3.2.46 \(\int \frac {a+b \text {ArcCos}(c x)}{x^3} \, dx\) [146]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {4724, 270} \begin {gather*} \frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {a+b \text {ArcCos}(c x)}{2 x^2} \end {gather*}

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 270

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]

Rule 4724

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]

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*}

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Mathematica [A]
time = 0.02, size = 44, normalized size = 1.13 \begin {gather*} -\frac {a}{2 x^2}+\frac {b c \sqrt {1-c^2 x^2}}{2 x}-\frac {b \text {ArcCos}(c x)}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])/x^3,x]

[Out]

-1/2*a/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 = 0.00, size = 50, normalized size = 1.28

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\arccos \left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\)	\(50\)
default	\(c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\arccos \left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.49, size = 37, normalized size = 0.95 \begin {gather*} \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 {gather*}

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 = 2.93, size = 37, normalized size = 0.95 \begin {gather*} \frac {\sqrt {-c^{2} x^{2} + 1} b c x + a x^{2} - b \arccos \left (c x\right ) - a}{2 \, x^{2}} \end {gather*}

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 = 0.88, size = 63, normalized size = 1.62 \begin {gather*} - \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (33) = 66\).
time = 0.47, size = 492, normalized size = 12.62 \begin {gather*} -\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 {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))/x^3,x)

[Out]

int((a + b*acos(c*x))/x^3, x)

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