Optimal. Leaf size=77 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a+a \cosh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sinh (c+d x)}{2 d (a+a \cosh (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.03, 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 = {2729, 2728,
212} \begin {gather*} \frac {\text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
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 \text {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 = 0.06, size = 63, normalized size = 0.82 \begin {gather*} \frac {\cosh ^2\left (\frac {1}{2} (c+d x)\right ) \left (\text {ArcTan}\left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh \left (\frac {1}{2} (c+d x)\right )+\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d (a (1+\cosh (c+d x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs.
\(2(62)=124\).
time = 1.08, size = 144, normalized size = 1.87
method | result | size |
default | \(-\frac {\sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}\, \left (\ln \left (\frac {2 \sqrt {-a}\, \sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}-2 a}{\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {-a}\, \sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}\right ) \sqrt {2}}{4 a^{2} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-a}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs.
\(2 (62) = 124\).
time = 0.54, size = 170, normalized size = 2.21 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs.
\(2 (62) = 124\).
time = 0.35, size = 219, normalized size = 2.84 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 70, normalized size = 0.91 \begin {gather*} \frac {\frac {\sqrt {2} \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (\sqrt {a} e^{\left (\frac {3}{2} \, d x + 2 \, c\right )} - \sqrt {a} e^{\left (\frac {1}{2} \, d x + c\right )}\right )} e^{\left (-\frac {1}{2} \, c\right )}}{a^{2} {\left (e^{\left (d x + c\right )} + 1\right )}^{2}}}{2 \, d} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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