3.145 \(\int \frac{a+b \cos ^{-1}(c x)}{x^2} \, dx\)

Optimal. Leaf size=32 \[ b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-\frac{a+b \cos ^{-1}(c x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0281992, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4628, 266, 63, 208} \[ b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-\frac{a+b \cos ^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[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 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 266

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 63

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 208

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

Rubi steps

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

Mathematica [A]  time = 0.0119583, size = 43, normalized size = 1.34 \[ -\frac{a}{x}+b c \log \left (\sqrt{1-c^2 x^2}+1\right )-b c \log (x)-\frac{b \cos ^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[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]]

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Maple [A]  time = 0.006, size = 41, normalized size = 1.3 \begin{align*} c \left ( -{\frac{a}{cx}}+b \left ( -{\frac{\arccos \left ( cx \right ) }{cx}}+{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42362, size = 63, normalized size = 1.97 \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.74335, size = 221, normalized size = 6.91 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 1.95527, size = 41, normalized size = 1.28 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.52024, size = 468, normalized size = 14.62 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))