Optimal. Leaf size=90 \[ \frac {5 \tanh ^{-1}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b} \]
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Rubi [A]
time = 0.07, 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 = {5545, 3853,
3855} \begin {gather*} \frac {5 \tanh ^{-1}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3853
Rule 3855
Rule 5545
Rubi steps
\begin {align*} \int x \text {csch}^7\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \text {csch}^7(a+b x) \, dx,x,x^2\right )\\ &=-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}-\frac {5}{12} \text {Subst}\left (\int \text {csch}^5(a+b x) \, dx,x,x^2\right )\\ &=\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}+\frac {5}{16} \text {Subst}\left (\int \text {csch}^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}-\frac {5}{32} \text {Subst}\left (\int \text {csch}(a+b x) \, dx,x,x^2\right )\\ &=\frac {5 \tanh ^{-1}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 147, normalized size = 1.63 \begin {gather*} -\frac {5 \text {csch}^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\text {csch}^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {csch}^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b}-\frac {5 \log \left (\tanh \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}-\frac {5 \text {sech}^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {sech}^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {sech}^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.14, size = 129, normalized size = 1.43
method | result | size |
risch | \(-\frac {{\mathrm e}^{x^{2} b +a} \left (15 \,{\mathrm e}^{10 x^{2} b +10 a}-85 \,{\mathrm e}^{8 x^{2} b +8 a}+198 \,{\mathrm e}^{6 x^{2} b +6 a}+198 \,{\mathrm e}^{4 x^{2} b +4 a}-85 \,{\mathrm e}^{2 x^{2} b +2 a}+15\right )}{48 b \left ({\mathrm e}^{2 x^{2} b +2 a}-1\right )^{6}}+\frac {5 \ln \left ({\mathrm e}^{x^{2} b +a}+1\right )}{32 b}-\frac {5 \ln \left ({\mathrm e}^{x^{2} b +a}-1\right )}{32 b}\) | \(129\) |
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. 205 vs.
\(2 (82) = 164\).
time = 0.30, size = 205, normalized size = 2.28 \begin {gather*} \frac {5 \, \log \left (e^{\left (-b x^{2} - a\right )} + 1\right )}{32 \, b} - \frac {5 \, \log \left (e^{\left (-b x^{2} - a\right )} - 1\right )}{32 \, 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 )}} \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. 2590 vs.
\(2 (82) = 164\).
time = 0.42, size = 2590, normalized size = 28.78 \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 x \operatorname {csch}^{7}{\left (a + b x^{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 158, normalized size = 1.76 \begin {gather*} \frac {5 \, \log \left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )} + 2\right )}{64 \, b} - \frac {5 \, \log \left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )} - 2\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 399, normalized size = 4.43 \begin {gather*} \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 (6\,{\mathrm {e}}^{4\,b\,x^2+4\,a}-4\,{\mathrm {e}}^{2\,b\,x^2+2\,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 ({\mathrm {e}}^{4\,b\,x^2+4\,a}-2\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+1\right )}-\frac {16\,{\mathrm {e}}^{5\,b\,x^2+5\,a}}{3\,b\,\left (15\,{\mathrm {e}}^{4\,b\,x^2+4\,a}-6\,{\mathrm {e}}^{2\,b\,x^2+2\,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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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