Optimal. Leaf size=581 \[ \frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}} \]
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Rubi [A]
time = 0.38, antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2526, 2505,
211, 2520, 12, 5048, 4966, 2449, 2352, 2497} \begin {gather*} -\frac {i \sqrt {g} p \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2}}-\frac {i \sqrt {g} p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2}}+\frac {i \sqrt {g} p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {\sqrt {g} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {\sqrt {g} p \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{3/2}}+\frac {2 \sqrt {e} p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 2352
Rule 2449
Rule 2497
Rule 2505
Rule 2520
Rule 2526
Rule 4966
Rule 5048
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2 \left (f+g x^2\right )} \, dx &=\int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x^2}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx}{f}-\frac {g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx}{f}\\ &=-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {(2 e p) \int \frac {1}{d+e x^2} \, dx}{f}+\frac {(2 e g p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (d+e x^2\right )} \, dx}{f}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {\left (2 e \sqrt {g} p\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{d+e x^2} \, dx}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {\left (2 e \sqrt {g} p\right ) \int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac {\left (\sqrt {e} \sqrt {g} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{f^{3/2}}+\frac {\left (\sqrt {e} \sqrt {g} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+2 \frac {(g p) \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{f^2}-\frac {(g p) \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {f} \left (-i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f^2}-\frac {(g p) \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {f} \left (i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f^2}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}}+2 \frac {\left (i \sqrt {g} p\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{f^{3/2}}\\ &=\frac {2 \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f}-\frac {2 \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}+\frac {\sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{3/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x}-\frac {\sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{3/2}}+\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}}-\frac {i \sqrt {g} p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 673, normalized size = 1.16 \begin {gather*} \frac {\frac {4 \sqrt {e} \sqrt {f} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+i \sqrt {g} p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+i \sqrt {g} p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )-i \sqrt {g} p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )-i \sqrt {g} p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\frac {2 \sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{x}-2 \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-i \sqrt {g} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )}{2 f^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.49, size = 755, normalized size = 1.30
method | result | size |
risch | \(\text {Expression too large to display}\) | \(755\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{2} \left (f + g 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 {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{x^2\,\left (g\,x^2+f\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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