Optimal. Leaf size=90 \[ -\frac{\text{CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )}{b}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a+b x)\right )}{b}-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)} \]
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Rubi [A] time = 0.283546, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4807, 4659, 4721, 4661, 3312, 3302, 4723, 4406} \[ -\frac{\text{CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )}{b}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a+b x)\right )}{b}-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 4807
Rule 4659
Rule 4721
Rule 4661
Rule 3312
Rule 3302
Rule 4723
Rule 4406
Rubi steps
\begin{align*} \int \frac{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}}{\sin ^{-1}(a+b x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{3/2}}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}-\frac{2 \operatorname{Subst}\left (\int \frac{x \left (1-x^2\right )}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}+\frac{8 \operatorname{Subst}\left (\int \frac{x^2 \sqrt{1-x^2}}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}+\frac{8 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{1}{8 x}-\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac{\left (1-(a+b x)^2\right )^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left (1-(a+b x)^2\right )^{3/2}}{b \sin ^{-1}(a+b x)}-\frac{\text{Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b}-\frac{\text{Ci}\left (4 \sin ^{-1}(a+b x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.43503, size = 110, normalized size = 1.22 \[ -\frac{\frac{\left (a^2+2 a b x+b^2 x^2-1\right ) \left (4 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+a^2+2 a b x+b^2 x^2-1\right )}{\sin ^{-1}(a+b x)^2}+2 \text{CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )+2 \text{CosIntegral}\left (4 \sin ^{-1}(a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 108, normalized size = 1.2 \begin{align*} -{\frac{16\,{\it Ci} \left ( 2\,\arcsin \left ( bx+a \right ) \right ) \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}+16\,{\it Ci} \left ( 4\,\arcsin \left ( bx+a \right ) \right ) \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}-8\,\sin \left ( 2\,\arcsin \left ( bx+a \right ) \right ) \arcsin \left ( bx+a \right ) -4\,\sin \left ( 4\,\arcsin \left ( bx+a \right ) \right ) \arcsin \left ( bx+a \right ) +4\,\cos \left ( 2\,\arcsin \left ( bx+a \right ) \right ) +\cos \left ( 4\,\arcsin \left ( bx+a \right ) \right ) +3}{16\,b \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}}{\arcsin \left (b x + a\right )^{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 (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac{3}{2}}}{\operatorname{asin}^{3}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30654, size = 136, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}{\left (b x + a\right )}}{b \arcsin \left (b x + a\right )} - \frac{\operatorname{Ci}\left (4 \, \arcsin \left (b x + a\right )\right )}{b} - \frac{\operatorname{Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{b} - \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{2 \, b \arcsin \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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