Optimal. Leaf size=95 \[ \frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2} d}-\frac {b \cosh (c+d x) \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3263, 12, 3260,
214} \begin {gather*} \frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^{3/2}}-\frac {b \sinh (c+d x) \cosh (c+d x)}{2 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 3260
Rule 3263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=-\frac {b \cosh (c+d x) \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {-2 a+b}{a+b \sinh ^2(c+d x)} \, dx}{2 a (a-b)}\\ &=-\frac {b \cosh (c+d x) \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}+\frac {(2 a-b) \int \frac {1}{a+b \sinh ^2(c+d x)} \, dx}{2 a (a-b)}\\ &=-\frac {b \cosh (c+d x) \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}+\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a (a-b) d}\\ &=\frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2} d}-\frac {b \cosh (c+d x) \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 96, normalized size = 1.01 \begin {gather*} \frac {(2 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2} d}-\frac {b \sinh (2 (c+d x))}{2 a (a-b) d (2 a-b+b \cosh (2 (c+d x)))} \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. \(284\) vs.
\(2(83)=166\).
time = 1.20, size = 285, normalized size = 3.00
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (a -b \right )}\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {\left (-b +2 a \right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a -b}}{d}\) | \(285\) |
default | \(\frac {-\frac {2 \left (\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (a -b \right )}\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {\left (-b +2 a \right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a -b}}{d}\) | \(285\) |
risch | \(\frac {2 a \,{\mathrm e}^{2 d x +2 c}-b \,{\mathrm e}^{2 d x +2 c}+b}{d a \left (a -b \right ) \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right ) d}\) | \(421\) |
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. 681 vs.
\(2 (83) = 166\).
time = 0.43, size = 1617, normalized size = 17.02 \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 = 0.57, size = 144, normalized size = 1.52 \begin {gather*} \frac {\frac {{\left (2 \, a - b\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{2} - a b\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}}{{\left (a^{2} - a b\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, 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 (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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