Optimal. Leaf size=687 \[ \frac{2 i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{598}{225} b^2 d^2 \sqrt{d-c^2 d x^2}-\frac{2}{125} b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}-\frac{74}{675} b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \]
[Out]
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Rubi [A] time = 0.885807, antiderivative size = 687, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 16, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.552, Rules used = {4699, 4697, 4709, 4183, 2531, 2282, 6589, 4619, 261, 4645, 444, 43, 194, 12, 1247, 698} \[ \frac{2 i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{598}{225} b^2 d^2 \sqrt{d-c^2 d x^2}-\frac{2}{125} b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}-\frac{74}{675} b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4699
Rule 4697
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4619
Rule 261
Rule 4645
Rule 444
Rule 43
Rule 194
Rule 12
Rule 1247
Rule 698
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1-c^2 x^2}} \, dx}{5 \sqrt{1-c^2 x^2}}\\ &=-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{75 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c d^2 \sqrt{d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{15-10 c^2 x+3 c^4 x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{c^2 x}{3}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (2 b d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1-c^2 x}}+4 \sqrt{1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-c^2 x}}+\frac{1}{3} \sqrt{1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{598}{225} b^2 d^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{74}{675} b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}-\frac{2}{125} b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 i b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (2 i b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{598}{225} b^2 d^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{74}{675} b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}-\frac{2}{125} b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{598}{225} b^2 d^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{74}{675} b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}-\frac{2}{125} b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}+\frac{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 4.53179, size = 775, normalized size = 1.13 \[ \frac{d^2 \left (-108000 a b \sqrt{d-c^2 d x^2} \left (-i \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right )-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x-\sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )-54000 b^2 \sqrt{d-c^2 d x^2} \left (-2 i \sin ^{-1}(c x) \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right )+2 \left (\text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )\right )+2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 c x \sin ^{-1}(c x)+\sin ^{-1}(c x)^2 \left (-\left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )\right )+3600 a^2 \sqrt{1-c^2 x^2} \left (3 c^4 x^4-11 c^2 x^2+23\right ) \sqrt{d-c^2 d x^2}+54000 a^2 \sqrt{d} \sqrt{1-c^2 x^2} \log (c x)-54000 a^2 \sqrt{d} \sqrt{1-c^2 x^2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-6000 a b \sqrt{d-c^2 d x^2} \left (-3 \sin ^{-1}(c x) \left (3 \sqrt{1-c^2 x^2}+\cos \left (3 \sin ^{-1}(c x)\right )\right )+9 c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )+30 a b \sqrt{d-c^2 d x^2} \left (-15 \sin ^{-1}(c x) \left (30 \sqrt{1-c^2 x^2}+5 \cos \left (3 \sin ^{-1}(c x)\right )-3 \cos \left (5 \sin ^{-1}(c x)\right )\right )+450 c x+25 \sin \left (3 \sin ^{-1}(c x)\right )-9 \sin \left (5 \sin ^{-1}(c x)\right )\right )+1000 b^2 \sqrt{d-c^2 d x^2} \left (27 \sqrt{1-c^2 x^2} \left (\sin ^{-1}(c x)^2-2\right )-6 \sin ^{-1}(c x) \left (9 c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )+\left (9 \sin ^{-1}(c x)^2-2\right ) \cos \left (3 \sin ^{-1}(c x)\right )\right )-b^2 \sqrt{d-c^2 d x^2} \left (6750 \sqrt{1-c^2 x^2} \left (\sin ^{-1}(c x)^2-2\right )+30 \sin ^{-1}(c x) \left (9 \left (\sin \left (5 \sin ^{-1}(c x)\right )-50 c x\right )-25 \sin \left (3 \sin ^{-1}(c x)\right )\right )+125 \left (9 \sin ^{-1}(c x)^2-2\right ) \cos \left (3 \sin ^{-1}(c x)\right )-27 \left (25 \sin ^{-1}(c x)^2-2\right ) \cos \left (5 \sin ^{-1}(c x)\right )\right )\right )}{54000 \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.415, size = 1574, normalized size = 2.3 \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^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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