Optimal. Leaf size=133 \[ \frac {\left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2} d}-\frac {b \sinh (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^2}-\frac {3 a b \sinh (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))} \]
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Rubi [A]
time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12,
2738, 211} \begin {gather*} \frac {\left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac {3 a b \sinh (c+d x)}{2 d \left (a^2-b^2\right )^2 (a+b \cosh (c+d x))}-\frac {b \sinh (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \cosh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 2738
Rule 2743
Rule 2833
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cosh (c+d x))^3} \, dx &=-\frac {b \sinh (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^2}-\frac {\int \frac {-2 a+b \cosh (c+d x)}{(a+b \cosh (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )}\\ &=-\frac {b \sinh (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^2}-\frac {3 a b \sinh (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))}+\frac {\int \frac {2 a^2+b^2}{a+b \cosh (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=-\frac {b \sinh (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^2}-\frac {3 a b \sinh (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))}+\frac {\left (2 a^2+b^2\right ) \int \frac {1}{a+b \cosh (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=-\frac {b \sinh (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^2}-\frac {3 a b \sinh (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))}-\frac {\left (i \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2} d}-\frac {b \sinh (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \cosh (c+d x))^2}-\frac {3 a b \sinh (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \cosh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 113, normalized size = 0.85 \begin {gather*} \frac {-\frac {2 \left (2 a^2+b^2\right ) \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {b \left (-4 a^2+b^2-3 a b \cosh (c+d x)\right ) \sinh (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cosh (c+d x))^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.94, size = 186, normalized size = 1.40
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (4 a +b \right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (4 a -b \right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(186\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (4 a +b \right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (4 a -b \right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(186\) |
risch | \(\frac {2 a^{2} b \,{\mathrm e}^{3 d x +3 c}+b^{3} {\mathrm e}^{3 d x +3 c}+6 a^{3} {\mathrm e}^{2 d x +2 c}+3 a \,b^{2} {\mathrm e}^{2 d x +2 c}+10 a^{2} b \,{\mathrm e}^{d x +c}-{\mathrm e}^{d x +c} b^{3}+3 a \,b^{2}}{d \left (a^{2}-b^{2}\right )^{2} \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(431\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 1239 vs.
\(2 (120) = 240\).
time = 0.42, size = 2591, normalized size = 19.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 195, normalized size = 1.47 \begin {gather*} \frac {\frac {{\left (2 \, a^{2} + b^{2}\right )} \arctan \left (\frac {b e^{\left (d x + c\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a^{2} b e^{\left (d x + c\right )} - b^{3} e^{\left (d x + c\right )} + 3 \, a b^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} + b\right )}^{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+b\,\mathrm {cosh}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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