Optimal. Leaf size=111 \[ -\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^3 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^3 c}+\frac {x}{2 b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2} \]
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Rubi [A] time = 0.17, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4621, 4719, 4623, 3303, 3299, 3302} \[ -\frac {\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^3 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^3 c}+\frac {x}{2 b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 4621
Rule 4623
Rule 4719
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin ^{-1}(c x)\right )^3} \, dx &=-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {c \int \frac {x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx}{2 b}\\ &=-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}+\frac {x}{2 b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\int \frac {1}{a+b \sin ^{-1}(c x)} \, dx}{2 b^2}\\ &=-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}+\frac {x}{2 b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{2 b^3 c}\\ &=-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}+\frac {x}{2 b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{2 b^3 c}-\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{2 b^3 c}\\ &=-\frac {\sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}+\frac {x}{2 b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^3 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^3 c}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 93, normalized size = 0.84 \[ -\frac {\frac {b \left (\frac {b \sqrt {1-c^2 x^2}}{c}-x \left (a+b \sin ^{-1}(c x)\right )\right )}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{c}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{c}}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.07, size = 482, normalized size = 4.34 \[ -\frac {b^{2} \arcsin \left (c x\right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {b^{2} \arcsin \left (c x\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac {b^{2} c x \arcsin \left (c x\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c} - \frac {a b \arcsin \left (c x\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c} + \frac {a b c x}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {\sqrt {-c^{2} x^{2} + 1} b^{2}}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 138, normalized size = 1.24 \[ \frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{2 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{2 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c} \]
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 \[ \frac {b c x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a c x - \sqrt {c x + 1} \sqrt {-c x + 1} b - {\left (b^{4} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b^{3} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a^{2} b^{2} c\right )} \int \frac {1}{b^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b^{2}}\,{d x}}{2 \, {\left (b^{4} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b^{3} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a^{2} b^{2} c\right )}} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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