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

Optimal. Leaf size=119 \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c}+x \left (a+b \tan ^{-1}(c x)\right )^3+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{c}+\frac{3 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c} \]

[Out]

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

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Rubi [A]  time = 0.209268, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4846, 4920, 4854, 4884, 4994, 6610} \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c}+x \left (a+b \tan ^{-1}(c x)\right )^3+\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{c}+\frac{3 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])^3,x]

[Out]

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

Rule 4846

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

Rule 4920

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

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[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 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]

Rule 4994

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=x \left (a+b \tan ^{-1}(c x)\right )^3-(3 b c) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{c}+x \left (a+b \tan ^{-1}(c x)\right )^3+(3 b) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{c}+x \left (a+b \tan ^{-1}(c x)\right )^3+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c}-\left (6 b^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{c}+x \left (a+b \tan ^{-1}(c x)\right )^3+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c}-\left (3 i b^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^3}{c}+x \left (a+b \tan ^{-1}(c x)\right )^3+\frac{3 b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c}+\frac{3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c}+\frac{3 b^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c}\\ \end{align*}

Mathematica [A]  time = 0.0921695, size = 192, normalized size = 1.61 \[ \frac{3 a b^2 \left (-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+c x \tan ^{-1}(c x)^2-i \tan ^{-1}(c x)^2+2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{c}+\frac{b^3 \left (-3 i \tan ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )+c x \tan ^{-1}(c x)^3-i \tan ^{-1}(c x)^3+3 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{c}-\frac{3 a^2 b \log \left (c^2 x^2+1\right )}{2 c}+3 a^2 b x \tan ^{-1}(c x)+a^3 x \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c*x])^3,x]

[Out]

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

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Maple [B]  time = 0.134, size = 270, normalized size = 2.3 \begin{align*} x{a}^{3}-{\frac{i{b}^{3} \left ( \arctan \left ( cx \right ) \right ) ^{3}}{c}}+{b}^{3}x \left ( \arctan \left ( cx \right ) \right ) ^{3}+3\,{\frac{{b}^{3} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{c}\ln \left ({\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}}+1 \right ) }-{\frac{3\,i{b}^{3}\arctan \left ( cx \right ) }{c}{\it polylog} \left ( 2,-{\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}} \right ) }+{\frac{3\,{b}^{3}}{2\,c}{\it polylog} \left ( 3,-{\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,i \left ( \arctan \left ( cx \right ) \right ) ^{2}a{b}^{2}}{c}}+3\,xa{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}+6\,{\frac{\arctan \left ( cx \right ) a{b}^{2}}{c}\ln \left ({\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}}+1 \right ) }-{\frac{3\,ia{b}^{2}}{c}{\it polylog} \left ( 2,-{\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}} \right ) }+3\,x{a}^{2}b\arctan \left ( cx \right ) -{\frac{3\,{a}^{2}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*a^3-I/c*b^3*arctan(c*x)^3+b^3*x*arctan(c*x)^3+3/c*b^3*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-3*I/c*b^3*
arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/2/c*b^3*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3*I/c*arctan(c*x
)^2*a*b^2+3*x*a*b^2*arctan(c*x)^2+6/c*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)*a*b^2-3*I/c*polylog(2,-(1+I*c*
x)^2/(c^2*x^2+1))*a*b^2+3*x*a^2*b*arctan(c*x)-3/2/c*a^2*b*ln(c^2*x^2+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, b^{3} x \arctan \left (c x\right )^{3} - \frac{3}{32} \, b^{3} x \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2} + \frac{7 \, b^{3} \arctan \left (c x\right )^{4}}{32 \, c} + 28 \, b^{3} c^{2} \int \frac{x^{2} \arctan \left (c x\right )^{3}}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 3 \, b^{3} c^{2} \int \frac{x^{2} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2}}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 96 \, a b^{2} c^{2} \int \frac{x^{2} \arctan \left (c x\right )^{2}}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 12 \, b^{3} c^{2} \int \frac{x^{2} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + \frac{a b^{2} \arctan \left (c x\right )^{3}}{c} - 12 \, b^{3} c \int \frac{x \arctan \left (c x\right )^{2}}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 3 \, b^{3} c \int \frac{x \log \left (c^{2} x^{2} + 1\right )^{2}}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + a^{3} x + 3 \, b^{3} \int \frac{\arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2}}{32 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + \frac{3 \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} a^{2} b}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*x*arctan(c*x)^3 - 3/32*b^3*x*arctan(c*x)*log(c^2*x^2 + 1)^2 + 7/32*b^3*arctan(c*x)^4/c + 28*b^3*c^2*in
tegrate(1/32*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 3*b^3*c^2*integrate(1/32*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2
/(c^2*x^2 + 1), x) + 96*a*b^2*c^2*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^3*c^2*integrate(1/
32*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + a*b^2*arctan(c*x)^3/c - 12*b^3*c*integrate(1/32*x*arct
an(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c*integrate(1/32*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + a^3*x + 3*b^3*in
tegrate(1/32*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a^2
*b/c

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atan(c*x))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c*x) + a)^3, x)