Optimal. Leaf size=555 \[ \frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^3 \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )} \]
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Rubi [A]
time = 0.76, antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5544, 4276,
3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 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 )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {b^2 \log \left (a \cosh \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 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 )}{a^2 d \sqrt {b^2-a^2}}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^2\right )+b\right )}+\frac {b^3 \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {b^3 \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {b^2-a^2}}+1\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {b^2-a^2}+b}+1\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {x^4}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3401
Rule 3405
Rule 4276
Rule 5544
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b \text {sech}(c+d x))^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \cosh (c+d x))^2}-\frac {2 b x}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac {x^4}{4 a^2}-\frac {b \text {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \text {Subst}\left (\int \frac {x}{(b+a \cosh (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {x^4}{4 a^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac {(2 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^2}-\frac {b^3 \text {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \text {Subst}\left (\int \frac {\sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2-b^2\right ) d}\\ &=\frac {x^4}{4 a^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac {b^3 \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^2 \left (a^2-b^2\right )}-\frac {(2 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 )}{a \sqrt {-a^2+b^2}}+\frac {(2 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 )}{a \sqrt {-a^2+b^2}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}\\ &=\frac {x^4}{4 a^2}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \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 )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac {b^3 \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 )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {b^3 \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 )}{a \left (-a^2+b^2\right )^{3/2}}+\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 )}{a^2 \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 )}{a^2 \sqrt {-a^2+b^2} d}\\ &=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\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 )}{a^2 \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 )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^3 \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^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b^3 \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^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}-\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \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 )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac {b^3 \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^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {b^3 \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^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \log \left (b+a \cosh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^3 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^2 x^2 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 4.09, size = 755, normalized size = 1.36 \begin {gather*} \frac {\left (b+a \cosh \left (c+d x^2\right )\right ) \text {sech}^2\left (c+d x^2\right ) \left (\left (-c+d x^2\right ) \left (c+d x^2\right ) \left (b+a \cosh \left (c+d x^2\right )\right )-\frac {2 b \left (a^2-b^2\right ) \left (b+a \cosh \left (c+d x^2\right )\right ) \left (b \sqrt {-\left (a^2-b^2\right )^2} \left (c+d x^2\right )-2 b^2 \sqrt {-a^2+b^2} \text {ArcTan}\left (\frac {b+a e^{-c-d x^2}}{\sqrt {a^2-b^2}}\right )-4 a^2 \sqrt {-a^2+b^2} c \text {ArcTan}\left (\frac {b+a e^{-c-d x^2}}{\sqrt {a^2-b^2}}\right )+2 b^2 \sqrt {-a^2+b^2} c \text {ArcTan}\left (\frac {b+a e^{-c-d x^2}}{\sqrt {a^2-b^2}}\right )-2 b^2 \sqrt {-a^2+b^2} \text {ArcTan}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2-b^2}}\right )-2 a^2 \sqrt {a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )+b^2 \sqrt {a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )+2 a^2 \sqrt {a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )-b^2 \sqrt {a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )-b \sqrt {-\left (a^2-b^2\right )^2} \log \left (a+2 b e^{c+d x^2}+a e^{2 \left (c+d x^2\right )}\right )+\sqrt {a^2-b^2} \left (-2 a^2+b^2\right ) \text {PolyLog}\left (2,\frac {a e^{c+d x^2}}{-b+\sqrt {-a^2+b^2}}\right )+\sqrt {a^2-b^2} \left (2 a^2-b^2\right ) \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )\right )}{\left (-\left (a^2-b^2\right )^2\right )^{3/2}}+\frac {2 a b^2 d x^2 \sinh \left (c+d x^2\right )}{(a-b) (a+b)}\right )}{4 a^2 d^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.32, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )\right )^{2}}\, dx\]
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. 2473 vs.
\(2 (497) = 994\).
time = 0.40, size = 2473, normalized size = 4.46 \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 \frac {x^{3}}{\left (a + b \operatorname {sech}{\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 = -0.00 \begin {gather*} \int \frac {x^3}{{\left (a+\frac {b}{\mathrm {cosh}\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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