\(\int (c+a^2 c x^2) \arctan (a x)^2 \, dx\) [261]

Optimal result

Integrand size = 17, antiderivative size = 128 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c x}{3}-\frac {c \left (1+a^2 x^2\right ) \arctan (a x)}{3 a}+\frac {2 i c \arctan (a x)^2}{3 a}+\frac {2}{3} c x \arctan (a x)^2+\frac {1}{3} c x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {4 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a}+\frac {2 i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5000, 4930, 5040, 4964, 2449, 2352, 8} \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {1}{3} c x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {c \left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {2 i c \arctan (a x)^2}{3 a}+\frac {2}{3} c x \arctan (a x)^2+\frac {4 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a}+\frac {2 i c \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{3 a}+\frac {c x}{3} \]

[In]

Int[(c + a^2*c*x^2)*ArcTan[a*x]^2,x]

[Out]

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

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]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (a x+\left (-2 i+3 a x+a^3 x^3\right ) \arctan (a x)^2-\arctan (a x) \left (1+a^2 x^2-4 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-2 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{3 a} \]

[In]

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

[Out]

(c*(a*x + (-2*I + 3*a*x + a^3*x^3)*ArcTan[a*x]^2 - 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])]))/(3*a)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right )^{2} a^{3} x^{3}}{3}+a c x \arctan \left (a x \right )^{2}-\frac {2 c \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}\right )}{3}}{a}\)	\(200\)
default	\(\frac {\frac {c \arctan \left (a x \right )^{2} a^{3} x^{3}}{3}+a c x \arctan \left (a x \right )^{2}-\frac {2 c \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}\right )}{3}}{a}\)	\(200\)
parts	\(\frac {a^{2} c \,x^{3} \arctan \left (a x \right )^{2}}{3}+c x \arctan \left (a x \right )^{2}-\frac {2 c \left (\frac {a \arctan \left (a x \right ) x^{2}}{2}+\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {a x -\arctan \left (a x \right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2 a}\right )}{3}\)	\(201\)
risch	\(\frac {i c a \ln \left (i a x +1\right ) x^{2}}{6}-\frac {c \ln \left (i a x +1\right )^{2} x}{4}-\frac {c \ln \left (-i a x +1\right )^{2} x}{4}+\frac {c x}{3}-\frac {i c \ln \left (-i a x +1\right )^{2}}{6 a}-\frac {c \arctan \left (a x \right )}{3 a}+\frac {2 i c \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a}+\frac {2 i c \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a}-\frac {2 i c \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{3 a}+\frac {c \,a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {c \,a^{2} \ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {c \,a^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{12}+\frac {i c \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{3 a}-\frac {i c a \ln \left (-i a x +1\right ) x^{2}}{6}+\frac {c \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}+\frac {37 i c}{54 a}+\frac {i c \ln \left (i a x +1\right )^{2}}{6 a}\)	\(284\)

[In]

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

[Out]

1/a*(1/3*c*arctan(a*x)^2*a^3*x^3+a*c*x*arctan(a*x)^2-2/3*c*(1/2*a^2*arctan(a*x)*x^2+arctan(a*x)*ln(a^2*x^2+1)-
1/2*a*x+1/2*arctan(a*x)+1/2*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)-1/2*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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

36*a^4*c*integrate(1/48*x^4*arctan(a*x)^2/(a^2*x^2 + 1), x) + 3*a^4*c*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a
^2*x^2 + 1), x) + 4*a^4*c*integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 8*a^3*c*integrate(1/48*x^3*a
rctan(a*x)/(a^2*x^2 + 1), x) + 72*a^2*c*integrate(1/48*x^2*arctan(a*x)^2/(a^2*x^2 + 1), x) + 6*a^2*c*integrate
(1/48*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12*a^2*c*integrate(1/48*x^2*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x
) + 1/12*(a^2*c*x^3 + 3*c*x)*arctan(a*x)^2 + 1/4*c*arctan(a*x)^3/a - 24*a*c*integrate(1/48*x*arctan(a*x)/(a^2*
x^2 + 1), x) - 1/48*(a^2*c*x^3 + 3*c*x)*log(a^2*x^2 + 1)^2 + 3*c*integrate(1/48*log(a^2*x^2 + 1)^2/(a^2*x^2 +
1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(atan(a*x)^2*(c + a^2*c*x^2),x)

[Out]

int(atan(a*x)^2*(c + a^2*c*x^2), x)