Optimal. Leaf size=351 \[ \frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}+\frac {x \text {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {x \text {PolyLog}\left (2,-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {\text {PolyLog}\left (3,-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3}+\frac {\text {PolyLog}\left (3,-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3} \]
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Rubi [A]
time = 0.63, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5751, 3401,
2296, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {\text {Li}_3\left (-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{4 \sqrt {-a} \sqrt {b} d^3}+\frac {\text {Li}_3\left (-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{4 \sqrt {-a} \sqrt {b} d^3}+\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b} d^2}+\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{-2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (\frac {(a+b) e^{2 c+2 d x}}{2 \sqrt {-a} \sqrt {b}+a-b}+1\right )}{2 \sqrt {-a} \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3401
Rule 5751
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2 \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=2 \int \frac {x^2}{a-b+(a+b) \cosh (2 c+2 d x)} \, dx\\ &=4 \int \frac {e^{2 c+2 d x} x^2}{a+b+2 (a-b) e^{2 c+2 d x}+(a+b) e^{2 (2 c+2 d x)}} \, dx\\ &=\frac {(2 (a+b)) \int \frac {e^{2 c+2 d x} x^2}{2 (a-b)-4 \sqrt {-a} \sqrt {b}+2 (a+b) e^{2 c+2 d x}} \, dx}{\sqrt {-a} \sqrt {b}}-\frac {(2 (a+b)) \int \frac {e^{2 c+2 d x} x^2}{2 (a-b)+4 \sqrt {-a} \sqrt {b}+2 (a+b) e^{2 c+2 d x}} \, dx}{\sqrt {-a} \sqrt {b}}\\ &=\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {\int x \log \left (1+\frac {2 (a+b) e^{2 c+2 d x}}{2 (a-b)-4 \sqrt {-a} \sqrt {b}}\right ) \, dx}{\sqrt {-a} \sqrt {b} d}+\frac {\int x \log \left (1+\frac {2 (a+b) e^{2 c+2 d x}}{2 (a-b)+4 \sqrt {-a} \sqrt {b}}\right ) \, dx}{\sqrt {-a} \sqrt {b} d}\\ &=\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}+\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {\int \text {Li}_2\left (-\frac {2 (a+b) e^{2 c+2 d x}}{2 (a-b)-4 \sqrt {-a} \sqrt {b}}\right ) \, dx}{2 \sqrt {-a} \sqrt {b} d^2}+\frac {\int \text {Li}_2\left (-\frac {2 (a+b) e^{2 c+2 d x}}{2 (a-b)+4 \sqrt {-a} \sqrt {b}}\right ) \, dx}{2 \sqrt {-a} \sqrt {b} d^2}\\ &=\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}+\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(a+b) x}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 \sqrt {-a} \sqrt {b} d^3}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(a+b) x}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 \sqrt {-a} \sqrt {b} d^3}\\ &=\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}-\frac {x^2 \log \left (1+\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d}+\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {x \text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{2 \sqrt {-a} \sqrt {b} d^2}-\frac {\text {Li}_3\left (-\frac {(a+b) e^{2 c+2 d x}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3}+\frac {\text {Li}_3\left (-\frac {(a+b) e^{2 c+2 d x}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{4 \sqrt {-a} \sqrt {b} d^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.65, size = 316, normalized size = 0.90 \begin {gather*} \frac {i \left (2 d^2 x^2 \log \left (1+\frac {\left (\sqrt {a}-i \sqrt {b}\right ) e^{2 (c+d x)}}{\sqrt {a}+i \sqrt {b}}\right )-2 d^2 x^2 \log \left (1+\frac {\left (\sqrt {a}+i \sqrt {b}\right ) e^{2 (c+d x)}}{\sqrt {a}-i \sqrt {b}}\right )+2 d x \text {PolyLog}\left (2,-\frac {\left (\sqrt {a}-i \sqrt {b}\right ) e^{2 (c+d x)}}{\sqrt {a}+i \sqrt {b}}\right )-2 d x \text {PolyLog}\left (2,-\frac {\left (\sqrt {a}+i \sqrt {b}\right ) e^{2 (c+d x)}}{\sqrt {a}-i \sqrt {b}}\right )-\text {PolyLog}\left (3,-\frac {\left (\sqrt {a}-i \sqrt {b}\right ) e^{2 (c+d x)}}{\sqrt {a}+i \sqrt {b}}\right )+\text {PolyLog}\left (3,-\frac {\left (\sqrt {a}+i \sqrt {b}\right ) e^{2 (c+d x)}}{\sqrt {a}-i \sqrt {b}}\right )\right )}{4 \sqrt {a} \sqrt {b} d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1185\) vs.
\(2(285)=570\).
time = 2.86, size = 1186, normalized size = 3.38
method | result | size |
risch | \(-\frac {\polylog \left (3, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right )}{4 d^{3} \sqrt {-a b}}-\frac {\polylog \left (3, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right )}{2 d^{3} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {2 c^{3}}{3 d^{3} \sqrt {-a b}}+\frac {4 c^{3}}{3 d^{3} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b \polylog \left (3, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right )}{4 d^{3} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {a \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x^{2}}{2 d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) c^{2}}{2 d^{3} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {b \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x^{2}}{2 d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {c^{2} a x}{d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {a \polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {b \polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {2 c^{3} b}{3 d^{3} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {2 c^{3} a}{3 d^{3} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {c^{2} b x}{d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) c^{2}}{2 d^{3} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a \polylog \left (3, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right )}{4 d^{3} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {2 c^{2} x}{d^{2} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x^{2}}{d \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) x^{2}}{2 d \sqrt {-a b}}+\frac {c^{2} x}{d^{2} \sqrt {-a b}}+\frac {\polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x}{d^{2} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) x}{2 d^{2} \sqrt {-a b}}+\frac {b \,x^{3}}{3 \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a \,x^{3}}{3 \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {c^{2} \arctan \left (\frac {2 \left (a +b \right ) {\mathrm e}^{2 d x +2 c}+2 a -2 b}{4 \sqrt {a b}}\right )}{d^{3} \sqrt {a b}}-\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) c^{2}}{2 d^{3} \sqrt {-a b}}-\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) c^{2}}{d^{3} \left (-2 \sqrt {-a b}-a +b \right )}-\frac {x^{3}}{3 \sqrt {-a b}}-\frac {2 x^{3}}{3 \left (-2 \sqrt {-a b}-a +b \right )}\) | \(1186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 2110 vs.
\(2 (283) = 566\).
time = 0.40, size = 2110, normalized size = 6.01 \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^{2} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \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^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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