3.2.47 \(\int \frac {a+b \text {ArcCos}(c x)}{x^4} \, dx\) [147]

Optimal. Leaf size=62 \[ \frac {b c \sqrt {1-c^2 x^2}}{6 x^2}-\frac {a+b \text {ArcCos}(c x)}{3 x^3}+\frac {1}{6} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]

[Out]

1/3*(-a-b*arccos(c*x))/x^3+1/6*b*c^3*arctanh((-c^2*x^2+1)^(1/2))+1/6*b*c*(-c^2*x^2+1)^(1/2)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4724, 272, 44, 65, 214} \begin {gather*} -\frac {a+b \text {ArcCos}(c x)}{3 x^3}+\frac {b c \sqrt {1-c^2 x^2}}{6 x^2}+\frac {1}{6} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])/x^4,x]

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(6*x^2) - (a + b*ArcCos[c*x])/(3*x^3) + (b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/6

Rule 44

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

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*} \int \frac {a+b \cos ^{-1}(c x)}{x^4} \, dx &=-\frac {a+b \cos ^{-1}(c x)}{3 x^3}-\frac {1}{3} (b c) \int \frac {1}{x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {a+b \cos ^{-1}(c x)}{3 x^3}-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {b c \sqrt {1-c^2 x^2}}{6 x^2}-\frac {a+b \cos ^{-1}(c x)}{3 x^3}-\frac {1}{12} \left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {b c \sqrt {1-c^2 x^2}}{6 x^2}-\frac {a+b \cos ^{-1}(c x)}{3 x^3}+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )\\ &=\frac {b c \sqrt {1-c^2 x^2}}{6 x^2}-\frac {a+b \cos ^{-1}(c x)}{3 x^3}+\frac {1}{6} b c^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 79, normalized size = 1.27 \begin {gather*} -\frac {a}{3 x^3}+\frac {b c \sqrt {1-c^2 x^2}}{6 x^2}-\frac {b \text {ArcCos}(c x)}{3 x^3}-\frac {1}{6} b c^3 \log (x)+\frac {1}{6} b c^3 \log \left (1+\sqrt {1-c^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])/x^4,x]

[Out]

-1/3*a/x^3 + (b*c*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*ArcCos[c*x])/(3*x^3) - (b*c^3*Log[x])/6 + (b*c^3*Log[1 + Sqr
t[1 - c^2*x^2]])/6

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 65, normalized size = 1.05

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(65\)
default	\(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(-1/3*a/c^3/x^3+b*(-1/3/c^3/x^3*arccos(c*x)+1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)+1/6*arctanh(1/(-c^2*x^2+1)^(1/2
))))

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 69, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \arccos \left (c x\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^4,x, algorithm="maxima")

[Out]

1/6*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)*c - 2*arccos(c*x)/x^3)*b - 1/3
*a/x^3

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (52) = 104\).
time = 1.90, size = 121, normalized size = 1.95 \begin {gather*} \frac {b c^{3} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c^{3} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) - 4 \, b x^{3} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-c^{2} x^{2} + 1} b c x + 4 \, {\left (b x^{3} - b\right )} \arccos \left (c x\right ) - 4 \, a}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^4,x, algorithm="fricas")

[Out]

1/12*(b*c^3*x^3*log(sqrt(-c^2*x^2 + 1) + 1) - b*c^3*x^3*log(sqrt(-c^2*x^2 + 1) - 1) - 4*b*x^3*arctan(sqrt(-c^2
*x^2 + 1)*c*x/(c^2*x^2 - 1)) + 2*sqrt(-c^2*x^2 + 1)*b*c*x + 4*(b*x^3 - b)*arccos(c*x) - 4*a)/x^3

________________________________________________________________________________________

Sympy [A]
time = 1.79, size = 119, normalized size = 1.92 \begin {gather*} - \frac {a}{3 x^{3}} - \frac {b c \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {c}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 c x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} - \frac {b \operatorname {acos}{\left (c x \right )}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))/x**4,x)

