Optimal. Leaf size=935 \[ -\frac{i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 d (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 d (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 (e f-d g) \log (d+e x) c^2}{e^2 \left (c^2 d^2-e^2\right )}+\frac{b^2 (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x) c}{e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b \left (2 e^2 g-c^2 d (e f+d g)\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) c}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 g \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 g \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{a b (e f-d g) \sqrt{1-c^2 x^2} c}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)} \]
[Out]
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Rubi [A] time = 2.93444, antiderivative size = 935, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 20, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.87, Rules used = {37, 4755, 12, 1651, 844, 216, 725, 204, 4799, 4797, 4641, 4773, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 d (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{b^2 d (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 (e f-d g) \log (d+e x) c^2}{e^2 \left (c^2 d^2-e^2\right )}+\frac{b^2 (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x) c}{e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{a b \left (2 e^2 g-c^2 d (e f+d g)\right ) \tan ^{-1}\left (\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right ) c}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 g \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 g \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right ) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{a b (e f-d g) \sqrt{1-c^2 x^2} c}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 4755
Rule 12
Rule 1651
Rule 844
Rule 216
Rule 725
Rule 204
Rule 4799
Rule 4797
Rule 4641
Rule 4773
Rule 3324
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-(2 b c) \int -\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{2 (e f-d g) (d+e x)^2 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{(b c) \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e f-d g}\\ &=-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{(b c) \int \left (\frac{a (f+g x)^2}{(d+e x)^2 \sqrt{1-c^2 x^2}}+\frac{b (f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^2 \sqrt{1-c^2 x^2}}\right ) \, dx}{e f-d g}\\ &=-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{(a b c) \int \frac{(f+g x)^2}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e f-d g}+\frac{\left (b^2 c\right ) \int \frac{(f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e f-d g}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{\left (b^2 c\right ) \int \left (\frac{g^2 \sin ^{-1}(c x)}{e^2 \sqrt{1-c^2 x^2}}+\frac{(e f-d g)^2 \sin ^{-1}(c x)}{e^2 (d+e x)^2 \sqrt{1-c^2 x^2}}+\frac{2 g (e f-d g) \sin ^{-1}(c x)}{e^2 (d+e x) \sqrt{1-c^2 x^2}}\right ) \, dx}{e f-d g}+\frac{(a b c) \int \frac{c^2 d f^2-g (2 e f-d g)+\left (\frac{c^2 d^2}{e}-e\right ) g^2 x}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{\left (c^2 d^2-e^2\right ) (e f-d g)}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{\left (2 b^2 c g\right ) \int \frac{\sin ^{-1}(c x)}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^2}+\frac{\left (a b c g^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{e^2 (e f-d g)}+\frac{\left (b^2 c g^2\right ) \int \frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{e^2 (e f-d g)}+\frac{\left (b^2 c (e f-d g)\right ) \int \frac{\sin ^{-1}(c x)}{(d+e x)^2 \sqrt{1-c^2 x^2}} \, dx}{e^2}-\frac{\left (a b c \left (2 e^2 g-c^2 d (e f+d g)\right )\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac{\left (2 b^2 c g\right ) \operatorname{Subst}\left (\int \frac{x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{\left (b^2 c^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x}{(c d+e \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac{\left (a b c \left (2 e^2 g-c^2 d (e f+d g)\right )\right ) \operatorname{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^2 \left (c^2 d^2-e^2\right )}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 c (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-\frac{a b c \left (2 e^2 g-c^2 d (e f+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^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{\left (4 b^2 c g\right ) \operatorname{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^2}+\frac{\left (b^2 c^3 d (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{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 (b^2 c^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 c (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-\frac{a b c \left (2 e^2 g-c^2 d (e f+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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (4 i b^2 c g\right ) \operatorname{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 \sqrt{c^2 d^2-e^2}}+\frac{\left (4 i b^2 c g\right ) \operatorname{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 \sqrt{c^2 d^2-e^2}}-\frac{\left (b^2 c^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{c d+x} \, dx,x,c e x\right )}{e^2 \left (c^2 d^2-e^2\right )}+\frac{\left (2 b^2 c^3 d (e f-d g)\right ) \operatorname{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^2 \left (c^2 d^2-e^2\right )}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 c (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-\frac{a b c \left (2 e^2 g-c^2 d (e f+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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}-\frac{b^2 c^2 (e f-d g) \log (d+e x)}{e^2 \left (c^2 d^2-e^2\right )}+\frac{\left (2 i b^2 c g\right ) \operatorname{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^2 \sqrt{c^2 d^2-e^2}}-\frac{\left (2 i b^2 c g\right ) \operatorname{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^2 \sqrt{c^2 d^2-e^2}}-\frac{\left (2 i b^2 c^3 d (e f-d g)\right ) \operatorname{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 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{\left (2 i b^2 c^3 d (e f-d g)\right ) \operatorname{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 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 c (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-\frac{a b c \left (2 e^2 g-c^2 d (e f+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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}-\frac{i b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{i b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 c^2 (e f-d g) \log (d+e x)}{e^2 \left (c^2 d^2-e^2\right )}+\frac{\left (2 b^2 c g\right ) \operatorname{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^2 \sqrt{c^2 d^2-e^2}}-\frac{\left (2 b^2 c g\right ) \operatorname{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^2 \sqrt{c^2 d^2-e^2}}+\frac{\left (i b^2 c^3 d (e f-d g)\right ) \operatorname{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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (i b^2 c^3 d (e f-d g)\right ) \operatorname{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^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 c (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-\frac{a b c \left (2 e^2 g-c^2 d (e f+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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}-\frac{i b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{i b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 c^2 (e f-d g) \log (d+e x)}{e^2 \left (c^2 d^2-e^2\right )}-\frac{2 b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{\left (b^2 c^3 d (e f-d g)\right ) \operatorname{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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{\left (b^2 c^3 d (e f-d g)\right ) \operatorname{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^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=\frac{a b c (e f-d g) \sqrt{1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}+\frac{b^2 c (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}-\frac{a b c \left (2 e^2 g-c^2 d (e f+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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}-\frac{i b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 i b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{i b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac{b^2 c^2 (e f-d g) \log (d+e x)}{e^2 \left (c^2 d^2-e^2\right )}-\frac{2 b^2 c 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^2 \sqrt{c^2 d^2-e^2}}-\frac{b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac{2 b^2 c 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^2 \sqrt{c^2 d^2-e^2}}+\frac{b^2 c^3 d (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^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.60271, size = 574, normalized size = 0.61 \[ \frac{\frac{2 b c (e f-d g) \left (-i c^2 d (d+e x) \left (-i b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+i b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )+e \sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2} \left (a+b \sin ^{-1}(c x)\right )-b c \sqrt{c^2 d^2-e^2} (d+e x) \log (d+e x)\right )}{\left (c^2 d^2-e^2\right )^{3/2} (d+e x)}+\frac{4 b c g \left (-b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+b \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )}{\sqrt{c^2 d^2-e^2}}-\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2}-\frac{2 g \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}}{2 e^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.931, size = 3105, normalized size = 3.3 \begin{align*} \text{output 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{a^{2} g x + a^{2} f +{\left (b^{2} g x + b^{2} f\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g x + a b f\right )} \arcsin \left (c x\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, 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 (f + g x\right )}{\left (d + e x\right )^{3}}\, 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 (g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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