Optimal. Leaf size=130 \[ \frac {2 a x}{b}-\frac {(a+b x)^2}{4 b^2}-\frac {2 a \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{b^2}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{2 b^2}-\frac {\text {ArcSin}(a+b x)^2}{4 b^2}-\frac {a^2 \text {ArcSin}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \text {ArcSin}(a+b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4889, 4827,
4847, 4737, 4767, 8, 4795, 30} \begin {gather*} -\frac {a^2 \text {ArcSin}(a+b x)^2}{2 b^2}+\frac {\sqrt {1-(a+b x)^2} (a+b x) \text {ArcSin}(a+b x)}{2 b^2}-\frac {\text {ArcSin}(a+b x)^2}{4 b^2}-\frac {2 a \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{b^2}+\frac {1}{2} x^2 \text {ArcSin}(a+b x)^2-\frac {(a+b x)^2}{4 b^2}+\frac {2 a x}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 4737
Rule 4767
Rule 4795
Rule 4827
Rule 4847
Rule 4889
Rubi steps
\begin {align*} \int x \sin ^{-1}(a+b x)^2 \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2-\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2-\text {Subst}\left (\int \left (\frac {a^2 \sin ^{-1}(x)}{b^2 \sqrt {1-x^2}}-\frac {2 a x \sin ^{-1}(x)}{b^2 \sqrt {1-x^2}}+\frac {x^2 \sin ^{-1}(x)}{b^2 \sqrt {1-x^2}}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2-\frac {\text {Subst}\left (\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^2}+\frac {(2 a) \text {Subst}\left (\int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^2}-\frac {a^2 \text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac {2 a \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^2}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^2}-\frac {a^2 \sin ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2-\frac {\text {Subst}(\int x \, dx,x,a+b x)}{2 b^2}-\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b^2}+\frac {(2 a) \text {Subst}(\int 1 \, dx,x,a+b x)}{b^2}\\ &=\frac {2 a x}{b}-\frac {(a+b x)^2}{4 b^2}-\frac {2 a \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^2}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^2}-\frac {\sin ^{-1}(a+b x)^2}{4 b^2}-\frac {a^2 \sin ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \sin ^{-1}(a+b x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 83, normalized size = 0.64 \begin {gather*} \frac {b x (6 a-b x)-2 (3 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \text {ArcSin}(a+b x)+\left (-1-2 a^2+2 b^2 x^2\right ) \text {ArcSin}(a+b x)^2}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 124, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\arcsin \left (b x +a \right )^{2} \left (-1+\left (b x +a \right )^{2}\right )}{2}+\frac {\arcsin \left (b x +a \right ) \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{2}-\frac {\arcsin \left (b x +a \right )^{2}}{4}-\frac {\left (b x +a \right )^{2}}{4}-a \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{2}}\) | \(124\) |
default | \(\frac {\frac {\arcsin \left (b x +a \right )^{2} \left (-1+\left (b x +a \right )^{2}\right )}{2}+\frac {\arcsin \left (b x +a \right ) \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{2}-\frac {\arcsin \left (b x +a \right )^{2}}{4}-\frac {\left (b x +a \right )^{2}}{4}-a \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{2}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.66, size = 80, normalized size = 0.62 \begin {gather*} -\frac {b^{2} x^{2} - 6 \, a b x - {\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2} - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - 3 \, a\right )} \arcsin \left (b x + a\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.17, size = 138, normalized size = 1.06 \begin {gather*} \begin {cases} - \frac {a^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a x}{2 b} - \frac {3 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{2 b^{2}} + \frac {x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{2} - \frac {x^{2}}{4} + \frac {x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{2 b} - \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asin}^{2}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 139, normalized size = 1.07 \begin {gather*} -\frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{b^{2}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{2 \, b^{2}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{2 \, b^{2}} - \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left (b x + a\right )} a}{b^{2}} + \frac {\arcsin \left (b x + a\right )^{2}}{4 \, b^{2}} - \frac {{\left (b x + a\right )}^{2} - 1}{4 \, b^{2}} - \frac {1}{8 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\mathrm {asin}\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________