Optimal. Leaf size=287 \[ \frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {i b p q \text {Li}_2\left (-\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {i b p q \text {Li}_2\left (-\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{-\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{h}+\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h} \]
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Rubi [A] time = 1.06, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {216, 2405, 4741, 4521, 2190, 2279, 2391, 2445} \[ \frac {i b p q \text {PolyLog}\left (2,-\frac {2 f e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{-\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{h}+\frac {i b p q \text {PolyLog}\left (2,-\frac {2 f e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{h}+\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{-\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{h}+\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h} \]
Antiderivative was successfully verified.
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Rule 216
Rule 2190
Rule 2279
Rule 2391
Rule 2405
Rule 2445
Rule 4521
Rule 4741
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx &=\operatorname {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname {Subst}\left ((b f p q) \int \frac {\sin ^{-1}\left (\frac {h x}{2}\right )}{e h+f h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname {Subst}\left ((b f p q) \operatorname {Subst}\left (\int \frac {x \cos (x)}{\frac {e h^2}{2}+f h \sin (x)} \, dx,x,\sin ^{-1}\left (\frac {h x}{2}\right )\right ),c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h}+\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname {Subst}\left ((i b f p q) \operatorname {Subst}\left (\int \frac {e^{i x} x}{e^{i x} f h+\frac {1}{2} i e h^2-\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}} \, dx,x,\sin ^{-1}\left (\frac {h x}{2}\right )\right ),c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left ((i b f p q) \operatorname {Subst}\left (\int \frac {e^{i x} x}{e^{i x} f h+\frac {1}{2} i e h^2+\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}} \, dx,x,\sin ^{-1}\left (\frac {h x}{2}\right )\right ),c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \log \left (1+\frac {e^{i x} f h}{\frac {1}{2} i e h^2-\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}}\right ) \, dx,x,\sin ^{-1}\left (\frac {h x}{2}\right )\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \log \left (1+\frac {e^{i x} f h}{\frac {1}{2} i e h^2+\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}}\right ) \, dx,x,\sin ^{-1}\left (\frac {h x}{2}\right )\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\operatorname {Subst}\left (\frac {(i b p q) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f h x}{\frac {1}{2} i e h^2-\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(i b p q) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f h x}{\frac {1}{2} i e h^2+\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {\sin ^{-1}\left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {i b p q \text {Li}_2\left (-\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {i b p q \text {Li}_2\left (-\frac {2 e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 316, normalized size = 1.10 \[ \frac {a \sin ^{-1}\left (\frac {h x}{2}\right )}{h}+\frac {b \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )}{h}+\frac {i b p q \text {Li}_2\left (\frac {2 i e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{e h-i \sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {i b p q \text {Li}_2\left (\frac {2 i e^{i \sin ^{-1}\left (\frac {h x}{2}\right )} f}{e h+i \sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {f h e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{-\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}+\frac {1}{2} i e h^2}\right )}{h}-\frac {b p q \sin ^{-1}\left (\frac {h x}{2}\right ) \log \left (1+\frac {f h e^{i \sin ^{-1}\left (\frac {h x}{2}\right )}}{\frac {1}{2} h \sqrt {4 f^2-e^2 h^2}+\frac {1}{2} i e h^2}\right )}{h}+\frac {i b p q \sin ^{-1}\left (\frac {h x}{2}\right )^2}{2 h} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {h x + 2} \sqrt {-h x + 2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt {h x + 2} \sqrt {-h x + 2} a}{h^{2} x^{2} - 4}, x\right ) \]
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 \[ \int \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + 2} \sqrt {-h x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\sqrt {-h x +2}\, \sqrt {h x +2}}\, dx \]
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 \[ b \int \frac {q \log \relax (d) + \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \relax (c)}{\sqrt {h x + 2} \sqrt {-h x + 2}}\,{d x} + \frac {a \arcsin \left (\frac {1}{2} \, h x\right )}{h} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {2-h\,x}\,\sqrt {h\,x+2}} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {- h x + 2} \sqrt {h x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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