\(\int (a+b \arctan (c x))^3 \, dx\) [29]

Optimal result

Integrand size = 10, antiderivative size = 119 \[ \int (a+b \arctan (c x))^3 \, dx=\frac {i (a+b \arctan (c x))^3}{c}+x (a+b \arctan (c x))^3+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4930, 5040, 4964, 5004, 5114, 6745} \[ \int (a+b \arctan (c x))^3 \, dx=\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c}+x (a+b \arctan (c x))^3+\frac {i (a+b \arctan (c x))^3}{c}+\frac {3 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c} \]

[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 4930

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

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 5004

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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[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 6745

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*} \text {integral}& = x (a+b \arctan (c x))^3-(3 b c) \int \frac {x (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx \\ & = \frac {i (a+b \arctan (c x))^3}{c}+x (a+b \arctan (c x))^3+(3 b) \int \frac {(a+b \arctan (c x))^2}{i-c x} \, dx \\ & = \frac {i (a+b \arctan (c x))^3}{c}+x (a+b \arctan (c x))^3+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}-\left (6 b^2\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = \frac {i (a+b \arctan (c x))^3}{c}+x (a+b \arctan (c x))^3+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\left (3 i b^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = \frac {i (a+b \arctan (c x))^3}{c}+x (a+b \arctan (c x))^3+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.61 \[ \int (a+b \arctan (c x))^3 \, dx=a^3 x+3 a^2 b x \arctan (c x)-\frac {3 a^2 b \log \left (1+c^2 x^2\right )}{2 c}+\frac {3 a b^2 \left (-i \arctan (c x)^2+c x \arctan (c x)^2+2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{c}+\frac {b^3 \left (-i \arctan (c x)^3+c x \arctan (c x)^3+3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-3 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{c} \]

[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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (112 ) = 224\).

Time = 5.71 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.02

method	result	size
derivativedivides	\(\frac {c \,a^{3} x +b^{3} \left (\arctan \left (c x \right )^{3} \left (c x +i\right )-2 i \arctan \left (c x \right )^{3}+3 \arctan \left (c x \right )^{2} \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-3 i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )+3 a^{2} b \left (c x \arctan \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )+3 b^{2} a \left (\arctan \left (c x \right )^{2} \left (c x +i\right )+2 \arctan \left (c x \right ) \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-2 i \arctan \left (c x \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )\right )}{c}\)	\(240\)
default	\(\frac {c \,a^{3} x +b^{3} \left (\arctan \left (c x \right )^{3} \left (c x +i\right )-2 i \arctan \left (c x \right )^{3}+3 \arctan \left (c x \right )^{2} \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-3 i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )+3 a^{2} b \left (c x \arctan \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )+3 b^{2} a \left (\arctan \left (c x \right )^{2} \left (c x +i\right )+2 \arctan \left (c x \right ) \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-2 i \arctan \left (c x \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )\right )}{c}\)	\(240\)
parts	\(x \,a^{3}+\frac {b^{3} \left (\arctan \left (c x \right )^{3} \left (c x +i\right )-2 i \arctan \left (c x \right )^{3}+3 \arctan \left (c x \right )^{2} \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-3 i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{c}+3 a^{2} b \arctan \left (c x \right ) x -\frac {3 a^{2} b \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {3 b^{2} a \left (\arctan \left (c x \right )^{2} \left (c x +i\right )+2 \arctan \left (c x \right ) \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-2 i \arctan \left (c x \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )\right )}{c}\)	\(245\)

[In]

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

[Out]

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

Fricas [F]

\[ \int (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]

[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)

Sympy [F]

\[ \int (a+b \arctan (c x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, dx \]

[In]

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

[Out]

Integral((a + b*atan(c*x))**3, x)

Maxima [F]

\[ \int (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]

[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

Giac [F]

\[ \int (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \]

[In]

int((a + b*atan(c*x))^3,x)

[Out]

int((a + b*atan(c*x))^3, x)