Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3181, 208} \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3181
Rubi steps
\begin {align*} \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a^2-\left (a^2+b^2\right ) x^2} \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 31, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.40, size = 288, normalized size = 9.29 \[ \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b^{4} \cosh \relax (x)^{4} + 4 \, b^{4} \cosh \relax (x) \sinh \relax (x)^{3} + b^{4} \sinh \relax (x)^{4} + 8 \, a^{4} + 8 \, a^{2} b^{2} + b^{4} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{4} \cosh \relax (x)^{2} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)^{2} + 4 \, {\left (b^{4} \cosh \relax (x)^{3} + {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 4 \, {\left (a b^{2} \cosh \relax (x)^{2} + 2 \, a b^{2} \cosh \relax (x) \sinh \relax (x) + a b^{2} \sinh \relax (x)^{2} + 2 \, a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}}}{b^{2} \cosh \relax (x)^{4} + 4 \, b^{2} \cosh \relax (x) \sinh \relax (x)^{3} + b^{2} \sinh \relax (x)^{4} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} + 2 \, a^{2} + b^{2}\right )} \sinh \relax (x)^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} + {\left (2 \, a^{2} + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}\right )}{2 \, {\left (a^{3} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 79, normalized size = 2.55 \[ \frac {\log \left (\frac {b^{2} e^{\left (2 \, x\right )} + 2 \, a^{2} + b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}}{b^{2} e^{\left (2 \, x\right )} + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 98, normalized size = 3.16
method | result | size |
default | \(\frac {\ln \left (\sqrt {a^{2}+b^{2}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+\sqrt {a^{2}+b^{2}}\right )}{2 a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left (\sqrt {a^{2}+b^{2}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a \tanh \left (\frac {x}{2}\right )+\sqrt {a^{2}+b^{2}}\right )}{2 a \sqrt {a^{2}+b^{2}}}\) | \(98\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}+b^{2} \sqrt {a^{2}+b^{2}}-2 a^{3}-2 b^{2} a}{b^{2} \sqrt {a^{2}+b^{2}}}\right )}{2 a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}+b^{2} \sqrt {a^{2}+b^{2}}+2 a^{3}+2 b^{2} a}{b^{2} \sqrt {a^{2}+b^{2}}}\right )}{2 a \sqrt {a^{2}+b^{2}}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 76, normalized size = 2.45 \[ -\frac {\log \left (\frac {b^{2} e^{\left (-2 \, x\right )} + 2 \, a^{2} + b^{2} - 2 \, \sqrt {a^{2} + b^{2}} a}{b^{2} e^{\left (-2 \, x\right )} + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} a}\right )}{2 \, \sqrt {a^{2} + b^{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 109, normalized size = 3.52 \[ \frac {\mathrm {atan}\left (\frac {2\,a^2\,{\left (-a^4-a^2\,b^2\right )}^{3/2}+b^2\,{\left (-a^4-a^2\,b^2\right )}^{3/2}+b^2\,{\mathrm {e}}^{2\,x}\,{\left (-a^4-a^2\,b^2\right )}^{3/2}}{2\,a^8+4\,a^6\,b^2+2\,a^4\,b^4}\right )}{\sqrt {-a^4-a^2\,b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 46.11, size = 605, normalized size = 19.52 \[ \begin {cases} \frac {\tilde {\infty } \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\tanh {\left (\frac {x}{2} \right )}}{2 b^{2}} - \frac {1}{2 b^{2} \tanh {\left (\frac {x}{2} \right )}} & \text {for}\: a = - i b \vee a = i b \\\frac {2 \tanh {\left (\frac {x}{2} \right )}}{b^{2} \left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 1\right )} & \text {for}\: a = 0 \\\frac {a \log {\left (- \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} - \frac {a \log {\left (\sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} - \frac {i b \log {\left (- \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} + \frac {i b \log {\left (\sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} + \frac {\sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} \log {\left (- \sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 i a b \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} - \frac {\sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} \log {\left (\sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 i a b \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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