[Out]

-a/(3*x**3) - b*c*Piecewise((-c**2*acosh(1/(c*x))/2 + c/(2*x*sqrt(-1 + 1/(c**2*x**2))) - 1/(2*c*x**3*sqrt(-1 +
1/(c**2*x**2))), 1/Abs(c**2*x**2) > 1), (I*c**2*asin(1/(c*x))/2 - I*c*sqrt(1 - 1/(c**2*x**2))/(2*x), True))/3
- b*acos(c*x)/(3*x**3)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1634 vs. \(2 (52) = 104\).
time = 0.72, size = 1634, normalized size = 26.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^4,x, algorithm="giac")

[Out]

-1/3*b*c^3*arccos(c*x)/(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1
)^6 + 1) + 1/6*b*c^3*log(abs(c*x + sqrt(-c^2*x^2 + 1) + 1))/(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(
c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1) - 1/6*b*c^3*log(abs(-c*x + sqrt(-c^2*x^2 + 1) - 1))/(3*(c^2*x^2
- 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1) - 1/3*a*c^3/(3*(c^2*x^2 -
1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1) + (c^2*x^2 - 1)*b*c^3*arccos
(c*x)/((c*x + 1)^2*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6
+ 1)) + 1/2*(c^2*x^2 - 1)*b*c^3*log(abs(c*x + sqrt(-c^2*x^2 + 1) + 1))/((c*x + 1)^2*(3*(c^2*x^2 - 1)/(c*x + 1)
^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) - 1/2*(c^2*x^2 - 1)*b*c^3*log(abs(-c*x
+ sqrt(-c^2*x^2 + 1) - 1))/((c*x + 1)^2*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^
2 - 1)^3/(c*x + 1)^6 + 1)) + 1/3*sqrt(-c^2*x^2 + 1)*b*c^3/((c*x + 1)*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2
- 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) + (c^2*x^2 - 1)*a*c^3/((c*x + 1)^2*(3*(c^2*x^2 - 1)/(c
*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) - (c^2*x^2 - 1)^2*b*c^3*arccos(c
*x)/((c*x + 1)^4*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 +
1)) + 1/2*(c^2*x^2 - 1)^2*b*c^3*log(abs(c*x + sqrt(-c^2*x^2 + 1) + 1))/((c*x + 1)^4*(3*(c^2*x^2 - 1)/(c*x + 1)
^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) - 1/2*(c^2*x^2 - 1)^2*b*c^3*log(abs(-c*
x + sqrt(-c^2*x^2 + 1) - 1))/((c*x + 1)^4*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*
x^2 - 1)^3/(c*x + 1)^6 + 1)) - (c^2*x^2 - 1)^2*a*c^3/((c*x + 1)^4*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 -
1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) + 1/3*(c^2*x^2 - 1)^3*b*c^3*arccos(c*x)/((c*x + 1)^6*(3*(
c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) + 1/6*(c^2*x^2 -
1)^3*b*c^3*log(abs(c*x + sqrt(-c^2*x^2 + 1) + 1))/((c*x + 1)^6*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^
2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) - 1/6*(c^2*x^2 - 1)^3*b*c^3*log(abs(-c*x + sqrt(-c^2*x^2 + 1
) - 1))/((c*x + 1)^6*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^
6 + 1)) - 1/3*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*c^3/((c*x + 1)^5*(3*(c^2*x^2 - 1)/(c*x + 1)^2 + 3*(c^2*x^2
- 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1)) + 1/3*(c^2*x^2 - 1)^3*a*c^3/((c*x + 1)^6*(3*(c^2*x^2 -
1)/(c*x + 1)^2 + 3*(c^2*x^2 - 1)^2/(c*x + 1)^4 + (c^2*x^2 - 1)^3/(c*x + 1)^6 + 1))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))/x^4,x)

[Out]

int((a + b*acos(c*x))/x^4, x)

________________________________________________________________________________________