relax (x) + {\left (a b^{2} + b^{3}\right )} \sinh \relax (x)\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 96, normalized size = 2.53 \[ -\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {3}{2}} \sqrt {a +b}}-\frac {a \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {3}{2}} \sqrt {a +b}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}}{2 \, b} - \frac {1}{8} \, \int \frac {16 \, {\left (a e^{\left (3 \, x\right )} + a e^{x}\right )}}{b^{2} e^{\left (4 \, x\right )} + b^{2} + 2 \, {\left (2 \, a b + b^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 204, normalized size = 5.37 \[ \frac {{\mathrm {e}}^x}{2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,b}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^x\,\sqrt {b^3\,\left (a+b\right )}}{2\,b\,\left (a+b\right )\,{\left (a^2\right )}^{3/2}}\right )-2\,\mathrm {atan}\left (\left (\frac {b^5\,\sqrt {b^4+a\,b^3}}{4}+\frac {a\,b^4\,\sqrt {b^4+a\,b^3}}{4}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^3}{b^5\,{\left (a+b\right )}^2\,{\left (a^2\right )}^{3/2}}-\frac {4\,\left (2\,b^2\,{\left (a^2\right )}^{3/2}+2\,a\,b\,{\left (a^2\right )}^{3/2}\right )}{a^3\,b^4\,\left (a+b\right )\,\sqrt {b^3\,\left (a+b\right )}\,\sqrt {b^4+a\,b^3}}\right )-\frac {2\,a^3\,{\mathrm {e}}^{3\,x}}{b^5\,{\left (a+b\right )}^2\,{\left (a^2\right )}^{3/2}}\right )\right )\right )\,\sqrt {a^2}}{2\,\sqrt {b^4+a\,b^3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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