Optimal. Leaf size=1113 \[ \frac {i c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.01, antiderivative size = 1113, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4861, 4857,
3405, 3404, 2296, 2221, 2611, 2320, 6724, 4615, 2317, 2438} \begin {gather*} \frac {2 i c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 i c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 i c^2 f \sqrt {1-c^2 x^2} \text {Li}_3\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b^2}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i c^2 f \sqrt {1-c^2 x^2} \text {Li}_3\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b^2}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) b}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) b}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {i c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3404
Rule 3405
Rule 4615
Rule 4857
Rule 4861
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{(c f+g \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {(a+b x)^2}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=\frac {i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {\left (2 c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{c f-i e^{i x} g-\sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{c f-i e^{i x} g+\sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=\frac {i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i c^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i c^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {i e^{i x} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {i e^{i x} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=\frac {i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i g x}{c f-\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=\frac {i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ &=\frac {i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i g x}{c f-\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ &=\frac {i c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 f \sqrt {1-c^2 x^2} \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 f \sqrt {1-c^2 x^2} \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.50, size = 651, normalized size = 0.58 \begin {gather*} \frac {c \sqrt {1-c^2 x^2} \left (i (a+b \text {ArcSin}(c x))^2+\frac {g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{c f+c g x}-2 b (a+b \text {ArcSin}(c x)) \log \left (1+\frac {i e^{i \text {ArcSin}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-2 b (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 i b^2 \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b^2 \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-\frac {i c f \left ((a+b \text {ArcSin}(c x))^2 \log \left (1+\frac {i e^{i \text {ArcSin}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2}}+\frac {c f \left (2 b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+i \left ((a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (g x +f \right )^{2} \sqrt {-c^{2} d \,x^{2}+d}}\, dx\]
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 {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, 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 {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (f+g\,x\right )}^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________