Optimal. Leaf size=170 \[ -\frac{2 d \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}},x\right )}{b c}-\frac{2 \sqrt{\pi } d \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{b^{3/2}}-\frac{2 \sqrt{\pi } d \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2}}-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.778696, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(2 d) \int \frac{\sqrt{1-c^2 x^2}}{x^2 \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{(4 c d) \int \frac{\sqrt{1-c^2 x^2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(4 d) \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \left (-\frac{c^2}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}+\frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(4 d) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}+\frac{\cos (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}+\frac{(2 c d) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (2 d \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac{\left (2 d \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (4 d \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}-\frac{\left (4 d \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2}\\ &=-\frac{2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}-\frac{2 d \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2}}-\frac{2 d \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{b^{3/2}}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ \end{align*}
Mathematica [A] time = 1.8356, size = 0, normalized size = 0. \[ \int \frac{d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.3, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{c}^{2}d{x}^{2}+d}{x} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} - d \left (\int \frac{c^{2} x^{2}}{a x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int - \frac{1}{a x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]