Optimal. Leaf size=1073 \[ \frac {b d x^3 \sqrt {d-c^2 d x^2} c^3}{9 g \sqrt {1-c^2 x^2}}-\frac {b d f x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) c^2}{2 g^2}-\frac {d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt {1-c^2 x^2}}+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {b d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} c}{g^3 \sqrt {1-c^2 x^2}}-\frac {b d x \sqrt {d-c^2 d x^2} c}{3 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt {1-c^2 x^2} c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.22, antiderivative size = 1073, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 23, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.742, Rules used = {4777, 4767, 4647, 4641, 30, 4677, 4765, 683, 4757, 6742, 725, 204, 1654, 12, 4799, 4797, 8, 4773, 3323, 2264, 2190, 2279, 2391} \[ \frac {b d x^3 \sqrt {d-c^2 d x^2} c^3}{9 g \sqrt {1-c^2 x^2}}-\frac {b d f x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) c^2}{2 g^2}-\frac {d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt {1-c^2 x^2}}+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {b d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} c}{g^3 \sqrt {1-c^2 x^2}}-\frac {b d x \sqrt {d-c^2 d x^2} c}{3 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt {1-c^2 x^2} c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 204
Rule 683
Rule 725
Rule 1654
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3323
Rule 4641
Rule 4647
Rule 4677
Rule 4757
Rule 4765
Rule 4767
Rule 4773
Rule 4777
Rule 4797
Rule 4799
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^2}-\frac {c^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g}+\frac {\left (-c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^2 (f+g x)}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g \sqrt {1-c^2 x^2}}\\ &=\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1-c^2 x^2}}+\frac {\left (c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (b c^2 d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (b c d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1-c^2 x^2}}+\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \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 )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \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 )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d 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 )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d 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 )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.46, size = 507, normalized size = 0.47 \[ \frac {d \sqrt {d-c^2 d x^2} \left (-\frac {18 \left (c^2 f^2-g^2\right ) \left (-2 b c (f+g x) \left (-i \sqrt {c^2 f^2-g^2} \left (\left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )\right )-i b \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+i b \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+b c g x\right )+\left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+c^2 g x (f+g x) \left (a+b \sin ^{-1}(c x)\right )^2\right )}{b c g^2 (f+g x)}+\frac {18 \left (c^2 x^2-1\right ) \left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{b c (f+g x)}+18 c^2 f x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+12 g \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {9 c f \left (a+b \sin ^{-1}(c x)\right )^2}{b}-9 b c^3 f x^2+4 b c g x \left (c^2 x^2-3\right )\right )}{36 g^2 \sqrt {1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a c^{2} d x^{2} - a d + {\left (b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.55, size = 2742, normalized size = 2.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________