Optimal. Leaf size=166 \[ -\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 (a-b)^{5/2} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.18, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3265, 425, 541,
536, 212, 211} \begin {gather*} -\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b (7 a-4 b) \cosh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 d (a-b)^{5/2}}-\frac {b \cosh (c+d x)}{4 a d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 425
Rule 536
Rule 541
Rule 3265
Rubi steps
\begin {align*} \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-4 a+b+3 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {8 a^2-9 a b+4 b^2-(7 a-4 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^3 (a-b)^2 d}\\ &=-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 (a-b)^{5/2} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.23, size = 237, normalized size = 1.43 \begin {gather*} -\frac {\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {8 a^2 b \cosh (c+d x)}{(a-b) (2 a-b+b \cosh (2 (c+d x)))^2}+\frac {2 a (7 a-4 b) b \cosh (c+d x)}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))}-8 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^3 d} \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. \(307\) vs.
\(2(152)=304\).
time = 1.74, size = 308, normalized size = 1.86
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-28 a b +16 b^{2}\right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (9 a^{3}-30 a^{2} b +40 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}-68 a b +32 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a^{2} \left (3 a -2 b \right )}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{16 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}\right )}{a^{3}}}{d}\) | \(308\) |
default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-28 a b +16 b^{2}\right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (9 a^{3}-30 a^{2} b +40 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}-68 a b +32 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a^{2} \left (3 a -2 b \right )}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{16 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}\right )}{a^{3}}}{d}\) | \(308\) |
risch | \(-\frac {{\mathrm e}^{d x +c} b \left (7 a b \,{\mathrm e}^{6 d x +6 c}-4 b^{2} {\mathrm e}^{6 d x +6 c}+36 a^{2} {\mathrm e}^{4 d x +4 c}-31 a b \,{\mathrm e}^{4 d x +4 c}+4 b^{2} {\mathrm e}^{4 d x +4 c}+36 a^{2} {\mathrm e}^{2 d x +2 c}-31 a b \,{\mathrm e}^{2 d x +2 c}+4 b^{2} {\mathrm e}^{2 d x +2 c}+7 a b -4 b^{2}\right )}{4 d \,a^{2} \left (a -b \right )^{2} \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{3} d}+\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{16 \left (a -b \right )^{3} d a}-\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}-\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{16 \left (a -b \right )^{3} d a}+\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}\) | \(571\) |
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. 5242 vs.
\(2 (152) = 304\).
time = 0.53, size = 9815, normalized size = 59.13 \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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