Optimal. Leaf size=510 \[ \frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {Li}_2\left (-\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {Li}_2\left (-\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e f e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}}{-\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e f e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}}{\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}} \]
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Rubi [A] time = 0.64, antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2407, 216, 2404, 12, 4741, 4521, 2190, 2279, 2391} \[ \frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {PolyLog}\left (2,-\frac {e f e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}}{-\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {PolyLog}\left (2,-\frac {e f e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}}{\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e f e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}}{-\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e f e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}}{\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 2190
Rule 2279
Rule 2391
Rule 2404
Rule 2407
Rule 4521
Rule 4741
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx &=\frac {\sqrt {1-\frac {g^2 x^2}{f^2}} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {1-\frac {g^2 x^2}{f^2}}} \, dx}{\sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (b e n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \int \frac {f \sin ^{-1}\left (\frac {g x}{f}\right )}{d g+e g x} \, dx}{\sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \int \frac {\sin ^{-1}\left (\frac {g x}{f}\right )}{d g+e g x} \, dx}{\sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \frac {x \cos (x)}{\frac {d g^2}{f}+e g \sin (x)} \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{\sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{e e^{i x} g+\frac {i d g^2}{f}-\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}} \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{\sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{e e^{i x} g+\frac {i d g^2}{f}+\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}} \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{\sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {\left (b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {e e^{i x} g}{\frac {i d g^2}{f}-\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right ) \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {\left (b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {e e^{i x} g}{\frac {i d g^2}{f}+\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right ) \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e g x}{\frac {i d g^2}{f}-\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e g x}{\frac {i d g^2}{f}+\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}\\ &=\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {Li}_2\left (-\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {Li}_2\left (-\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}\\ \end {align*}
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Mathematica [B] time = 4.78, size = 1077, normalized size = 2.11 \[ \frac {\tan ^{-1}\left (\frac {g x}{\sqrt {f-g x} \sqrt {f+g x}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {i b n \sqrt {f-g x} \sqrt {\frac {f+g x}{f-g x}} \left (\log ^2\left (i-\sqrt {\frac {f+g x}{f-g x}}\right )+2 \log (d+e x) \log \left (i-\sqrt {\frac {f+g x}{f-g x}}\right )+2 \log \left (\frac {1}{2} \left (1-i \sqrt {\frac {f+g x}{f-g x}}\right )\right ) \log \left (i-\sqrt {\frac {f+g x}{f-g x}}\right )-2 \log \left (\frac {\sqrt {e f-d g}-\sqrt {e f+d g} \sqrt {\frac {f+g x}{f-g x}}}{\sqrt {e f-d g}-i \sqrt {e f+d g}}\right ) \log \left (i-\sqrt {\frac {f+g x}{f-g x}}\right )-2 \log \left (\frac {\sqrt {e f-d g}+\sqrt {e f+d g} \sqrt {\frac {f+g x}{f-g x}}}{\sqrt {e f-d g}+i \sqrt {e f+d g}}\right ) \log \left (i-\sqrt {\frac {f+g x}{f-g x}}\right )-\log ^2\left (\sqrt {\frac {f+g x}{f-g x}}+i\right )-2 \log (d+e x) \log \left (\sqrt {\frac {f+g x}{f-g x}}+i\right )-2 \log \left (\frac {1}{2} \left (i \sqrt {\frac {f+g x}{f-g x}}+1\right )\right ) \log \left (\sqrt {\frac {f+g x}{f-g x}}+i\right )+2 \log \left (\sqrt {\frac {f+g x}{f-g x}}+i\right ) \log \left (\frac {\sqrt {e f-d g}-\sqrt {e f+d g} \sqrt {\frac {f+g x}{f-g x}}}{\sqrt {e f-d g}+i \sqrt {e f+d g}}\right )+2 \log \left (\sqrt {\frac {f+g x}{f-g x}}+i\right ) \log \left (\frac {\sqrt {e f-d g}+\sqrt {e f+d g} \sqrt {\frac {f+g x}{f-g x}}}{\sqrt {e f-d g}-i \sqrt {e f+d g}}\right )-2 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} i \sqrt {\frac {f+g x}{f-g x}}\right )+2 \text {Li}_2\left (\frac {1}{2} i \sqrt {\frac {f+g x}{f-g x}}+\frac {1}{2}\right )+2 \text {Li}_2\left (\frac {\sqrt {e f+d g} \left (1-i \sqrt {\frac {f+g x}{f-g x}}\right )}{i \sqrt {e f-d g}+\sqrt {e f+d g}}\right )-2 \text {Li}_2\left (\frac {\sqrt {e f+d g} \left (i \sqrt {\frac {f+g x}{f-g x}}+1\right )}{\sqrt {e f+d g}-i \sqrt {e f-d g}}\right )-2 \text {Li}_2\left (\frac {\sqrt {e f+d g} \left (i \sqrt {\frac {f+g x}{f-g x}}+1\right )}{i \sqrt {e f-d g}+\sqrt {e f+d g}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e f+d g} \left (\sqrt {\frac {f+g x}{f-g x}}+i\right )}{\sqrt {e f-d g}+i \sqrt {e f+d g}}\right )\right )}{2 g \sqrt {f+g x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {g x + f} \sqrt {-g x + f} b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt {g x + f} \sqrt {-g x + f} a}{g^{2} x^{2} - f^{2}}, x\right ) \]
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 \[ \int \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x + f} \sqrt {-g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e x +d \right )^{n}\right )+a}{\sqrt {-g x +f}\, \sqrt {g x +f}}\, dx \]
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 \[ b \int \frac {\log \left ({\left (e x + d\right )}^{n}\right ) + \log \relax (c)}{\sqrt {g x + f} \sqrt {-g x + f}}\,{d x} + \frac {a \arcsin \left (\frac {g x}{f}\right )}{g} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {f+g\,x}\,\sqrt {f-g\,x}} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\sqrt {f - g x} \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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