Optimal. Leaf size=1073 \[ \frac{b d x^3 \sqrt{d-c^2 d x^2} c^3}{9 g \sqrt{1-c^2 x^2}}-\frac{b d f x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^2 \sqrt{1-c^2 x^2}}+\frac{d f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) c^2}{2 g^2}-\frac{d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt{1-c^2 x^2}}+\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt{1-c^2 x^2}}+\frac{b d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} c}{g^3 \sqrt{1-c^2 x^2}}-\frac{b d x \sqrt{d-c^2 d x^2} c}{3 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac{a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}+\frac{i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}+\frac{b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{a d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}-\frac{d (c f-g) (c f+g) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac{d \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt{1-c^2 x^2} c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.22419, antiderivative size = 1073, normalized size of antiderivative = 1., 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 4777
Rule 4767
Rule 4647
Rule 4641
Rule 30
Rule 4677
Rule 4765
Rule 683
Rule 4757
Rule 6742
Rule 725
Rule 204
Rule 1654
Rule 12
Rule 4799
Rule 4797
Rule 8
Rule 4773
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
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 = 1.46182, size = 507, normalized size = 0.47 \[ \frac{d \sqrt{d-c^2 d x^2} \left (-\frac{18 \left (c^2 f^2-g^2\right ) \left (-2 b c (f+g x) \left (-i \sqrt{c^2 f^2-g^2} \left (-i b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )+i b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )\right )-g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+b c g x\right )+\left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+c^2 g x (f+g x) \left (a+b \sin ^{-1}(c x)\right )^2\right )}{b c g^2 (f+g x)}+\frac{18 \left (c^2 x^2-1\right ) \left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{b c (f+g x)}+18 c^2 f x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+12 g \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{9 c f \left (a+b \sin ^{-1}(c x)\right )^2}{b}-9 b c^3 f x^2+4 b c g x \left (c^2 x^2-3\right )\right )}{36 g^2 \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.305, size = 2742, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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