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

Optimal. Leaf size=133 \[ -\frac{3}{2} i b^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )+3 b^2 c^2 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{3}{2} i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 x} \]

[Out]

((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2 - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*x) - (c^2*(a + b*ArcTan[c*x])^3)/2 -
(a + b*ArcTan[c*x])^3/(2*x^2) + 3*b^2*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - ((3*I)/2)*b^3*c^2*Poly
Log[2, -1 + 2/(1 - I*c*x)]

________________________________________________________________________________________

Rubi [A]  time = 0.28473, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4852, 4918, 4924, 4868, 2447, 4884} \[ -\frac{3}{2} i b^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )+3 b^2 c^2 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{3}{2} i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} c^2 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])^3/x^3,x]

[Out]

((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2 - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*x) - (c^2*(a + b*ArcTan[c*x])^3)/2 -
(a + b*ArcTan[c*x])^3/(2*x^2) + 3*b^2*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - ((3*I)/2)*b^3*c^2*Poly
Log[2, -1 + 2/(1 - I*c*x)]

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4918

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcTan[c*x])^p)/(d + e*
x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*d*(p + 1)), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)
])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4884

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x^3} \, dx &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}+\frac{1}{2} (3 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}+\frac{1}{2} (3 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx-\frac{1}{2} \left (3 b c^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 x}-\frac{1}{2} c^2 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}+\left (3 b^2 c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{3}{2} i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 x}-\frac{1}{2} c^2 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}+\left (3 i b^2 c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx\\ &=-\frac{3}{2} i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 x}-\frac{1}{2} c^2 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-\left (3 b^3 c^3\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{3}{2} i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 x}-\frac{1}{2} c^2 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 x^2}+3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-\frac{3}{2} i b^3 c^2 \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )\\ \end{align*}

Mathematica [A]  time = 0.322897, size = 176, normalized size = 1.32 \[ -\frac{3 i b^3 c^2 x^2 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+a \left (a (a+3 b c x)-6 b^2 c^2 x^2 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )\right )+3 b^2 \tan ^{-1}(c x)^2 \left (a c^2 x^2+a+b c x (1+i c x)\right )+3 b \tan ^{-1}(c x) \left (a \left (a c^2 x^2+a+2 b c x\right )-2 b^2 c^2 x^2 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )+b^3 \left (c^2 x^2+1\right ) \tan ^{-1}(c x)^3}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c*x])^3/x^3,x]

[Out]

-(3*b^2*(a + a*c^2*x^2 + b*c*x*(1 + I*c*x))*ArcTan[c*x]^2 + b^3*(1 + c^2*x^2)*ArcTan[c*x]^3 + 3*b*ArcTan[c*x]*
(a*(a + 2*b*c*x + a*c^2*x^2) - 2*b^2*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c*x])]) + a*(a*(a + 3*b*c*x) - 6*b^2*c^2*
x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + (3*I)*b^3*c^2*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])])/(2*x^2)

