Optimal. Leaf size=128 \[ \frac {b^2 (6 a+b) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2} d}+\frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {b^3 \sinh (c+d x)}{2 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.13, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3757, 398, 393,
211} \begin {gather*} \frac {b^2 (6 a+b) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a+b)^{7/2}}+\frac {b^3 \sinh (c+d x)}{2 a d (a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sinh ^3(c+d x)}{3 d (a+b)^2}+\frac {(a+3 b) \sinh (c+d x)}{d (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 398
Rule 3757
Rubi steps
\begin {align*} \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a+3 b}{(a+b)^3}+\frac {x^2}{(a+b)^2}+\frac {b^2 (3 a+b)+3 b^2 (a+b) x^2}{(a+b)^3 \left (a+(a+b) x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {\text {Subst}\left (\int \frac {b^2 (3 a+b)+3 b^2 (a+b) x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{(a+b)^3 d}\\ &=\frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {b^3 \sinh (c+d x)}{2 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {\left (b^2 (6 a+b)\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{2 a (a+b)^3 d}\\ &=\frac {b^2 (6 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2} d}+\frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {b^3 \sinh (c+d x)}{2 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 111, normalized size = 0.87 \begin {gather*} \frac {-\frac {6 b^2 (6 a+b) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{3/2} (a+b)^{7/2}}+\frac {3 \left (3 a+11 b+\frac {4 b^3}{a (a-b+(a+b) \cosh (2 (c+d x)))}\right ) \sinh (c+d x)}{(a+b)^3}+\frac {\sinh (3 (c+d x))}{(a+b)^2}}{12 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. \(377\) vs.
\(2(114)=228\).
time = 3.00, size = 378, normalized size = 2.95
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b^{2} \left (\frac {-\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (6 a +b \right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{3}}}{d}\) | \(378\) |
default | \(\frac {-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b^{2} \left (\frac {-\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (6 a +b \right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{3}}}{d}\) | \(378\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {3 \,{\mathrm e}^{d x +c} a}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) d}+\frac {11 \,{\mathrm e}^{d x +c} b}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {11 \,{\mathrm e}^{-d x -c} b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3} {\mathrm e}^{d x +c}}{\left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right ) d \left (a +b \right ) a \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}\) | \(522\) |
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. 3642 vs.
\(2 (114) = 228\).
time = 0.44, size = 6934, normalized size = 54.17 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \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 {{\mathrm {cosh}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\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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