Optimal. Leaf size=374 \[ -\frac {i m (a+b \text {ArcCos}(c x))^3}{6 b^2 c}+\frac {m (a+b \text {ArcCos}(c x))^2 \log \left (1+\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \text {ArcCos}(c x))^2 \log \left (1+\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {(a+b \text {ArcCos}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (2,-\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \text {ArcCos}(c x)) \text {PolyLog}\left (2,-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \text {PolyLog}\left (3,-\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \text {PolyLog}\left (3,-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.42, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4738, 4864,
4826, 4616, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {i m (a+b \text {ArcCos}(c x))^3}{6 b^2 c}-\frac {i m (a+b \text {ArcCos}(c x)) \text {Li}_2\left (-\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \text {ArcCos}(c x)) \text {Li}_2\left (-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {m (a+b \text {ArcCos}(c x))^2 \log \left (1+\frac {g e^{i \text {ArcCos}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m (a+b \text {ArcCos}(c x))^2 \log \left (1+\frac {g e^{i \text {ArcCos}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{2 b c}-\frac {(a+b \text {ArcCos}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {b m \text {Li}_3\left (-\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \text {Li}_3\left (-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 4616
Rule 4738
Rule 4826
Rule 4864
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {(g m) \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{f+g x} \, dx}{2 b c}\\ &=-\frac {\left (a+b \cos ^{-1}(c x)\right )^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,\cos ^{-1}(c x)\right )}{2 b c}\\ &=-\frac {i m \left (a+b \cos ^{-1}(c x)\right )^3}{6 b^2 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^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,\cos ^{-1}(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,\cos ^{-1}(c x)\right )}{2 b c}\\ &=-\frac {i m \left (a+b \cos ^{-1}(c x)\right )^3}{6 b^2 c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^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,\cos ^{-1}(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,\cos ^{-1}(c x)\right )}{c}\\ &=-\frac {i m \left (a+b \cos ^{-1}(c x)\right )^3}{6 b^2 c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {(i b m) \text {Subst}\left (\int \text {Li}_2\left (-\frac {e^{i x} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c}+\frac {(i b m) \text {Subst}\left (\int \text {Li}_2\left (-\frac {e^{i x} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c}\\ &=-\frac {i m \left (a+b \cos ^{-1}(c x)\right )^3}{6 b^2 c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {g x}{-c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{c}\\ &=-\frac {i m \left (a+b \cos ^{-1}(c x)\right )^3}{6 b^2 c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {m \left (a+b \cos ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {i m \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m \left (a+b \cos ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \text {Li}_3\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {b m \text {Li}_3\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1248\) vs. \(2(374)=748\).
time = 3.66, size = 1248, normalized size = 3.34 \begin {gather*} \frac {-3 i a m \text {ArcCos}(c x)^2-i b m \text {ArcCos}(c x)^3+24 i a m \text {ArcSin}\left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(c f-g) \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {c^2 f^2-g^2}}\right )+3 b m \text {ArcCos}(c x)^2 \log \left (1+\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+6 a m \text {ArcCos}(c x) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+3 b m \text {ArcCos}(c x)^2 \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+12 a m \text {ArcSin}\left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+12 b m \text {ArcCos}(c x) \text {ArcSin}\left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f-\sqrt {c^2 f^2-g^2}\right )}{g}\right )+3 b m \text {ArcCos}(c x)^2 \log \left (1+\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+6 a m \text {ArcCos}(c x) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )+3 b m \text {ArcCos}(c x)^2 \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-12 a m \text {ArcSin}\left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-12 b m \text {ArcCos}(c x) \text {ArcSin}\left (\frac {\sqrt {1+\frac {c f}{g}}}{\sqrt {2}}\right ) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-6 a m \text {ArcCos}(c x) \log (f+g x)-6 a m \text {ArcSin}(c x) \log (f+g x)-3 b \text {ArcCos}(c x)^2 \log \left (h (f+g x)^m\right )+6 a \text {ArcSin}(c x) \log \left (h (f+g x)^m\right )-3 b m \text {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 \text {ArcCos}(c x) \text {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 \text {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 \text {ArcCos}(c x) \text {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 \text {ArcCos}(c x) \text {PolyLog}\left (2,\frac {e^{i \text {ArcCos}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-6 i a m \text {PolyLog}\left (2,\frac {e^{i \text {ArcCos}(c x)} \left (-c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )-6 i b m \text {ArcCos}(c x) \text {PolyLog}\left (2,-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-6 i a m \text {PolyLog}\left (2,-\frac {e^{i \text {ArcCos}(c x)} \left (c f+\sqrt {c^2 f^2-g^2}\right )}{g}\right )+6 b m \text {PolyLog}\left (3,\frac {e^{i \text {ArcCos}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )+6 b m \text {PolyLog}\left (3,-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{6 c} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 3.33, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arccos \left (c x \right )\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________