Optimal. Leaf size=287 \[ -\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d}+\frac {p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}-\frac {p \text {Li}_2\left (\frac {b}{a x^2}+1\right )}{2 d}-\frac {2 p \text {Li}_2\left (\frac {e x}{d}+1\right )}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
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Rubi [A] time = 0.46, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2466, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ \frac {p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}-\frac {p \text {PolyLog}\left (2,\frac {b}{a x^2}+1\right )}{2 d}-\frac {2 p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d}+\frac {p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2454
Rule 2462
Rule 2466
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}-\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x^2}\right )}{2 d}-\frac {(2 b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}+\frac {(b p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x^2}\right )}{2 d}-\frac {(2 b p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a x \log (d+e x)}{b \left (b+a x^2\right )}\right ) \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {(2 p) \int \frac {\log (d+e x)}{x} \, dx}{d}+\frac {(2 a p) \int \frac {x \log (d+e x)}{b+a x^2} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}+\frac {(2 a p) \int \left (-\frac {\sqrt {-a} \log (d+e x)}{2 a \left (\sqrt {b}-\sqrt {-a} x\right )}+\frac {\sqrt {-a} \log (d+e x)}{2 a \left (\sqrt {b}+\sqrt {-a} x\right )}\right ) \, dx}{d}+\frac {(2 e p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {\left (\sqrt {-a} p\right ) \int \frac {\log (d+e x)}{\sqrt {b}-\sqrt {-a} x} \, dx}{d}+\frac {\left (\sqrt {-a} p\right ) \int \frac {\log (d+e x)}{\sqrt {b}+\sqrt {-a} x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d+e x} \, dx}{d}-\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a} x}{-\sqrt {-a} d+\sqrt {b} e}\right )}{x} \, dx,x,d+e x\right )}{d}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a} x}{\sqrt {-a} d+\sqrt {b} e}\right )}{x} \, dx,x,d+e x\right )}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}+\frac {p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 264, normalized size = 0.92 \[ -\frac {2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )-2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )-2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {b} e-\sqrt {-a} d}\right )+p \text {Li}_2\left (\frac {b}{a x^2}+1\right )+4 p \text {Li}_2\left (\frac {e x}{d}+1\right )+4 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (c \left (\frac {a x^{2} + b}{x^{2}}\right )^{p}\right )}{e x^{2} + d x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{\left (e x +d \right ) x}\, dx \]
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 \[ \int \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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