________________________________________________________________________________________

Maple [B]  time = 0.043, size = 457, normalized size = 3.4 \begin{align*} -{\frac{3\,{c}^{2}a{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2}}-{\frac{3\,c{b}^{3} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{2\,x}}-{\frac{3\,{a}^{2}b\arctan \left ( cx \right ) }{2\,{x}^{2}}}+{\frac{3\,i}{8}}{c}^{2}{b}^{3} \left ( \ln \left ( cx-i \right ) \right ) ^{2}-{\frac{3\,i}{4}}{c}^{2}{b}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) +{\frac{3\,i}{2}}{c}^{2}{b}^{3}{\it dilog} \left ( 1+icx \right ) -{\frac{3\,i}{2}}{c}^{2}{b}^{3}{\it dilog} \left ( 1-icx \right ) +{\frac{3\,i}{4}}{c}^{2}{b}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) -{\frac{3\,a{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac{3\,{c}^{2}a{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{2}}-{\frac{3\,{c}^{2}{b}^{3}\arctan \left ( cx \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{2}}+3\,{c}^{2}{b}^{3}\ln \left ( cx \right ) \arctan \left ( cx \right ) -{\frac{3\,{c}^{2}{a}^{2}b\arctan \left ( cx \right ) }{2}}+3\,{c}^{2}a{b}^{2}\ln \left ( cx \right ) -{\frac{3\,{a}^{2}cb}{2\,x}}-{\frac{3\,i}{8}}{c}^{2}{b}^{3} \left ( \ln \left ( cx+i \right ) \right ) ^{2}-{\frac{{b}^{3} \left ( \arctan \left ( cx \right ) \right ) ^{3}}{2\,{x}^{2}}}-{\frac{{c}^{2}{b}^{3} \left ( \arctan \left ( cx \right ) \right ) ^{3}}{2}}-3\,{\frac{ca{b}^{2}\arctan \left ( cx \right ) }{x}}+{\frac{3\,i}{4}}{c}^{2}{b}^{3}\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) +{\frac{3\,i}{2}}{c}^{2}{b}^{3}\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -{\frac{3\,i}{4}}{c}^{2}{b}^{3}\ln \left ( cx+i \right ) \ln \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) -{\frac{3\,i}{4}}{c}^{2}{b}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) \ln \left ( cx-i \right ) +{\frac{3\,i}{4}}{c}^{2}{b}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) \ln \left ( cx+i \right ) -{\frac{3\,i}{2}}{c}^{2}{b}^{3}\ln \left ( cx \right ) \ln \left ( 1-icx \right ) -{\frac{{a}^{3}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*c^2*a*b^2*ln(c^2*x^2+1)-3/2*c*b^3*arctan(c*x)^2/x-3/2*a^2*b/x^2*arctan(c*x)+3/8*I*c^2*b^3*ln(c*x-I)^2-3/4
*I*c^2*b^3*dilog(1/2*I*(c*x-I))+3/2*I*c^2*b^3*dilog(1+I*c*x)-3/2*I*c^2*b^3*dilog(1-I*c*x)+3/4*I*c^2*b^3*dilog(
-1/2*I*(c*x+I))-3/2*a*b^2/x^2*arctan(c*x)^2-3/2*c^2*a*b^2*arctan(c*x)^2-3/2*c^2*b^3*arctan(c*x)*ln(c^2*x^2+1)+
3*c^2*b^3*ln(c*x)*arctan(c*x)-3/2*c^2*a^2*b*arctan(c*x)+3*c^2*a*b^2*ln(c*x)-3/2*c*a^2*b/x-3/8*I*c^2*b^3*ln(c*x
+I)^2-1/2*b^3/x^2*arctan(c*x)^3-1/2*c^2*b^3*arctan(c*x)^3-3*c*a*b^2*arctan(c*x)/x+3/4*I*c^2*b^3*ln(c*x-I)*ln(-
1/2*I*(c*x+I))+3/2*I*c^2*b^3*ln(c*x)*ln(1+I*c*x)-3/4*I*c^2*b^3*ln(c*x+I)*ln(1/2*I*(c*x-I))-3/4*I*c^2*b^3*ln(c^
2*x^2+1)*ln(c*x-I)+3/4*I*c^2*b^3*ln(c^2*x^2+1)*ln(c*x+I)-3/2*I*c^2*b^3*ln(c*x)*ln(1-I*c*x)-1/2*a^3/x^2

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \arctan \left (c x\right )^{3} + 3 \, a b^{2} \arctan \left (c x\right )^{2} + 3 \, a^{2} b \arctan \left (c x\right ) + a^{3}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*arctan(c*x)^3 + 3*a*b^2*arctan(c*x)^2 + 3*a^2*b*arctan(c*x) + a^3)/x^3, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atan(c*x))**3/x**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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