Optimal. Leaf size=77 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sinh (c+d x)}{2 d (a \cosh (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2650, 2649, 206} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sinh (c+d x)}{2 d (a \cosh (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rubi steps
\begin {align*} \int \frac {1}{(a+a \cosh (c+d x))^{3/2}} \, dx &=\frac {\sinh (c+d x)}{2 d (a+a \cosh (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+a \cosh (c+d x)}} \, dx}{4 a}\\ &=\frac {\sinh (c+d x)}{2 d (a+a \cosh (c+d x))^{3/2}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {i a \sinh (c+d x)}{\sqrt {a+a \cosh (c+d x)}}\right )}{2 a d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a+a \cosh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sinh (c+d x)}{2 d (a+a \cosh (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 63, normalized size = 0.82 \[ \frac {\cosh ^2\left (\frac {1}{2} (c+d x)\right ) \left (\tanh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \tan ^{-1}\left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d (a (\cosh (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 219, normalized size = 2.84 \[ -\frac {\sqrt {2} {\left (\cosh \left (d x + c\right )^{2} + 2 \, {\left (\cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{\sqrt {a}}\right ) - 2 \, \sqrt {\frac {1}{2}} {\left (\cosh \left (d x + c\right )^{2} + {\left (2 \, \cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - \cosh \left (d x + c\right )\right )} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{2 \, {\left (a^{2} d \cosh \left (d x + c\right )^{2} + a^{2} d \sinh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) + a^{2} d + 2 \, {\left (a^{2} d \cosh \left (d x + c\right ) + a^{2} d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 67, normalized size = 0.87 \[ \frac {\sqrt {2} {\left (\frac {\arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a}} + \frac {a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} - a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (a e^{\left (d x + c\right )} + a\right )}^{2}}\right )}}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 144, normalized size = 1.87 \[ -\frac {\sqrt {a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\ln \left (\frac {2 \sqrt {a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {-a}-2 a}{\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {-a}\right ) \sqrt {2}}{4 a^{2} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-a}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 170, normalized size = 2.21 \[ \frac {1}{6} \, \sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )} + 8 \, e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} - 3 \, e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (a^{\frac {3}{2}} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{\frac {3}{2}} e^{\left (d x + c\right )} + a^{\frac {3}{2}}\right )} d} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{a^{\frac {3}{2}} d}\right )} - \frac {4 \, \sqrt {2} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}}{3 \, {\left (a^{\frac {3}{2}} d e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a^{\frac {3}{2}} d e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{\frac {3}{2}} d e^{\left (d x + c\right )} + a^{\frac {3}{2}} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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