Optimal. Leaf size=108 \[ \frac {a \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac {\text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}-\frac {1-2 (a+b x)^2}{2 b^2 \sin ^{-1}(a+b x)}-\frac {x \sqrt {1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2} \]
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Rubi [A] time = 0.27, antiderivative size = 151, normalized size of antiderivative = 1.40, number of steps used = 14, number of rules used = 12, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {4805, 4745, 4621, 4719, 4623, 3302, 4633, 4635, 4406, 12, 3299, 4641} \[ \frac {a \text {CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac {\text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}+\frac {(a+b x)^2}{b^2 \sin ^{-1}(a+b x)}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}-\frac {\sqrt {1-(a+b x)^2} (a+b x)}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {1}{2 b^2 \sin ^{-1}(a+b x)}+\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 4406
Rule 4621
Rule 4623
Rule 4633
Rule 4635
Rule 4641
Rule 4719
Rule 4745
Rule 4805
Rubi steps
\begin {align*} \int \frac {x}{\sin ^{-1}(a+b x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-\frac {a}{b}+\frac {x}{b}}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b \sin ^{-1}(x)^3}+\frac {x}{b \sin ^{-1}(x)^3}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b^2}-\frac {\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^2}+\frac {a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {1}{2 b^2 \sin ^{-1}(a+b x)}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}+\frac {(a+b x)^2}{b^2 \sin ^{-1}(a+b x)}-\frac {2 \operatorname {Subst}\left (\int \frac {x}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^2}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {1}{2 b^2 \sin ^{-1}(a+b x)}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}+\frac {(a+b x)^2}{b^2 \sin ^{-1}(a+b x)}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}+\frac {a \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {1}{2 b^2 \sin ^{-1}(a+b x)}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}+\frac {(a+b x)^2}{b^2 \sin ^{-1}(a+b x)}+\frac {a \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {1}{2 b^2 \sin ^{-1}(a+b x)}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}+\frac {(a+b x)^2}{b^2 \sin ^{-1}(a+b x)}+\frac {a \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {a \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{2 b^2 \sin ^{-1}(a+b x)^2}-\frac {1}{2 b^2 \sin ^{-1}(a+b x)}-\frac {a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}+\frac {(a+b x)^2}{b^2 \sin ^{-1}(a+b x)}+\frac {a \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^2}-\frac {\text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 121, normalized size = 1.12 \[ -\frac {x \sqrt {-a^2-2 a b x-b^2 x^2+1}}{2 b \sin ^{-1}(a+b x)^2}+\frac {a^2+3 a b x+2 b^2 x^2-1}{2 b^2 \sin ^{-1}(a+b x)}-2 \left (\frac {\text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{2 b^2}-\frac {a \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^2}\right )-\frac {3 a \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\arcsin \left (b x + a\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 139, normalized size = 1.29 \[ \frac {a \operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b^{2}} - \frac {{\left (b x + a\right )} a}{2 \, b^{2} \arcsin \left (b x + a\right )} - \frac {\operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{2}} + \frac {{\left (b x + a\right )}^{2} - 1}{b^{2} \arcsin \left (b x + a\right )} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{2 \, b^{2} \arcsin \left (b x + a\right )^{2}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a}{2 \, b^{2} \arcsin \left (b x + a\right )^{2}} + \frac {1}{2 \, b^{2} \arcsin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 109, normalized size = 1.01 \[ \frac {-\frac {\sin \left (2 \arcsin \left (b x +a \right )\right )}{4 \arcsin \left (b x +a \right )^{2}}-\frac {\cos \left (2 \arcsin \left (b x +a \right )\right )}{2 \arcsin \left (b x +a \right )}-\Si \left (2 \arcsin \left (b x +a \right )\right )+\frac {a \left (\Ci \left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{2}} \]
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 \[ -\frac {b \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2} \int \frac {4 \, b x + 3 \, a}{\arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}\,{d x} + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x - {\left (2 \, b^{2} x^{2} + 3 \, a b x + a^{2} - 1\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}{2 \, b^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2}} \]
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 \[ \int \frac {x}{{\mathrm {asin}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {x}{\operatorname {asin}^{3}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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