Optimal. Leaf size=1137 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.99994, antiderivative size = 1137, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 14, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.424, Rules used = {4777, 4775, 4773, 3318, 4184, 3717, 2190, 2279, 2391, 3323, 2264, 2531, 2282, 6589} \[ \frac{2 i \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{-i \sin ^{-1}(c x)}\right ) b^2}{d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{2 i \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) b^2}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 i g^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) b^2}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 i g^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) b^2}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right ) b}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) b}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) b}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) b}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4777
Rule 4775
Rule 4773
Rule 3318
Rule 4184
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 3323
Rule 2264
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x) \left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (-\frac{c \left (a+b \sin ^{-1}(c x)\right )^2}{2 (c f+g) (-1+c x) \sqrt{1-c^2 x^2}}+\frac{c \left (a+b \sin ^{-1}(c x)\right )^2}{2 (c f-g) (1+c x) \sqrt{1-c^2 x^2}}+\frac{g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(-c f+g) (c f+g) (f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (c \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt{1-c^2 x^2}} \, dx}{2 d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (c \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x) \sqrt{1-c^2 x^2}} \, dx}{2 d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{d (-c f+g) (c f+g) \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{-c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{d (-c f+g) (c f+g) \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{d-c^2 d x^2}}\\ &=-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 i g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}+\frac{\left (2 i g^3 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}\\ &=-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-i x} (a+b x)}{1-i e^{-i x}} \, dx,x,\sin ^{-1}(c x)\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}\\ &=-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{-i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}\\ &=-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{-i \sin ^{-1}(c x)}\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{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 )}{d (-c f+g) (c f+g) \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}\\ &=-\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 4.96684, size = 597, normalized size = 0.53 \[ \frac{\sqrt{1-c^2 x^2} \left (\frac{2 i g^2 \left (-2 i b \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )+2 i b \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+2 b^2 \text{PolyLog}\left (3,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )-2 b^2 \text{PolyLog}\left (3,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )}{(c f-g) (c f+g) \sqrt{c^2 f^2-g^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \left (a \tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )-i a+4 b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+b \sin ^{-1}(c x) \left (\tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )-i\right )\right )-4 i b^2 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c f+g}+\frac{4 i b^2 \text{PolyLog}\left (2,-i e^{-i \sin ^{-1}(c x)}\right )-\left (a+b \sin ^{-1}(c x)\right ) \left (a \cot \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )-i a-4 b \log \left (1+i e^{-i \sin ^{-1}(c x)}\right )+b \sin ^{-1}(c x) \left (\cot \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )-i\right )\right )}{c f-g}\right )}{2 d \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 2.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{gx+f} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}}\,{d x} \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{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} g x^{5} + c^{4} d^{2} f x^{4} - 2 \, c^{2} d^{2} g x^{3} - 2 \, c^{2} d^{2} f x^{2} + d^{2} g x + d^{2} 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 (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (f + g x\right )}\, 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 (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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