Integrand size = 10, antiderivative size = 119 \[ \int (a+b \arctan (c x))^3 \, dx=\frac {i (a+b \arctan (c x))^3}{c}+x (a+b \arctan (c x))^3+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c} \]
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
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.61 \[ \int (a+b \arctan (c x))^3 \, dx=a^3 x+3 a^2 b x \arctan (c x)-\frac {3 a^2 b \log \left (1+c^2 x^2\right )}{2 c}+\frac {3 a b^2 \left (-i \arctan (c x)^2+c x \arctan (c x)^2+2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{c}+\frac {b^3 \left (-i \arctan (c x)^3+c x \arctan (c x)^3+3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-3 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{c} \]
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)*Arc Tan[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
Time = 0.62 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5345, 5455, 5379, 5529, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \arctan (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle x (a+b \arctan (c x))^3-3 b c \int \frac {x (a+b \arctan (c x))^2}{c^2 x^2+1}dx\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle x (a+b \arctan (c x))^3-3 b c \left (-\frac {\int \frac {(a+b \arctan (c x))^2}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle x (a+b \arctan (c x))^3-3 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}\right )}{c}-\frac {i (a+b \arctan (c x))^3}{3 b c^2}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle x (a+b \arctan (c x))^3-3 b c \left (-\frac {i (a+b \arctan (c x))^3}{3 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c}-2 b \left (-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{4 c}\right )}{c}\right )\) |
x*(a + b*ArcTan[c*x])^3 - 3*b*c*(((-1/3*I)*(a + b*ArcTan[c*x])^3)/(b*c^2) - (((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c - 2*b*(((-1/2*I)*(a + b*Ar cTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (b*PolyLog[3, 1 - 2/(1 + I*c *x)])/(4*c)))/c)
3.1.29.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[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])
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] + Simp[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 ]
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] - Si mp[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]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[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]
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]
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 | \(x \,a^{3}+\frac {b^{3} \left (\arctan \left (c x \right )^{3} \left (c x +i\right )-2 i \arctan \left (c x \right )^{3}+3 \arctan \left (c x \right )^{2} \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-3 i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {3 \operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{c}+3 a^{2} b \arctan \left (c x \right ) x -\frac {3 a^{2} b \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {3 b^{2} a \left (\arctan \left (c x \right )^{2} \left (c x +i\right )+2 \arctan \left (c x \right ) \ln \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-2 i \arctan \left (c x \right )^{2}-i \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )\right )}{c}\) | \(245\) |
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*arctan(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) )))
\[ \int (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
\[ \int (a+b \arctan (c x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, dx \]
\[ \int (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
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*integrate(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*arctan(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*integrate(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
\[ \int (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \]