Optimal. Leaf size=156 \[ -\frac {\text {ArcTan}(\sinh (c+d x))}{b^3 d}+\frac {\sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}+\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.16, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3757, 425, 541,
536, 209, 211} \begin {gather*} -\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}+\frac {(a+b) \sinh (c+d x)}{4 a b d \left ((a+b) \sinh ^2(c+d x)+a\right )^2}-\frac {\text {ArcTan}(\sinh (c+d x))}{b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 425
Rule 536
Rule 541
Rule 3757
Rubi steps
\begin {align*} \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a-3 b-3 (a+b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a b d}\\ &=\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {4 a^2-a b+3 b^2-(4 a-3 b) (a+b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 a^2 b^2 d}\\ &=\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b^3 d}+\frac {\left ((a+b) \left (8 a^2-4 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 b^3 d}\\ &=-\frac {\tan ^{-1}(\sinh (c+d x))}{b^3 d}+\frac {\sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}+\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.21, size = 317, normalized size = 2.03 \begin {gather*} -\frac {\frac {2 \sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{5/2}}+\frac {2 \left (8 a^3+4 a^2 b-a b^2+3 b^3\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b}}+64 \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\frac {i \sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \log (a-b+(a+b) \cosh (2 (c+d x)))}{a^{5/2}}-\frac {i \left (8 a^3+4 a^2 b-a b^2+3 b^3\right ) \log (a-b+(a+b) \cosh (2 (c+d x)))}{a^{5/2} \sqrt {a+b}}-\frac {32 b^2 (a+b) \sinh (c+d x)}{a (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {8 b \left (4 a^2+a b-3 b^2\right ) \sinh (c+d x)}{a^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{32 b^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. \(388\) vs.
\(2(142)=284\).
time = 2.72, size = 389, normalized size = 2.49
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (\frac {b \left (4 a^{2}-a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\left (4 a^{3}+23 a^{2} b +7 a \,b^{2}-12 b^{3}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {\left (4 a^{3}+23 a^{2} b +7 a \,b^{2}-12 b^{3}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {b \left (4 a^{2}-a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (8 a^{3}+4 a^{2} b -a \,b^{2}+3 b^{3}\right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4 a}}{b^{3}}-\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\) | \(389\) |
default | \(\frac {\frac {\frac {2 \left (\frac {b \left (4 a^{2}-a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\left (4 a^{3}+23 a^{2} b +7 a \,b^{2}-12 b^{3}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {\left (4 a^{3}+23 a^{2} b +7 a \,b^{2}-12 b^{3}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {b \left (4 a^{2}-a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (8 a^{3}+4 a^{2} b -a \,b^{2}+3 b^{3}\right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4 a}}{b^{3}}-\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\) | \(389\) |
risch | \(-\frac {\left (4 a^{3} {\mathrm e}^{6 d x +6 c}+5 a^{2} b \,{\mathrm e}^{6 d x +6 c}-2 a \,b^{2} {\mathrm e}^{6 d x +6 c}-3 b^{3} {\mathrm e}^{6 d x +6 c}+4 a^{3} {\mathrm e}^{4 d x +4 c}-19 a^{2} b \,{\mathrm e}^{4 d x +4 c}-14 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 b^{3} {\mathrm e}^{4 d x +4 c}-4 a^{3} {\mathrm e}^{2 d x +2 c}+19 a^{2} b \,{\mathrm e}^{2 d x +2 c}+14 a \,b^{2} {\mathrm e}^{2 d x +2 c}-9 b^{3} {\mathrm e}^{2 d x +2 c}-4 a^{3}-5 a^{2} b +2 a \,b^{2}+3 b^{3}\right ) {\mathrm e}^{d x +c}}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \,a^{2} b^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right )}{d \,b^{3}}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right )}{d \,b^{3}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{2 a d \,b^{3}}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{4 a^{2} d \,b^{2}}+\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{16 a^{3} d b}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{2 a d \,b^{3}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{4 a^{2} d \,b^{2}}-\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{16 a^{3} d b}\) | \(617\) |
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. 4457 vs.
\(2 (142) = 284\).
time = 0.49, size = 8070, normalized size = 51.73 \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 {cosh}\left (c+d\,x\right )}^7\,{\left (b\,{\mathrm {tanh}\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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