\(\int \frac {a+b \arccos (c x)}{x^2} \, dx\) [145]

Optimal result

Integrand size = 12, antiderivative size = 32 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=-\frac {a+b \arccos (c x)}{x}+b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4724, 272, 65, 214} \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {a+b \arccos (c x)}{x} \]

[In]

Int[(a + b*ArcCos[c*x])/x^2,x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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*} \text {integral}& = -\frac {a+b \arccos (c x)}{x}-(b c) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {a+b \arccos (c x)}{x}-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {a+b \arccos (c x)}{x}+\frac {b \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c} \\ & = -\frac {a+b \arccos (c x)}{x}+b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=-\frac {a}{x}-\frac {b \arccos (c x)}{x}-b c \log (x)+b c \log \left (1+\sqrt {1-c^2 x^2}\right ) \]

[In]

Integrate[(a + b*ArcCos[c*x])/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16

method	result	size
parts	\(-\frac {a}{x}+b c \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(37\)
derivativedivides	\(c \left (-\frac {a}{c x}+b \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(41\)
default	\(c \left (-\frac {a}{c x}+b \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(41\)

[In]

int((a+b*arccos(c*x))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (30) = 60\).

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.88 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=\frac {b c x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) - 2 \, b x \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, {\left (b x - b\right )} \arccos \left (c x\right ) - 2 \, a}{2 \, x} \]

[In]

integrate((a+b*arccos(c*x))/x^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=- \frac {a}{x} - b c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b \operatorname {acos}{\left (c x \right )}}{x} \]

[In]

integrate((a+b*acos(c*x))/x**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx={\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arccos \left (c x\right )}{x}\right )} b - \frac {a}{x} \]

[In]

integrate((a+b*arccos(c*x))/x^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (30) = 60\).

Time = 0.36 (sec) , antiderivative size = 347, normalized size of antiderivative = 10.84 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=-\frac {b c \arccos \left (c x\right )}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} + \frac {b c \log \left ({\left | c x + \sqrt {-c^{2} x^{2} + 1} + 1 \right |}\right )}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} - \frac {b c \log \left ({\left | -c x + \sqrt {-c^{2} x^{2} + 1} - 1 \right |}\right )}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} - \frac {a c}{\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1} + \frac {{\left (c^{2} x^{2} - 1\right )} b c \arccos \left (c x\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} b c \log \left ({\left | c x + \sqrt {-c^{2} x^{2} + 1} + 1 \right |}\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} - \frac {{\left (c^{2} x^{2} - 1\right )} b c \log \left ({\left | -c x + \sqrt {-c^{2} x^{2} + 1} - 1 \right |}\right )}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )} a c}{{\left (c x + 1\right )}^{2} {\left (\frac {c^{2} x^{2} - 1}{{\left (c x + 1\right )}^{2}} + 1\right )}} \]

[In]

integrate((a+b*arccos(c*x))/x^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arccos (c x)}{x^2} \, dx=b\,c\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,\mathrm {acos}\left (c\,x\right )}{x}-\frac {a}{x} \]

[In]

int((a + b*acos(c*x))/x^2,x)

[Out]

b*c*atanh(1/(1 - c^2*x^2)^(1/2)) - (b*acos(c*x))/x - a/x