Optimal. Leaf size=55 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \text {Si}\left (\sin ^{-1}(a+b x)\right )}{b^2}-\frac {x \sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)} \]
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Rubi [A] time = 0.14, antiderivative size = 87, normalized size of antiderivative = 1.58, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4805, 4745, 4621, 4723, 3299, 4631, 3302} \[ \frac {\text {CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \text {Si}\left (\sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)} \]
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}{\sin ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-\frac {a}{b}+\frac {x}{b}}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b \sin ^{-1}(x)^2}+\frac {x}{b \sin ^{-1}(x)^2}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}+\frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}+\frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \text {Si}\left (\sin ^{-1}(a+b x)\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 63, normalized size = 1.15 \[ \frac {\sin ^{-1}(a+b x) \text {Ci}\left (2 \sin ^{-1}(a+b x)\right )+a \sin ^{-1}(a+b x) \text {Si}\left (\sin ^{-1}(a+b x)\right )-b x \sqrt {1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\arcsin \left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.58, size = 83, normalized size = 1.51 \[ \frac {a \operatorname {Si}\left (\arcsin \left (b x + a\right )\right )}{b^{2}} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{b^{2} \arcsin \left (b x + a\right )} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2} \arcsin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 72, normalized size = 1.31 \[ \frac {-\frac {\sin \left (2 \arcsin \left (b x +a \right )\right )}{2 \arcsin \left (b x +a \right )}+\Ci \left (2 \arcsin \left (b x +a \right )\right )+\frac {a \left (\Si \left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{\arcsin \left (b x +a \right )}}{b^{2}} \]
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 = -0.02 \[ \int \frac {x}{{\mathrm {asin}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {x}{\operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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