Optimal. Leaf size=142 \[ \frac {\left (8 a^2+8 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2} d}-\frac {b \cosh ^2(c+d x) \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4232, 424, 393,
211} \begin {gather*} -\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{5/2} d (a+b)^{5/2}}-\frac {b \sinh (c+d x) \cosh ^2(c+d x)}{4 a d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 424
Rule 4232
Rubi steps
\begin {align*} \int \frac {\text {sech}(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (a+b+a x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {b \cosh ^2(c+d x) \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {4 a+b+(4 a+3 b) x^2}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a (a+b) d}\\ &=-\frac {b \cosh ^2(c+d x) \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=\frac {\left (8 a^2+8 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2} d}-\frac {b \cosh ^2(c+d x) \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.99, size = 214, normalized size = 1.51 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^5(c+d x) \left (\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \text {csch}(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} (\cosh (c)+\sinh (c))}{\sqrt {a}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x) (-\cosh (c)+\sinh (c))}{\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}}+8 \sqrt {a} b^2 (a+b) \tanh (c+d x)-2 \sqrt {a} b (8 a+5 b) (a+2 b+a \cosh (2 (c+d x))) \tanh (c+d x)\right )}{64 a^{5/2} (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^3} \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.
\(2(128)=256\).
time = 2.40, size = 308, normalized size = 2.17
method | result | size |
derivativedivides | \(\frac {\frac {\frac {b \left (8 a +3 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right ) a^{2}}+\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a^{2}}-\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a^{2}}-\frac {b \left (8 a +3 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 \left (a +b \right ) a^{2}}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (8 a^{2}+8 a b +3 b^{2}\right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{4 a^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(308\) |
default | \(\frac {\frac {\frac {b \left (8 a +3 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right ) a^{2}}+\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a^{2}}-\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a^{2}}-\frac {b \left (8 a +3 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 \left (a +b \right ) a^{2}}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (8 a^{2}+8 a b +3 b^{2}\right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{4 a^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(308\) |
risch | \(-\frac {{\mathrm e}^{d x +c} b \left (8 a^{2} {\mathrm e}^{6 d x +6 c}+5 a b \,{\mathrm e}^{6 d x +6 c}+8 a^{2} {\mathrm e}^{4 d x +4 c}+29 a b \,{\mathrm e}^{4 d x +4 c}+12 b^{2} {\mathrm e}^{4 d x +4 c}-8 a^{2} {\mathrm e}^{2 d x +2 c}-29 a b \,{\mathrm e}^{2 d x +2 c}-12 b^{2} {\mathrm e}^{2 d x +2 c}-8 a^{2}-5 a b \right )}{4 a^{2} \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{2}}\) | \(542\) |
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. 3753 vs.
\(2 (128) = 256\).
time = 0.49, size = 6806, normalized size = 47.93 \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 {\operatorname {sech}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, 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 {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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