Optimal. Leaf size=333 \[ -\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{4 b c \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.437249, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {4701, 4653, 4675, 3719, 2190, 2279, 2391, 4679, 4419, 4183} \[ -\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{4 b c \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4701
Rule 4653
Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 4679
Rule 4419
Rule 4183
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\left (2 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (2 b c \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (8 i b c \sqrt{1-c^2 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 )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (i b^2 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (i b^2 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{4 b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.703594, size = 322, normalized size = 0.97 \[ \frac{-i b^2 c x \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )-i b^2 c x \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+2 a^2 c^2 x^2-a^2+2 a b c x \sqrt{1-c^2 x^2} \log (c x)+a b c x \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )+4 a b c^2 x^2 \sin ^{-1}(c x)-2 a b \sin ^{-1}(c x)+2 b^2 c^2 x^2 \sin ^{-1}(c x)^2-2 i b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+2 b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )-b^2 \sin ^{-1}(c x)^2}{d x \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.207, size = 807, normalized size = 2.4 \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{\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} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} 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 (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{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 (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]