Optimal. Leaf size=60 \[ \frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^3}+\frac {\text {Ci}\left (\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {\text {Ci}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^3} \]
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Rubi [A] time = 0.61, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4805, 4747, 6741, 12, 6742, 3302, 4406, 3299} \[ \frac {a^2 \text {CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{b^3}+\frac {\text {CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {\text {CosIntegral}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 4406
Rule 4747
Rule 4805
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2}{\sin ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \left (-\frac {a}{b}+\frac {\sin (x)}{b}\right )^2}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) (a-\sin (x))^2}{b^2 x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) (a-\sin (x))^2}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 \cos (x)}{x}-\frac {2 a \cos (x) \sin (x)}{x}+\frac {\cos (x) \sin ^2(x)}{x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^3}+\frac {\operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^3}+\frac {\operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {\operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\text {Ci}\left (\sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Ci}\left (3 \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 45, normalized size = 0.75 \[ -\frac {-\left (\left (4 a^2+1\right ) \text {Ci}\left (\sin ^{-1}(a+b x)\right )\right )+\text {Ci}\left (3 \sin ^{-1}(a+b x)\right )+4 a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arcsin \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 56, normalized size = 0.93 \[ \frac {a^{2} \operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{b^{3}} - \frac {a \operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{3}} - \frac {\operatorname {Ci}\left (3 \, \arcsin \left (b x + a\right )\right )}{4 \, b^{3}} + \frac {\operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 49, normalized size = 0.82 \[ \frac {-a \Si \left (2 \arcsin \left (b x +a \right )\right )+\frac {\Ci \left (\arcsin \left (b x +a \right )\right )}{4}-\frac {\Ci \left (3 \arcsin \left (b x +a \right )\right )}{4}+a^{2} \Ci \left (\arcsin \left (b x +a \right )\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\arcsin \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\mathrm {asin}\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {asin}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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