Integrand size = 33, antiderivative size = 374 \[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=-\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]
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Time = 0.39 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4738, 4864, 4826, 4616, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=-\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{2 b c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 4616
Rule 4738
Rule 4826
Rule 4864
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {(g m) \int \frac {(a+b \arccos (c x))^2}{f+g x} \, dx}{2 b c} \\ & = -\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {(g m) \text {Subst}\left (\int \frac {(a+b x)^2 \sin (x)}{c f+g \cos (x)} \, dx,x,\arccos (c x)\right )}{2 b c} \\ & = -\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {(i g m) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{c f+e^{i x} g-\sqrt {c^2 f^2-g^2}} \, dx,x,\arccos (c x)\right )}{2 b c}+\frac {(i g m) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{c f+e^{i x} g+\sqrt {c^2 f^2-g^2}} \, dx,x,\arccos (c x)\right )}{2 b c} \\ & = -\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {m \text {Subst}\left (\int (a+b x) \log \left (1+\frac {e^{i x} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arccos (c x)\right )}{c}-\frac {m \text {Subst}\left (\int (a+b x) \log \left (1+\frac {e^{i x} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arccos (c x)\right )}{c} \\ & = -\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {(i b m) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e^{i x} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arccos (c x)\right )}{c}+\frac {(i b m) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e^{i x} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arccos (c x)\right )}{c} \\ & = -\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {g x}{-c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arccos (c x)}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arccos (c x)}\right )}{c} \\ & = -\frac {i m (a+b \arccos (c x))^3}{6 b^2 c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \arccos (c x))^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1248\) vs. \(2(374)=748\).
Time = 6.29 (sec) , antiderivative size = 1248, normalized size of antiderivative = 3.34 \[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\frac {-3 i a m \arccos (c x)^2-i b m \arccos (c x)^3+24 i a m \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \arctan \left (\frac {(c f-g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {c^2 f^2-g^2}}\right )+3 b m \arccos (c x)^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+6 a m \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+3 b m \arccos (c x)^2 \log \left (1+\frac {e^{i \arccos (c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+12 a m \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \arccos (c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+12 b m \arccos (c x) \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \arccos (c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+3 b m \arccos (c x)^2 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+6 a m \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )+3 b m \arccos (c x)^2 \log \left (1+\frac {e^{i \arccos (c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-12 a m \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \arccos (c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-12 b m \arccos (c x) \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \arccos (c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-6 a m \arccos (c x) \log (f+g x)-6 a m \arcsin (c x) \log (f+g x)-3 b \arccos (c x)^2 \log \left (h (f+g x)^m\right )+6 a \arcsin (c x) \log \left (h (f+g x)^m\right )-3 b m \arccos (c x)^2 \log \left (1+\frac {\left (c f-\sqrt {c^2 f^2-g^2}\right ) \left (c x+i \sqrt {1-c^2 x^2}\right )}{g}\right )-12 b m \arccos (c x) \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (c f-\sqrt {c^2 f^2-g^2}\right ) \left (c x+i \sqrt {1-c^2 x^2}\right )}{g}\right )-3 b m \arccos (c x)^2 \log \left (1+\frac {\left (c f+\sqrt {c^2 f^2-g^2}\right ) \left (c x+i \sqrt {1-c^2 x^2}\right )}{g}\right )+12 b m \arccos (c x) \arcsin \left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (c f+\sqrt {c^2 f^2-g^2}\right ) \left (c x+i \sqrt {1-c^2 x^2}\right )}{g}\right )-6 i b m \arccos (c x) \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-6 i a m \operatorname {PolyLog}\left (2,\frac {e^{i \arccos (c x)} \left (-c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-6 i b m \arccos (c x) \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-6 i a m \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )+6 b m \operatorname {PolyLog}\left (3,\frac {e^{i \arccos (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )+6 b m \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{6 c} \]
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\[\int \frac {\left (a +b \arccos \left (c x \right )\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]
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\[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right ) \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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\[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arccos (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{\sqrt {1-c^2\,x^2}} \,d x \]
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