Optimal. Leaf size=424 \[ -\frac{i b^2 c d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}-\frac{i c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac{2 b c d \sqrt{d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.404278, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {4695, 4647, 4641, 4627, 321, 216, 4683, 4625, 3717, 2190, 2279, 2391, 195} \[ -\frac{i b^2 c d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}-\frac{i c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac{2 b c d \sqrt{d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4695
Rule 4647
Rule 4641
Rule 4627
Rule 321
Rule 216
Rule 4683
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 195
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (3 c^2 d\right ) \int \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx}{\sqrt{1-c^2 x^2}}\\ &=b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{1}{2} b^2 c^2 d x \sqrt{d-c^2 d x^2}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 c^4 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{2 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}-\frac{\left (4 i b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}+\frac{2 b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}+\frac{2 b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (i b^2 c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}+\frac{2 b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{i b^2 c d \sqrt{d-c^2 d x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 2.41895, size = 396, normalized size = 0.93 \[ \frac{-8 b^2 d \sqrt{d-c^2 d x^2} \left (3 i c x \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x) \left (3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x \left (\sin ^{-1}(c x)+3 i\right ) \sin ^{-1}(c x)-6 c x \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )\right )\right )+36 a^2 c d^{3/2} x \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-12 a^2 d \sqrt{1-c^2 x^2} \left (c^2 x^2+2\right ) \sqrt{d-c^2 d x^2}-24 a b d \sqrt{d-c^2 d x^2} \left (2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 c x \log (c x)+c x \sin ^{-1}(c x)^2\right )-6 a b c d x \sqrt{d-c^2 d x^2} \left (2 \sin ^{-1}(c x) \left (\sin ^{-1}(c x)+\sin \left (2 \sin ^{-1}(c x)\right )\right )+\cos \left (2 \sin ^{-1}(c x)\right )\right )-b^2 c d x \sqrt{d-c^2 d x^2} \left (4 \sin ^{-1}(c x)^3+\left (6 \sin ^{-1}(c x)^2-3\right ) \sin \left (2 \sin ^{-1}(c x)\right )+6 \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )\right )}{24 x \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.324, size = 1148, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )^{2}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]