Optimal. Leaf size=196 \[ -\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b^2 \text {PolyLog}\left (2,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {2 a b \text {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \text {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \]
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Rubi [A]
time = 0.27, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5545, 4275,
4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438} \begin {gather*} \frac {a^2 x^6}{6}+\frac {2 a b \text {Li}_3\left (-e^{d x^2+c}\right )}{d^3}-\frac {2 a b \text {Li}_3\left (e^{d x^2+c}\right )}{d^3}-\frac {2 a b x^2 \text {Li}_2\left (-e^{d x^2+c}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{d x^2+c}\right )}{d^2}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac {b^2 \text {Li}_2\left (e^{2 \left (d x^2+c\right )}\right )}{2 d^3}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {b^2 x^4}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3797
Rule 4267
Rule 4269
Rule 4275
Rule 5545
Rule 6724
Rubi steps
\begin {align*} \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 (a+b \text {csch}(c+d x))^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text {csch}(c+d x)+b^2 x^2 \text {csch}^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^6}{6}+(a b) \text {Subst}\left (\int x^2 \text {csch}(c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \text {Subst}\left (\int x^2 \text {csch}^2(c+d x) \, dx,x,x^2\right )\\ &=\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {(2 a b) \text {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {(2 a b) \text {Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {b^2 \text {Subst}\left (\int x \coth (c+d x) \, dx,x,x^2\right )}{d}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {(2 a b) \text {Subst}\left (\int \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {(2 a b) \text {Subst}\left (\int \text {Li}_2\left (e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x}{1-e^{2 (c+d x)}} \, dx,x,x^2\right )}{d}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {(2 a b) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {(2 a b) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {b^2 \text {Subst}\left (\int \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {2 a b \text {Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \text {Li}_3\left (e^{c+d x^2}\right )}{d^3}-\frac {b^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {b^2 \text {Li}_2\left (e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {2 a b \text {Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \text {Li}_3\left (e^{c+d x^2}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 4.19, size = 269, normalized size = 1.37 \begin {gather*} \frac {1}{12} \left (2 a^2 x^6+\frac {6 b \left (-\frac {2 b d^2 e^{2 c} x^4}{-1+e^{2 c}}+2 a d^2 x^4 \log \left (1-e^{c+d x^2}\right )-2 a d^2 x^4 \log \left (1+e^{c+d x^2}\right )+2 b d x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )-4 a d x^2 \text {PolyLog}\left (2,-e^{c+d x^2}\right )+4 a d x^2 \text {PolyLog}\left (2,e^{c+d x^2}\right )+b \text {PolyLog}\left (2,e^{2 \left (c+d x^2\right )}\right )+4 a \text {PolyLog}\left (3,-e^{c+d x^2}\right )-4 a \text {PolyLog}\left (3,e^{c+d x^2}\right )\right )}{d^3}+\frac {3 b^2 x^4 \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )}{d}-\frac {3 b^2 x^4 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.40, size = 0, normalized size = 0.00 \[\int x^{5} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.44, size = 271, normalized size = 1.38 \begin {gather*} \frac {1}{6} \, a^{2} x^{6} - \frac {b^{2} x^{4}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} - d} - \frac {{\left (d^{2} x^{4} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2} {\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac {{\left (d^{2} x^{4} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2} {\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac {{\left (d x^{2} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} + \frac {{\left (d x^{2} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} - \frac {2 \, a b d^{3} x^{6} + 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} + \frac {2 \, a b d^{3} x^{6} - 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} \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. 1031 vs.
\(2 (180) = 360\).
time = 0.43, size = 1031, normalized size = 5.26 \begin {gather*} -\frac {a^{2} d^{3} x^{6} + 6 \, b^{2} c^{2} - {\left (a^{2} d^{3} x^{6} - 6 \, b^{2} d^{2} x^{4} + 6 \, b^{2} c^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a^{2} d^{3} x^{6} - 6 \, b^{2} d^{2} x^{4} + 6 \, b^{2} c^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a^{2} d^{3} x^{6} - 6 \, b^{2} d^{2} x^{4} + 6 \, b^{2} c^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} + 6 \, {\left (2 \, a b d x^{2} - {\left (2 \, a b d x^{2} + b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a b d x^{2} + b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (2 \, a b d x^{2} + b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} + b^{2}\right )} {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 6 \, {\left (2 \, a b d x^{2} - {\left (2 \, a b d x^{2} - b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a b d x^{2} - b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} - b^{2}\right )} {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) - 6 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2} - {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) + 6 \, {\left (a b c^{2} - b^{2} c - {\left (a b c^{2} - b^{2} c\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a b c^{2} - b^{2} c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a b c^{2} - b^{2} c\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 6 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c - {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right ) + 12 \, {\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} - a b\right )} {\rm polylog}\left (3, \cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 12 \, {\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} - a b\right )} {\rm polylog}\left (3, -\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right )}{6 \, {\left (d^{3} \cosh \left (d x^{2} + c\right )^{2} + 2 \, d^{3} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + d^{3} \sinh \left (d x^{2} + c\right )^{2} - d^{3}\right )}} \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^{5} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 x^5\,{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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