Optimal. Leaf size=65 \[ \frac {(A+3 B) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a \cosh (x)+a}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(A-B) \sinh (x)}{2 (a \cosh (x)+a)^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2750, 2649, 206} \[ \frac {(A+3 B) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a \cosh (x)+a}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(A-B) \sinh (x)}{2 (a \cosh (x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2750
Rubi steps
\begin {align*} \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx &=\frac {(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}}+\frac {(A+3 B) \int \frac {1}{\sqrt {a+a \cosh (x)}} \, dx}{4 a}\\ &=\frac {(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}}+\frac {(i (A+3 B)) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {i a \sinh (x)}{\sqrt {a+a \cosh (x)}}\right )}{2 a}\\ &=\frac {(A+3 B) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a+a \cosh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 44, normalized size = 0.68 \[ \frac {\frac {1}{2} (A-B) \sinh (x)+(A+3 B) \cosh ^3\left (\frac {x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right )}{(a (\cosh (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 189, normalized size = 2.91 \[ -\frac {\sqrt {2} {\left ({\left (A + 3 \, B\right )} \cosh \relax (x)^{2} + {\left (A + 3 \, B\right )} \sinh \relax (x)^{2} + 2 \, {\left (A + 3 \, B\right )} \cosh \relax (x) + 2 \, {\left ({\left (A + 3 \, B\right )} \cosh \relax (x) + A + 3 \, B\right )} \sinh \relax (x) + A + 3 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{\sqrt {a}}\right ) - 2 \, \sqrt {\frac {1}{2}} {\left ({\left (A - B\right )} \cosh \relax (x)^{2} + {\left (A - B\right )} \sinh \relax (x)^{2} - {\left (A - B\right )} \cosh \relax (x) + {\left (2 \, {\left (A - B\right )} \cosh \relax (x) - A + B\right )} \sinh \relax (x)\right )} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{2 \, {\left (a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a^{2}\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 78, normalized size = 1.20 \[ \frac {{\left (\sqrt {2} A + 3 \, \sqrt {2} B\right )} \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{2 \, a^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (A a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - B a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - A a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} + B a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )}\right )}}{2 \, {\left (a e^{x} + a\right )}^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 159, normalized size = 2.45 \[ -\frac {\sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (A \ln \left (\frac {2 \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \left (\cosh ^{2}\left (\frac {x}{2}\right )\right ) a +3 B \ln \left (\frac {2 \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-A \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {-a}+B \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {-a}\right ) \sqrt {2}}{4 \cosh \left (\frac {x}{2}\right ) a^{2} \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 300, normalized size = 4.62 \[ \frac {1}{6} \, {\left (\sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} + 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} - \frac {8 \, \sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}}\right )} A + \frac {1}{20} \, {\left (\sqrt {2} {\left (\frac {15 \, e^{\left (\frac {5}{2} \, x\right )} + 40 \, e^{\left (\frac {3}{2} \, x\right )} + 33 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {15 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} + 5 \, \sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} - 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} - \frac {8 \, {\left (5 \, \sqrt {2} \sqrt {a} e^{\left (\frac {5}{2} \, x\right )} + \sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, x\right )}\right )}}{a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {A+B\,\mathrm {cosh}\relax (x)}{{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B \cosh {\relax (x )}}{\left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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