Optimal. Leaf size=130 \[ \frac {\sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b^3 c^2}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b^3 c^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4633, 4719, 4635, 4406, 12, 3303, 3299, 3302, 4641} \[ \frac {\sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^3 c^2}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^3 c^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 4633
Rule 4635
Rule 4641
Rule 4719
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sin ^{-1}(c x)\right )^3} \, dx &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}+\frac {\int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx}{2 b c}-\frac {c \int \frac {x^2}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx}{b}\\ &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 \int \frac {x}{a+b \sin ^{-1}(c x)} \, dx}{b^2}\\ &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\sin ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b^2 c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \sin ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \sin ^{-1}(c x)\right )}+\frac {\text {Ci}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 c^2}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^3 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.39, size = 108, normalized size = 0.83 \[ \frac {-\frac {b^2 c x \sqrt {1-c^2 x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac {b \left (2 c^2 x^2-1\right )}{a+b \sin ^{-1}(c x)}+2 \sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (2 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )}{2 b^3 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.52, size = 864, normalized size = 6.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 157, normalized size = 1.21 \[ \frac {-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{4 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {2 \Si \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) \arcsin \left (c x \right ) b -2 \Ci \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) \arcsin \left (c x \right ) b +2 \Si \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -2 \Ci \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +\cos \left (2 \arcsin \left (c x \right )\right ) b}{2 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, a c^{2} x^{2} - \sqrt {c x + 1} \sqrt {-c x + 1} b c x + {\left (2 \, b c^{2} x^{2} - b\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - a - \frac {4 \, {\left (b^{4} c^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b^{3} c^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a^{2} b^{2} c^{2}\right )} \int \frac {x}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x}}{b^{2}}}{2 \, {\left (b^{4} c^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b^{3} c^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a^{2} b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________