Optimal. Leaf size=90 \[ \frac {5 \text {ArcSin}(\tanh (a+b x))}{16 b}+\frac {5 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{16 b}+\frac {5 \text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{24 b}+\frac {\text {sech}^2(a+b x)^{5/2} \tanh (a+b x)}{6 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 201, 222}
\begin {gather*} \frac {5 \text {ArcSin}(\tanh (a+b x))}{16 b}+\frac {\tanh (a+b x) \text {sech}^2(a+b x)^{5/2}}{6 b}+\frac {5 \tanh (a+b x) \text {sech}^2(a+b x)^{3/2}}{24 b}+\frac {5 \tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{16 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 4207
Rubi steps
\begin {align*} \int \text {sech}^2(a+b x)^{7/2} \, dx &=\frac {\text {Subst}\left (\int \left (1-x^2\right )^{5/2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\text {sech}^2(a+b x)^{5/2} \tanh (a+b x)}{6 b}+\frac {5 \text {Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,\tanh (a+b x)\right )}{6 b}\\ &=\frac {5 \text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{24 b}+\frac {\text {sech}^2(a+b x)^{5/2} \tanh (a+b x)}{6 b}+\frac {5 \text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (a+b x)\right )}{8 b}\\ &=\frac {5 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{16 b}+\frac {5 \text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{24 b}+\frac {\text {sech}^2(a+b x)^{5/2} \tanh (a+b x)}{6 b}+\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{16 b}\\ &=\frac {5 \sin ^{-1}(\tanh (a+b x))}{16 b}+\frac {5 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{16 b}+\frac {5 \text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{24 b}+\frac {\text {sech}^2(a+b x)^{5/2} \tanh (a+b x)}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 81, normalized size = 0.90 \begin {gather*} \frac {\cosh (a+b x) \sqrt {\text {sech}^2(a+b x)} \left (15 \text {ArcTan}(\sinh (a+b x))+15 \text {sech}(a+b x) \tanh (a+b x)+10 \text {sech}^3(a+b x) \tanh (a+b x)+8 \text {sech}^5(a+b x) \tanh (a+b x)\right )}{48 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 2.60, size = 230, normalized size = 2.56
method | result | size |
risch | \(\frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}\, \left (15 \,{\mathrm e}^{10 b x +10 a}+85 \,{\mathrm e}^{8 b x +8 a}+198 \,{\mathrm e}^{6 b x +6 a}-198 \,{\mathrm e}^{4 b x +4 a}-85 \,{\mathrm e}^{2 b x +2 a}-15\right )}{24 \left ({\mathrm e}^{2 b x +2 a}+1\right )^{5} b}+\frac {5 i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}+1\right ) {\mathrm e}^{-b x -a}}{16 b}-\frac {5 i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}+1\right ) {\mathrm e}^{-b x -a}}{16 b}\) | \(230\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs.
\(2 (76) = 152\).
time = 0.49, size = 156, normalized size = 1.73 \begin {gather*} -\frac {5 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{8 \, b} + \frac {15 \, e^{\left (-b x - a\right )} + 85 \, e^{\left (-3 \, b x - 3 \, a\right )} + 198 \, e^{\left (-5 \, b x - 5 \, a\right )} - 198 \, e^{\left (-7 \, b x - 7 \, a\right )} - 85 \, e^{\left (-9 \, b x - 9 \, a\right )} - 15 \, e^{\left (-11 \, b x - 11 \, a\right )}}{24 \, b {\left (6 \, e^{\left (-2 \, b x - 2 \, a\right )} + 15 \, e^{\left (-4 \, b x - 4 \, a\right )} + 20 \, e^{\left (-6 \, b x - 6 \, a\right )} + 15 \, e^{\left (-8 \, b x - 8 \, a\right )} + 6 \, e^{\left (-10 \, b x - 10 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} + 1\right )}} \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. 1604 vs.
\(2 (76) = 152\).
time = 0.36, size = 1604, normalized size = 17.82 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 124, normalized size = 1.38 \begin {gather*} \frac {15 \, \pi + \frac {4 \, {\left (15 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{5} + 160 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 528 \, e^{\left (b x + a\right )} - 528 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{3}} + 30 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{96 \, b} \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 {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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