Optimal. Leaf size=15 \[ \frac {\sin ^{-1}(a+b x)^3}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4807, 4641} \[ \frac {\sin ^{-1}(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4641
Rule 4807
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)^2}{\sqrt {1-a^2-2 a b x-b^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sin ^{-1}(a+b x)^3}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \[ \frac {\sin ^{-1}(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 13, normalized size = 0.87 \[ \frac {\arcsin \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.32, size = 13, normalized size = 0.87 \[ \frac {\arcsin \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 14, normalized size = 0.93 \[ \frac {\arcsin \left (b x +a \right )^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.44, size = 130, normalized size = 8.67 \[ -\frac {\arcsin \left (b x + a\right )^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \frac {\arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{b} - \frac {\arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.28, size = 13, normalized size = 0.87 \[ \frac {{\mathrm {asin}\left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.75, size = 26, normalized size = 1.73 \[ \begin {cases} \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\\frac {x \operatorname {asin}^{2}{\relax (a )}}{\sqrt {1 - a^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________