Optimal. Leaf size=519 \[ \frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}} \]
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Rubi [A]
time = 0.93, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2454, 222,
2451, 12, 4825, 4617, 2221, 2317, 2438, 2495} \begin {gather*} \frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {PolyLog}\left (2,-\frac {f g e^{i \text {ArcSin}\left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {PolyLog}\left (2,-\frac {f g e^{i \text {ArcSin}\left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \text {ArcSin}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {ArcSin}\left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \text {ArcSin}\left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {ArcSin}\left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \text {ArcSin}\left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {ArcSin}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 2454
Rule 2495
Rule 4617
Rule 4825
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx &=\text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {1-\frac {h^2 x^2}{g^2}}} \, dx}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (b f p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \int \frac {g \sin ^{-1}\left (\frac {h x}{g}\right )}{e h+f h x} \, dx}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \int \frac {\sin ^{-1}\left (\frac {h x}{g}\right )}{e h+f h x} \, dx}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {x \cos (x)}{\frac {e h^2}{g}+f h \sin (x)} \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (i b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{e^{i x} f h+\frac {i e h^2}{g}-\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}} \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (i b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{e^{i x} f h+\frac {i e h^2}{g}+\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}} \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\text {Subst}\left (\frac {\left (b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \log \left (1+\frac {e^{i x} f h}{\frac {i e h^2}{g}-\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right ) \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \log \left (1+\frac {e^{i x} f h}{\frac {i e h^2}{g}+\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right ) \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {f h x}{\frac {i e h^2}{g}-\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {h x}{g}\right )}\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {f h x}{\frac {i e h^2}{g}+\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {h x}{g}\right )}\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1083\) vs. \(2(519)=1038\).
time = 3.13, size = 1083, normalized size = 2.09 \begin {gather*} \frac {\tan ^{-1}\left (\frac {h x}{\sqrt {g-h x} \sqrt {g+h x}}\right ) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {i b p q \sqrt {g-h x} \sqrt {\frac {g+h x}{g-h x}} \left (2 \log (e+f x) \log \left (i-\sqrt {\frac {g+h x}{g-h x}}\right )+\log ^2\left (i-\sqrt {\frac {g+h x}{g-h x}}\right )+2 \log \left (i-\sqrt {\frac {g+h x}{g-h x}}\right ) \log \left (\frac {1}{2} \left (1-i \sqrt {\frac {g+h x}{g-h x}}\right )\right )-2 \log (e+f x) \log \left (i+\sqrt {\frac {g+h x}{g-h x}}\right )-2 \log \left (\frac {1}{2} \left (1+i \sqrt {\frac {g+h x}{g-h x}}\right )\right ) \log \left (i+\sqrt {\frac {g+h x}{g-h x}}\right )-\log ^2\left (i+\sqrt {\frac {g+h x}{g-h x}}\right )-2 \log \left (i-\sqrt {\frac {g+h x}{g-h x}}\right ) \log \left (\frac {\sqrt {f g-e h}-\sqrt {f g+e h} \sqrt {\frac {g+h x}{g-h x}}}{\sqrt {f g-e h}-i \sqrt {f g+e h}}\right )+2 \log \left (i+\sqrt {\frac {g+h x}{g-h x}}\right ) \log \left (\frac {\sqrt {f g-e h}-\sqrt {f g+e h} \sqrt {\frac {g+h x}{g-h x}}}{\sqrt {f g-e h}+i \sqrt {f g+e h}}\right )+2 \log \left (i+\sqrt {\frac {g+h x}{g-h x}}\right ) \log \left (\frac {\sqrt {f g-e h}+\sqrt {f g+e h} \sqrt {\frac {g+h x}{g-h x}}}{\sqrt {f g-e h}-i \sqrt {f g+e h}}\right )-2 \log \left (i-\sqrt {\frac {g+h x}{g-h x}}\right ) \log \left (\frac {\sqrt {f g-e h}+\sqrt {f g+e h} \sqrt {\frac {g+h x}{g-h x}}}{\sqrt {f g-e h}+i \sqrt {f g+e h}}\right )-2 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} i \sqrt {\frac {g+h x}{g-h x}}\right )+2 \text {Li}_2\left (\frac {1}{2}+\frac {1}{2} i \sqrt {\frac {g+h x}{g-h x}}\right )+2 \text {Li}_2\left (\frac {\sqrt {f g+e h} \left (1-i \sqrt {\frac {g+h x}{g-h x}}\right )}{i \sqrt {f g-e h}+\sqrt {f g+e h}}\right )-2 \text {Li}_2\left (\frac {\sqrt {f g+e h} \left (1+i \sqrt {\frac {g+h x}{g-h x}}\right )}{-i \sqrt {f g-e h}+\sqrt {f g+e h}}\right )-2 \text {Li}_2\left (\frac {\sqrt {f g+e h} \left (1+i \sqrt {\frac {g+h x}{g-h x}}\right )}{i \sqrt {f g-e h}+\sqrt {f g+e h}}\right )+2 \text {Li}_2\left (\frac {\sqrt {f g+e h} \left (i+\sqrt {\frac {g+h x}{g-h x}}\right )}{\sqrt {f g-e h}+i \sqrt {f g+e h}}\right )\right )}{2 h \sqrt {g+h x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {-h x +g}\, \sqrt {h x +g}}\, dx\]
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 \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 {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g - h x} \sqrt {g + h x}}\, 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 {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {g+h\,x}\,\sqrt {g-h\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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