\(\int x (c+a^2 c x^2) \arctan (a x)^3 \, dx\) [365]

Optimal result

Integrand size = 18, antiderivative size = 160 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2} \]

[Out]

-1/4*c*x/a+1/4*c*(a^2*x^2+1)*arctan(a*x)/a^2-1/2*I*c*arctan(a*x)^2/a^2-1/2*c*x*arctan(a*x)^2/a-1/4*c*x*(a^2*x^
2+1)*arctan(a*x)^2/a+1/4*c*(a^2*x^2+1)^2*arctan(a*x)^3/a^2-c*arctan(a*x)*ln(2/(1+I*a*x))/a^2-1/2*I*c*polylog(2
,1-2/(1+I*a*x))/a^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5050, 5000, 4930, 5040, 4964, 2449, 2352, 8} \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c x \left (a^2 x^2+1\right ) \arctan (a x)^2}{4 a}+\frac {c \left (a^2 x^2+1\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {i c \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x}{4 a} \]

[In]

Int[x*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]

[Out]

-1/4*(c*x)/a + (c*(1 + a^2*x^2)*ArcTan[a*x])/(4*a^2) - ((I/2)*c*ArcTan[a*x]^2)/a^2 - (c*x*ArcTan[a*x]^2)/(2*a)
- (c*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(4*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(4*a^2) - (c*ArcTan[a*x]*Log[2/
(1 + I*a*x)])/a^2 - ((I/2)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

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 5000

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(-b)*p*(d + e*x^2)^
q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2*q + 1))), x] + (Dist[2*d*(q/(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a +
b*ArcTan[c*x])^p, x], x] + Dist[b^2*d*p*((p - 1)/(2*q*(2*q + 1))), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])
^(p - 2), x], x] + Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[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 5050

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

Rubi steps \begin{align*} \text {integral}& = \frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {3 \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx}{4 a} \\ & = \frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \int 1 \, dx}{4 a}-\frac {c \int \arctan (a x)^2 \, dx}{2 a} \\ & = -\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}+c \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \int \frac {\arctan (a x)}{i-a x} \, dx}{a} \\ & = -\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}+\frac {c \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a} \\ & = -\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {(i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^2} \\ & = -\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{4 a^2}-\frac {i c \arctan (a x)^2}{2 a^2}-\frac {c x \arctan (a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{4 a^2}-\frac {c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.63 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (-a x-\left (-2 i+3 a x+a^3 x^3\right ) \arctan (a x)^2+\left (1+a^2 x^2\right )^2 \arctan (a x)^3+\arctan (a x) \left (1+a^2 x^2-4 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{4 a^2} \]

[In]

Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]

[Out]

(c*(-(a*x) - (-2*I + 3*a*x + a^3*x^3)*ArcTan[a*x]^2 + (1 + a^2*x^2)^2*ArcTan[a*x]^3 + ArcTan[a*x]*(1 + a^2*x^2
- 4*Log[1 + E^((2*I)*ArcTan[a*x])]) + (2*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(4*a^2)

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.48

method	result	size
parts	\(\frac {c \arctan \left (a x \right )^{3} a^{2} x^{4}}{4}+\frac {c \arctan \left (a x \right )^{3} x^{2}}{2}+\frac {c \arctan \left (a x \right )^{3}}{4 a^{2}}-\frac {3 c \left (\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{2} \arctan \left (a x \right ) x^{2}}{3}-\frac {2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{3}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{3}\right )}{4 a^{2}}\)	\(237\)
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c \arctan \left (a x \right )^{3}}{4}-\frac {3 c \left (\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{2} \arctan \left (a x \right ) x^{2}}{3}-\frac {2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{3}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{3}\right )}{4}}{a^{2}}\)	\(238\)
default	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c \arctan \left (a x \right )^{3}}{4}-\frac {3 c \left (\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{2} \arctan \left (a x \right ) x^{2}}{3}-\frac {2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{3}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{3}\right )}{4}}{a^{2}}\)	\(238\)

[In]

int(x*(a^2*c*x^2+c)*arctan(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x*(a^2*c*x^2+c)*arctan(a*x)^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=c \left (\int x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

integrate(x*(a**2*c*x**2+c)*atan(a*x)**3,x)

[Out]

c*(Integral(x*atan(a*x)**3, x) + Integral(a**2*x**3*atan(a*x)**3, x))

Maxima [F]

\[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x*(a^2*c*x^2+c)*arctan(a*x)^3,x, algorithm="maxima")

[Out]

1/64*(8*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^3 + 4*(512*a^5*c*integrate(1/64*x^5*arctan(a*x)^3/(a^3*x^2 +
a), x) - 192*a^4*c*integrate(1/64*x^4*arctan(a*x)^2/(a^3*x^2 + a), x) - 48*a^4*c*integrate(1/64*x^4*log(a^2*x
^2 + 1)^2/(a^3*x^2 + a), x) - 64*a^4*c*integrate(1/64*x^4*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 1024*a^3*c*inte
grate(1/64*x^3*arctan(a*x)^3/(a^3*x^2 + a), x) + 128*a^3*c*integrate(1/64*x^3*arctan(a*x)/(a^3*x^2 + a), x) -
384*a^2*c*integrate(1/64*x^2*arctan(a*x)^2/(a^3*x^2 + a), x) - 96*a^2*c*integrate(1/64*x^2*log(a^2*x^2 + 1)^2/
(a^3*x^2 + a), x) - 192*a^2*c*integrate(1/64*x^2*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 512*a*c*integrate(1/64*x
*arctan(a*x)^3/(a^3*x^2 + a), x) + 384*a*c*integrate(1/64*x*arctan(a*x)/(a^3*x^2 + a), x) - c*arctan(a*x)^3/a^
2 - 48*c*integrate(1/64*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x))*a^2 - 4*(a^3*c*x^3 + 3*a*c*x)*arctan(a*x)^2 + (a
^3*c*x^3 + 3*a*c*x)*log(a^2*x^2 + 1)^2)/a^2

Giac [F]

\[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{3} \,d x } \]

[In]

integrate(x*(a^2*c*x^2+c)*arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \]

[In]

int(x*atan(a*x)^3*(c + a^2*c*x^2),x)

[Out]

int(x*atan(a*x)^3*(c + a^2*c*x^2), x)