Optimal. Leaf size=84 \[ -\frac {\left (4 a^2+1\right ) \text {Si}\left (\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {2 a \text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}+\frac {3 \text {Si}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {x^2 \sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)} \]
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Rubi [A] time = 0.23, antiderivative size = 161, normalized size of antiderivative = 1.92, number of steps used = 12, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4805, 4745, 4621, 4723, 3299, 4631, 3302} \[ -\frac {a^2 \text {Si}\left (\sin ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {2 a \text {CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {3 \text {Si}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {2 a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 4621
Rule 4631
Rule 4723
Rule 4745
Rule 4805
Rubi steps
\begin {align*} \int \frac {x^2}{\sin ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 \sin ^{-1}(x)^2}-\frac {2 a x}{b^2 \sin ^{-1}(x)^2}+\frac {x^2}{b^2 \sin ^{-1}(x)^2}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {x}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{4 x}+\frac {3 \sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {2 a \text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)}-\frac {2 a \text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a^2 \text {Si}\left (\sin ^{-1}(a+b x)\right )}{b^3}+\frac {3 \text {Si}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 86, normalized size = 1.02 \[ -\frac {\frac {4 b^2 x^2 \sqrt {-a^2-2 a b x-b^2 x^2+1}}{\sin ^{-1}(a+b x)}+\left (4 a^2+1\right ) \text {Si}\left (\sin ^{-1}(a+b x)\right )+8 a \text {Ci}\left (2 \sin ^{-1}(a+b x)\right )-3 \text {Si}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arcsin \left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 169, normalized size = 2.01 \[ -\frac {a^{2} \operatorname {Si}\left (\arcsin \left (b x + a\right )\right )}{b^{3}} - \frac {2 \, a \operatorname {Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{b^{3} \arcsin \left (b x + a\right )} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3} \arcsin \left (b x + a\right )} + \frac {3 \, \operatorname {Si}\left (3 \, \arcsin \left (b x + a\right )\right )}{4 \, b^{3}} - \frac {\operatorname {Si}\left (\arcsin \left (b x + a\right )\right )}{4 \, b^{3}} + \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{b^{3} \arcsin \left (b x + a\right )} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{b^{3} \arcsin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 149, normalized size = 1.77 \[ \frac {-\frac {a \left (2 \Ci \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )-\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{\arcsin \left (b x +a \right )}-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{4 \arcsin \left (b x +a \right )}-\frac {\Si \left (\arcsin \left (b x +a \right )\right )}{4}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{4 \arcsin \left (b x +a \right )}+\frac {3 \Si \left (3 \arcsin \left (b x +a \right )\right )}{4}-\frac {a^{2} \left (\Si \left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{\arcsin \left (b x +a \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {asin}\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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