Optimal. Leaf size=110 \[ -\frac {3 \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}-\frac {3 \sinh (c+d x)}{16 a d (a-a \cosh (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 110, 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 = {2729, 2728,
212} \begin {gather*} -\frac {3 \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {3 \sinh (c+d x)}{16 a d (a-a \cosh (c+d x))^{3/2}}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}} \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))^{5/2}} \, dx &=-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}+\frac {3 \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx}{8 a}\\ &=-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}-\frac {3 \sinh (c+d x)}{16 a d (a-a \cosh (c+d x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx}{32 a^2}\\ &=-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}-\frac {3 \sinh (c+d x)}{16 a d (a-a \cosh (c+d x))^{3/2}}+\frac {(3 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 )}{16 a^2 d}\\ &=-\frac {3 \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}-\frac {3 \sinh (c+d x)}{16 a d (a-a \cosh (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 115, normalized size = 1.05 \begin {gather*} \frac {\left (6 \text {csch}^2\left (\frac {1}{4} (c+d x)\right )-\text {csch}^4\left (\frac {1}{4} (c+d x)\right )+24 \log \left (\tanh \left (\frac {1}{4} (c+d x)\right )\right )+6 \text {sech}^2\left (\frac {1}{4} (c+d x)\right )+\text {sech}^4\left (\frac {1}{4} (c+d x)\right )\right ) \sinh ^5\left (\frac {1}{2} (c+d x)\right )}{32 a^2 d (-1+\cosh (c+d x))^2 \sqrt {a-a \cosh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 137, normalized size = 1.25
method | result | size |
default | \(\frac {6 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (3 \ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3 \ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right ) \left (\sinh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a^{2} \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}\, d}\) | \(137\) |
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 \begin {gather*} \text {Failed to integrate} \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. 580 vs.
\(2 (91) = 182\).
time = 0.36, size = 580, normalized size = 5.27 \begin {gather*} -\frac {3 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, {\left (\cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} - 4 \, \cosh \left (d x + c\right )^{3} + 6 \, {\left (\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right )^{2} + 6 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (\cosh \left (d x + c\right )^{3} - 3 \, \cosh \left (d x + c\right )^{2} + 3 \, \cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right ) - 4 \, \cosh \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-a} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (3 \, \cosh \left (d x + c\right )^{4} + {\left (12 \, \cosh \left (d x + c\right ) - 11\right )} \sinh \left (d x + c\right )^{3} + 3 \, \sinh \left (d x + c\right )^{4} - 11 \, \cosh \left (d x + c\right )^{3} + {\left (18 \, \cosh \left (d x + c\right )^{2} - 33 \, \cosh \left (d x + c\right ) - 11\right )} \sinh \left (d x + c\right )^{2} - 11 \, \cosh \left (d x + c\right )^{2} + {\left (12 \, \cosh \left (d x + c\right )^{3} - 33 \, \cosh \left (d x + c\right )^{2} - 22 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right )\right )} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{32 \, {\left (a^{3} d \cosh \left (d x + c\right )^{4} + a^{3} d \sinh \left (d x + c\right )^{4} - 4 \, a^{3} d \cosh \left (d x + c\right )^{3} + 6 \, a^{3} d \cosh \left (d x + c\right )^{2} - 4 \, a^{3} d \cosh \left (d x + c\right ) + a^{3} d + 4 \, {\left (a^{3} d \cosh \left (d x + c\right ) - a^{3} d\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (a^{3} d \cosh \left (d x + c\right )^{2} - 2 \, a^{3} d \cosh \left (d x + c\right ) + a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - 3 \, a^{3} d \cosh \left (d x + c\right )^{2} + 3 \, a^{3} d \cosh \left (d x + c\right ) - a^{3} 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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 177, normalized size = 1.61 \begin {gather*} -\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a e^{\left (d x + c\right )}}}{\sqrt {a}}\right )}{16 \, a^{\frac {5}{2}} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} + \frac {3 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3} e^{\left (3 \, d x + 3 \, c\right )} - 11 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 11 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3} e^{\left (d x + c\right )} + 3 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3}}{16 \, {\left (a e^{\left (d x + c\right )} - a\right )}^{4} a^{2} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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