3.372 \(\int x (a+b \sin ^{-1}(\frac{c}{x})) \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} b c x \sqrt{1-\frac{c^2}{x^2}} \]

[Out]

(b*c*Sqrt[1 - c^2/x^2]*x)/2 + (x^2*(a + b*ArcSin[c/x]))/2

________________________________________________________________________________________

Rubi [A]  time = 0.0174758, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4842, 12, 191} \[ \frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} b c x \sqrt{1-\frac{c^2}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c/x]),x]

[Out]

(b*c*Sqrt[1 - c^2/x^2]*x)/2 + (x^2*(a + b*ArcSin[c/x]))/2

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int x \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} b \int \frac{c}{\sqrt{1-\frac{c^2}{x^2}}} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{2} (b c) \int \frac{1}{\sqrt{1-\frac{c^2}{x^2}}} \, dx\\ &=\frac{1}{2} b c \sqrt{1-\frac{c^2}{x^2}} x+\frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0278053, size = 47, normalized size = 1.21 \[ \frac{a x^2}{2}+\frac{1}{2} b c x \sqrt{\frac{x^2-c^2}{x^2}}+\frac{1}{2} b x^2 \sin ^{-1}\left (\frac{c}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSin[c/x]),x]

[Out]

(a*x^2)/2 + (b*c*x*Sqrt[(-c^2 + x^2)/x^2])/2 + (b*x^2*ArcSin[c/x])/2

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 51, normalized size = 1.3 \begin{align*} -{c}^{2} \left ( -{\frac{a{x}^{2}}{2\,{c}^{2}}}+b \left ( -{\frac{{x}^{2}}{2\,{c}^{2}}\arcsin \left ({\frac{c}{x}} \right ) }-{\frac{x}{2\,c}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsin(c/x)),x)

[Out]

-c^2*(-1/2*a/c^2*x^2+b*(-1/2/c^2*x^2*arcsin(c/x)-1/2/c*x*(1-c^2/x^2)^(1/2)))

________________________________________________________________________________________

Maxima [A]  time = 1.42014, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{2} \,{\left (x^{2} \arcsin \left (\frac{c}{x}\right ) + c x \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c/x)),x, algorithm="maxima")

[Out]

1/2*a*x^2 + 1/2*(x^2*arcsin(c/x) + c*x*sqrt(-c^2/x^2 + 1))*b

________________________________________________________________________________________

Fricas [A]  time = 2.36696, size = 95, normalized size = 2.44 \begin{align*} \frac{1}{2} \, b x^{2} \arcsin \left (\frac{c}{x}\right ) + \frac{1}{2} \, b c x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c/x)),x, algorithm="fricas")

[Out]

1/2*b*x^2*arcsin(c/x) + 1/2*b*c*x*sqrt(-(c^2 - x^2)/x^2) + 1/2*a*x^2

________________________________________________________________________________________

Sympy [A]  time = 2.66627, size = 60, normalized size = 1.54 \begin{align*} \frac{a x^{2}}{2} + \frac{b c \left (\begin{cases} c \sqrt{-1 + \frac{x^{2}}{c^{2}}} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{c^{2}}\right |} > 1 \\i c \sqrt{1 - \frac{x^{2}}{c^{2}}} & \text{otherwise} \end{cases}\right )}{2} + \frac{b x^{2} \operatorname{asin}{\left (\frac{c}{x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asin(c/x)),x)

[Out]

a*x**2/2 + b*c*Piecewise((c*sqrt(-1 + x**2/c**2), Abs(x**2)/Abs(c**2) > 1), (I*c*sqrt(1 - x**2/c**2), True))/2
+ b*x**2*asin(c/x)/2

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (\frac{c}{x}\right ) + a\right )} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c/x)),x, algorithm="giac")

[Out]

integrate((b*arcsin(c/x) + a)*x, x)