Optimal. Leaf size=56 \[ \frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sinh (x)}{b^2}+\frac {\sinh ^3(x)}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3186, 390, 205} \[ \frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sinh (x)}{b^2}+\frac {\sinh ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rule 3186
Rubi steps
\begin {align*} \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{a+b+b x^2} \, dx,x,\sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {a-b}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b+b x^2\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=-\frac {(a-b) \sinh (x)}{b^2}+\frac {\sinh ^3(x)}{3 b}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{b^2}\\ &=\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sinh (x)}{b^2}+\frac {\sinh ^3(x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 61, normalized size = 1.09 \[ -\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \text {csch}(x)}{\sqrt {b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(4 a-3 b) \sinh (x)}{4 b^2}+\frac {\sinh (3 x)}{12 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 1184, normalized size = 21.14 \[ \text {result too large to display} \]
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 = 0.10, size = 176, normalized size = 3.14 \[ -\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {a}{b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {a}{b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a^{2} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {5}{2}} \sqrt {a +b}}+\frac {a^{2} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {5}{2}} \sqrt {a +b}} \]
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 \[ \frac {{\left (b e^{\left (6 \, x\right )} - 3 \, {\left (4 \, a - 3 \, b\right )} e^{\left (4 \, x\right )} + 3 \, {\left (4 \, a - 3 \, b\right )} e^{\left (2 \, x\right )} - b\right )} e^{\left (-3 \, x\right )}}{24 \, b^{2}} + \frac {1}{32} \, \int \frac {64 \, {\left (a^{2} e^{\left (3 \, x\right )} + a^{2} e^{x}\right )}}{b^{3} e^{\left (4 \, x\right )} + b^{3} + 2 \, {\left (2 \, a b^{2} + b^{3}\right )} e^{\left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 243, normalized size = 4.34 \[ \frac {{\mathrm {e}}^{3\,x}}{24\,b}-\frac {{\mathrm {e}}^{-3\,x}}{24\,b}+\frac {{\mathrm {e}}^{-x}\,\left (4\,a-3\,b\right )}{8\,b^2}+\frac {\sqrt {a^4}\,\left (2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^x\,\sqrt {b^5\,\left (a+b\right )}}{2\,b^2\,\left (a+b\right )\,\sqrt {a^4}}\right )-2\,\mathrm {atan}\left (\left (\frac {b^7\,\sqrt {b^6+a\,b^5}}{4}+\frac {a\,b^6\,\sqrt {b^6+a\,b^5}}{4}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^2}{b^8\,{\left (a+b\right )}^2\,\sqrt {a^4}}-\frac {4\,\left (2\,a^3\,b^3\,\sqrt {a^4}+2\,a^4\,b^2\,\sqrt {a^4}\right )}{a^5\,b^6\,\left (a+b\right )\,\sqrt {b^5\,\left (a+b\right )}\,\sqrt {b^6+a\,b^5}}\right )-\frac {2\,a^2\,{\mathrm {e}}^{3\,x}}{b^8\,{\left (a+b\right )}^2\,\sqrt {a^4}}\right )\right )\right )}{2\,\sqrt {b^6+a\,b^5}}-\frac {{\mathrm {e}}^x\,\left (4\,a-3\,b\right )}{8\,b^2} \]
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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