Optimal. Leaf size=1678 \[ -\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \text {ArcSin}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \text {ArcSin}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \text {ArcSin}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \text {ArcSin}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \text {ArcSin}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \text {ArcSin}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \text {ArcSin}(c x)^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \text {ArcSin}(c x)^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i a b g^2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 b^2 c g (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {b^2 c^3 d (e f-d g)^2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g^2 \text {ArcSin}(c x) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i a b g^2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 b^2 c g (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 c^3 d (e f-d g)^2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g^2 \text {ArcSin}(c x) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.73, antiderivative size = 1678, normalized size of antiderivative = 1.00, number of steps
used = 55, number of rules used = 25, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4843, 45,
4837, 12, 6874, 821, 739, 210, 222, 2451, 4825, 4615, 2221, 2317, 2438, 4827, 4857, 3405, 3404,
2296, 2747, 31, 2611, 2320, 6724} \begin {gather*} -\frac {i b^2 d (e f-d g)^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {i b^2 d (e f-d g)^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 (e f-d g)^2 \log (d+e x) c^2}{e^3 \left (c^2 d^2-e^2\right )}+\frac {b^2 (e f-d g)^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) c}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \text {ArcTan}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) c}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {4 i b^2 g (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {4 i b^2 g (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {4 b^2 g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {4 b^2 g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {a b (e f-d g)^2 \sqrt {1-c^2 x^2} c}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b^2 g^2 \text {ArcSin}(c x)^3}{3 e^3}-\frac {i a b g^2 \text {ArcSin}(c x)^2}{e^3}-\frac {2 b^2 g (e f-d g) \text {ArcSin}(c x)^2}{e^3 (d+e x)}-\frac {b^2 (e f-d g)^2 \text {ArcSin}(c x)^2}{2 e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \text {ArcSin}(c x)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \text {ArcSin}(c x)}{e^3 (d+e x)^2}+\frac {b^2 g^2 \text {ArcSin}(c x)^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b^2 g^2 \text {ArcSin}(c x)^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \text {ArcSin}(c x) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \text {ArcSin}(c x) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 45
Rule 210
Rule 222
Rule 739
Rule 821
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2451
Rule 2611
Rule 2747
Rule 3404
Rule 3405
Rule 4615
Rule 4825
Rule 4827
Rule 4837
Rule 4843
Rule 4857
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac {a^2 (f+g x)^2}{(d+e x)^3}+\frac {2 a b (f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^3}+\frac {b^2 (f+g x)^2 \sin ^{-1}(c x)^2}{(d+e x)^3}\right ) \, dx\\ &=a^2 \int \frac {(f+g x)^2}{(d+e x)^3} \, dx+(2 a b) \int \frac {(f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^3} \, dx+b^2 \int \frac {(f+g x)^2 \sin ^{-1}(c x)^2}{(d+e x)^3} \, dx\\ &=-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}+a^2 \int \left (\frac {(e f-d g)^2}{e^2 (d+e x)^3}+\frac {2 g (e f-d g)}{e^2 (d+e x)^2}+\frac {g^2}{e^2 (d+e x)}\right ) \, dx+b^2 \int \left (\frac {(e f-d g)^2 \sin ^{-1}(c x)^2}{e^2 (d+e x)^3}+\frac {2 g (e f-d g) \sin ^{-1}(c x)^2}{e^2 (d+e x)^2}+\frac {g^2 \sin ^{-1}(c x)^2}{e^2 (d+e x)}\right ) \, dx-(2 a b c) \int \frac {-(e f-d g) (3 d g+e (f+4 g x))+2 g^2 (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {a^2 g^2 \log (d+e x)}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac {(a b c) \int \frac {-(e f-d g) (3 d g+e (f+4 g x))+2 g^2 (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b^2 g^2\right ) \int \frac {\sin ^{-1}(c x)^2}{d+e x} \, dx}{e^2}+\frac {\left (2 b^2 g (e f-d g)\right ) \int \frac {\sin ^{-1}(c x)^2}{(d+e x)^2} \, dx}{e^2}+\frac {\left (b^2 (e f-d g)^2\right ) \int \frac {\sin ^{-1}(c x)^2}{(d+e x)^3} \, dx}{e^2}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}+\frac {a^2 g^2 \log (d+e x)}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac {(a b c) \int \left (-\frac {(e f-d g) (e f+3 d g+4 e g x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {2 g^2 \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^3}+\frac {\left (b^2 g^2\right ) \text {Subst}\left (\int \frac {x^2 \cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (4 b^2 c g (e f-d g)\right ) \int \frac {\sin ^{-1}(c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b^2 c (e f-d g)^2\right ) \int \frac {\sin ^{-1}(c x)}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{e^3}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac {\left (2 a b c g^2\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b^2 g^2\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{c d-\sqrt {c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (b^2 g^2\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{c d+\sqrt {c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {(a b c (e f-d g)) \int \frac {e f+3 d g+4 e g x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (4 b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (b^2 c^2 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {x}{(c d+e \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {\left (2 b^2 g^2\right ) \text {Subst}\left (\int x \log \left (1-\frac {i e e^{i x}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (2 b^2 g^2\right ) \text {Subst}\left (\int x \log \left (1-\frac {i e e^{i x}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (2 a b c g^2\right ) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^2}+\frac {\left (8 b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )}-\frac {\left (b^2 c^2 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {\cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )}-\frac {\left (a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right )}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {\left (2 i b^2 g^2\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {i e e^{i x}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (2 i b^2 g^2\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {i e e^{i x}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (2 a b c g^2\right ) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac {\left (8 i b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c d-2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (8 i b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c d+2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (b^2 c^2 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e^3 \left (c^2 d^2-e^2\right )}+\frac {\left (2 b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )}+\frac {\left (a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {\left (2 b^2 g^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i e x}{c d-\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (2 b^2 g^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i e x}{c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (2 a b c g^2\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (2 a b c g^2\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (4 i b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 i b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 i b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c d-2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (2 i b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c d+2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {\left (2 a b g^2\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (2 a b g^2\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (4 b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 b^2 c g (e f-d g)\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {\left (i b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (i b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {\left (2 i a b g^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (2 i a b g^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (b^2 c^3 d (e f-d g)^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {b^2 c^3 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 c^3 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A]
time = 2.82, size = 903, normalized size = 0.54 \begin {gather*} \frac {-\frac {3 (e f-d g)^2 (a+b \text {ArcSin}(c x))^2}{(d+e x)^2}+\frac {12 g (-e f+d g) (a+b \text {ArcSin}(c x))^2}{d+e x}-\frac {2 i g^2 (a+b \text {ArcSin}(c x))^3}{b}+6 g^2 (a+b \text {ArcSin}(c x))^2 \log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+6 g^2 (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+\frac {24 b c g (-e f+d g) \left (i (a+b \text {ArcSin}(c x)) \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+b \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}+\frac {6 b c^2 (e f-d g)^2 \left (\frac {e \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c d+c e x}-b \log (d+e x)+\frac {c d \left (-i (a+b \text {ArcSin}(c x)) \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-12 b g^2 \left (i (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )\right )-12 b g^2 \left (i (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{6 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.98, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (e x +d \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (d + e x\right )^{3}}\, 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 {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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