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]

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

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, 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

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fricas [A]  time = 0.56, size = 40, normalized size = 1.03 \[ \frac {1}{2} \, b x^{2} \arcsin \left (\frac {c}{x}\right ) + \frac {1}{2} \, b c x \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} + \frac {1}{2} \, a x^{2} \]

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

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giac [B]  time = 0.21, size = 174, normalized size = 4.46 \[ \frac {b c x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {c}{x}\right ) + a c x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} + 2 \, b c^{2} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} + 2 \, b c^{3} \arcsin \left (\frac {c}{x}\right ) + 2 \, a c^{3} - \frac {2 \, b c^{4}}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} + \frac {b c^{5} \arcsin \left (\frac {c}{x}\right )}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} + \frac {a c^{5}}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}}}{8 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*c*x^2*(sqrt(-c^2/x^2 + 1) + 1)^2*arcsin(c/x) + a*c*x^2*(sqrt(-c^2/x^2 + 1) + 1)^2 + 2*b*c^2*x*(sqrt(-c^
2/x^2 + 1) + 1) + 2*b*c^3*arcsin(c/x) + 2*a*c^3 - 2*b*c^4/(x*(sqrt(-c^2/x^2 + 1) + 1)) + b*c^5*arcsin(c/x)/(x^
2*(sqrt(-c^2/x^2 + 1) + 1)^2) + a*c^5/(x^2*(sqrt(-c^2/x^2 + 1) + 1)^2))/c

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maple [A]  time = 0.01, size = 51, normalized size = 1.31 \[ -c^{2} \left (-\frac {a \,x^{2}}{2 c^{2}}+b \left (-\frac {x^{2} \arcsin \left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{2 c}\right )\right ) \]

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)))

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maxima [A]  time = 0.66, size = 36, normalized size = 0.92 \[ \frac {1}{2} \, a x^{2} + \frac {1}{2} \, {\left (x^{2} \arcsin \left (\frac {c}{x}\right ) + c x \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )} b \]

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

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mupad [B]  time = 0.29, size = 36, normalized size = 0.92 \[ \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asin}\left (\frac {c}{x}\right )}{2}+\frac {b\,c\,x\,\sqrt {1-\frac {c^2}{x^2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c/x)),x)

[Out]

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

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sympy [A]  time = 1.93, size = 58, normalized size = 1.49 \[ \frac {a x^{2}}{2} + \frac {b c \left (\begin {cases} c \sqrt {-1 + \frac {x^{2}}{c^{2}}} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\i c \sqrt {1 - \frac {x^{2}}{c^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {b x^{2} \operatorname {asin}{\left (\frac {c}{x} \right )}}{2} \]

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

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