3.146 \(\int \frac{a+b \cos ^{-1}(c x)}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0194112, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4628, 264} \[ \frac{b c \sqrt{1-c^2 x^2}}{2 x}-\frac{a+b \cos ^{-1}(c x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])/x^3,x]

[Out]

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

Rule 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 264

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

Rubi steps

\begin{align*} \int \frac{a+b \cos ^{-1}(c x)}{x^3} \, dx &=-\frac{a+b \cos ^{-1}(c x)}{2 x^2}-\frac{1}{2} (b c) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b c \sqrt{1-c^2 x^2}}{2 x}-\frac{a+b \cos ^{-1}(c x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0212276, size = 44, normalized size = 1.13 \[ -\frac{a}{2 x^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 x}-\frac{b \cos ^{-1}(c x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])/x^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 50, normalized size = 1.3 \begin{align*}{c}^{2} \left ( -{\frac{a}{2\,{c}^{2}{x}^{2}}}+b \left ( -{\frac{\arccos \left ( cx \right ) }{2\,{c}^{2}{x}^{2}}}+{\frac{1}{2\,cx}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))/x^3,x)

[Out]

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

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Maxima [A]  time = 1.45622, size = 50, normalized size = 1.28 \begin{align*} \frac{1}{2} \, b{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c}{x} - \frac{\arccos \left (c x\right )}{x^{2}}\right )} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.84842, size = 86, normalized size = 2.21 \begin{align*} \frac{\sqrt{-c^{2} x^{2} + 1} b c x + a x^{2} - b \arccos \left (c x\right ) - a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.56326, size = 63, normalized size = 1.62 \begin{align*} - \frac{a}{2 x^{2}} - \frac{b c \left (\begin{cases} - \frac{i \sqrt{c^{2} x^{2} - 1}}{x} & \text{for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- c^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right )}{2} - \frac{b \operatorname{acos}{\left (c x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))/x**3,x)

[Out]

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

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Giac [B]  time = 1.18139, size = 664, normalized size = 17.03 \begin{align*} -\frac{b c^{2} \arccos \left (c x\right )}{2 \,{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac{a c^{2}}{2 \,{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} + \frac{{\left (c^{2} x^{2} - 1\right )} b c^{2} \arccos \left (c x\right )}{{\left (c x + 1\right )}^{2}{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} + \frac{\sqrt{-c^{2} x^{2} + 1} b c^{2}}{{\left (c x + 1\right )}{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} + \frac{{\left (c^{2} x^{2} - 1\right )} a c^{2}}{{\left (c x + 1\right )}^{2}{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b c^{2} \arccos \left (c x\right )}{2 \,{\left (c x + 1\right )}^{4}{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b c^{2}}{{\left (c x + 1\right )}^{3}{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a c^{2}}{2 \,{\left (c x + 1\right )}^{4}{\left (\frac{2 \,{\left (c^{2} x^{2} - 1\right )}}{{\left (c x + 1\right )}^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2}}{{\left (c x + 1\right )}^{4}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^3,x, algorithm="giac")

[Out]

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