Optimal. Leaf size=90 \[ \frac {5 \tan ^{-1}\left (\sinh \left (a+b x^2\right )\right )}{32 b}+\frac {\tanh \left (a+b x^2\right ) \text {sech}^5\left (a+b x^2\right )}{12 b}+\frac {5 \tanh \left (a+b x^2\right ) \text {sech}^3\left (a+b x^2\right )}{48 b}+\frac {5 \tanh \left (a+b x^2\right ) \text {sech}\left (a+b x^2\right )}{32 b} \]
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Rubi [A] time = 0.08, 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 = {5436, 3768, 3770} \[ \frac {5 \tan ^{-1}\left (\sinh \left (a+b x^2\right )\right )}{32 b}+\frac {\tanh \left (a+b x^2\right ) \text {sech}^5\left (a+b x^2\right )}{12 b}+\frac {5 \tanh \left (a+b x^2\right ) \text {sech}^3\left (a+b x^2\right )}{48 b}+\frac {5 \tanh \left (a+b x^2\right ) \text {sech}\left (a+b x^2\right )}{32 b} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rule 5436
Rubi steps
\begin {align*} \int x \text {sech}^7\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \text {sech}^7(a+b x) \, dx,x,x^2\right )\\ &=\frac {\text {sech}^5\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{12 b}+\frac {5}{12} \operatorname {Subst}\left (\int \text {sech}^5(a+b x) \, dx,x,x^2\right )\\ &=\frac {5 \text {sech}^3\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{48 b}+\frac {\text {sech}^5\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{12 b}+\frac {5}{16} \operatorname {Subst}\left (\int \text {sech}^3(a+b x) \, dx,x,x^2\right )\\ &=\frac {5 \text {sech}\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{32 b}+\frac {5 \text {sech}^3\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{48 b}+\frac {\text {sech}^5\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{12 b}+\frac {5}{32} \operatorname {Subst}\left (\int \text {sech}(a+b x) \, dx,x,x^2\right )\\ &=\frac {5 \tan ^{-1}\left (\sinh \left (a+b x^2\right )\right )}{32 b}+\frac {5 \text {sech}\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{32 b}+\frac {5 \text {sech}^3\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{48 b}+\frac {\text {sech}^5\left (a+b x^2\right ) \tanh \left (a+b x^2\right )}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 77, normalized size = 0.86 \[ \frac {15 \tan ^{-1}\left (\sinh \left (a+b x^2\right )\right )+8 \tanh \left (a+b x^2\right ) \text {sech}^5\left (a+b x^2\right )+10 \tanh \left (a+b x^2\right ) \text {sech}^3\left (a+b x^2\right )+15 \tanh \left (a+b x^2\right ) \text {sech}\left (a+b x^2\right )}{96 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 1918, normalized size = 21.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 146, normalized size = 1.62 \[ \frac {5 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x^{2} + 2 \, a\right )} - 1\right )} e^{\left (-b x^{2} - a\right )}\right )\right )}}{64 \, b} + \frac {15 \, {\left (e^{\left (b x^{2} + a\right )} - e^{\left (-b x^{2} - a\right )}\right )}^{5} + 160 \, {\left (e^{\left (b x^{2} + a\right )} - e^{\left (-b x^{2} - a\right )}\right )}^{3} + 528 \, e^{\left (b x^{2} + a\right )} - 528 \, e^{\left (-b x^{2} - a\right )}}{48 \, {\left ({\left (e^{\left (b x^{2} + a\right )} - e^{\left (-b x^{2} - a\right )}\right )}^{2} + 4\right )}^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 83, normalized size = 0.92 \[ \frac {\mathrm {sech}\left (b \,x^{2}+a \right )^{5} \tanh \left (b \,x^{2}+a \right )}{12 b}+\frac {5 \mathrm {sech}\left (b \,x^{2}+a \right )^{3} \tanh \left (b \,x^{2}+a \right )}{48 b}+\frac {5 \,\mathrm {sech}\left (b \,x^{2}+a \right ) \tanh \left (b \,x^{2}+a \right )}{32 b}+\frac {5 \arctan \left ({\mathrm e}^{b \,x^{2}+a}\right )}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 182, normalized size = 2.02 \[ -\frac {5 \, \arctan \left (e^{\left (-b x^{2} - a\right )}\right )}{16 \, b} + \frac {15 \, e^{\left (-b x^{2} - a\right )} + 85 \, e^{\left (-3 \, b x^{2} - 3 \, a\right )} + 198 \, e^{\left (-5 \, b x^{2} - 5 \, a\right )} - 198 \, e^{\left (-7 \, b x^{2} - 7 \, a\right )} - 85 \, e^{\left (-9 \, b x^{2} - 9 \, a\right )} - 15 \, e^{\left (-11 \, b x^{2} - 11 \, a\right )}}{48 \, b {\left (6 \, e^{\left (-2 \, b x^{2} - 2 \, a\right )} + 15 \, e^{\left (-4 \, b x^{2} - 4 \, a\right )} + 20 \, e^{\left (-6 \, b x^{2} - 6 \, a\right )} + 15 \, e^{\left (-8 \, b x^{2} - 8 \, a\right )} + 6 \, e^{\left (-10 \, b x^{2} - 10 \, a\right )} + e^{\left (-12 \, b x^{2} - 12 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 395, normalized size = 4.39 \[ \frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^a\,{\mathrm {e}}^{b\,x^2}\,\sqrt {b^2}}{b}\right )}{16\,\sqrt {b^2}}-\frac {8\,{\mathrm {e}}^{3\,b\,x^2+3\,a}}{3\,b\,\left (5\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+10\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+10\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+5\,{\mathrm {e}}^{8\,b\,x^2+8\,a}+{\mathrm {e}}^{10\,b\,x^2+10\,a}+1\right )}-\frac {{\mathrm {e}}^{b\,x^2+a}}{b\,\left (4\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+6\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+4\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+{\mathrm {e}}^{8\,b\,x^2+8\,a}+1\right )}+\frac {5\,{\mathrm {e}}^{b\,x^2+a}}{24\,b\,\left (2\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+{\mathrm {e}}^{4\,b\,x^2+4\,a}+1\right )}-\frac {16\,{\mathrm {e}}^{5\,b\,x^2+5\,a}}{3\,b\,\left (6\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+15\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+20\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+15\,{\mathrm {e}}^{8\,b\,x^2+8\,a}+6\,{\mathrm {e}}^{10\,b\,x^2+10\,a}+{\mathrm {e}}^{12\,b\,x^2+12\,a}+1\right )}+\frac {{\mathrm {e}}^{b\,x^2+a}}{6\,b\,\left (3\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+3\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+{\mathrm {e}}^{6\,b\,x^2+6\,a}+1\right )}+\frac {5\,{\mathrm {e}}^{b\,x^2+a}}{16\,b\,\left ({\mathrm {e}}^{2\,b\,x^2+2\,a}+1\right )} \]
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 \[ \int x \operatorname {sech}^{7}{\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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