Optimal. Leaf size=241 \[ \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2}+\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2} \]
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Rubi [A]
time = 0.36, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5544, 4276,
3401, 2296, 2221, 2317, 2438} \begin {gather*} -\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{2 a d^2 \sqrt {b^2-a^2}}+\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {b^2-a^2}}\right )}{2 a d^2 \sqrt {b^2-a^2}}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {b^2-a^2}}+1\right )}{2 a d \sqrt {b^2-a^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {b^2-a^2}+b}+1\right )}{2 a d \sqrt {b^2-a^2}}+\frac {x^4}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3401
Rule 4276
Rule 5544
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{a+b \text {sech}(c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {x}{a}-\frac {b x}{a (b+a \cosh (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {e^{c+d x} x}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a}\\ &=\frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt {-a^2+b^2}}+\frac {b \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt {-a^2+b^2}}\\ &=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt {-a^2+b^2} d}\\ &=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a \sqrt {-a^2+b^2} d^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a \sqrt {-a^2+b^2} d^2}\\ &=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2}+\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.23, size = 843, normalized size = 3.50 \begin {gather*} \frac {\left (b+a \cosh \left (c+d x^2\right )\right ) \left (x^4+\frac {2 b \left (2 \left (c+d x^2\right ) \text {ArcTan}\left (\frac {(a+b) \coth \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+2 \left (c-i \text {ArcCos}\left (-\frac {b}{a}\right )\right ) \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (\text {ArcCos}\left (-\frac {b}{a}\right )+2 \left (\text {ArcTan}\left (\frac {(a+b) \coth \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+\text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt {a^2-b^2} e^{-\frac {c}{2}-\frac {d x^2}{2}}}{\sqrt {2} \sqrt {a} \sqrt {b+a \cosh \left (c+d x^2\right )}}\right )+\left (\text {ArcCos}\left (-\frac {b}{a}\right )-2 \left (\text {ArcTan}\left (\frac {(a+b) \coth \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+\text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt {a^2-b^2} e^{\frac {1}{2} \left (c+d x^2\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \cosh \left (c+d x^2\right )}}\right )-\left (\text {ArcCos}\left (-\frac {b}{a}\right )+2 \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (-a+b+i \sqrt {a^2-b^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {b}{a}\right )-2 \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (a-b+i \sqrt {a^2-b^2}\right ) \left (1+\tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (b-i \sqrt {a^2-b^2}\right ) \left (a+b-i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (b+i \sqrt {a^2-b^2}\right ) \left (a+b-i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a \left (a+b+i \sqrt {a^2-b^2} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}\right )\right )\right )}{\sqrt {a^2-b^2} d^2}\right ) \text {sech}\left (c+d x^2\right )}{4 a \left (a+b \text {sech}\left (c+d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.34, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 536 vs.
\(2 (209) = 418\).
time = 0.39, size = 536, normalized size = 2.22 \begin {gather*} \frac {{\left (a^{2} - b^{2}\right )} d^{2} x^{4} + 2 \, a b c \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) + 2 \, a \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, a b c \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) - 2 \, a \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, a b \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a} + 1\right ) - 2 \, a b \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a} + 1\right ) + 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a}\right ) - 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a}\right )}{4 \, {\left (a^{3} - a b^{2}\right )} d^{2}} \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 {x^{3}}{a + b \operatorname {sech}{\left (c + d x^{2} \right )}}\, 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 = -0.00 \begin {gather*} \int \frac {x^3}{a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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