\(\int (c+a^2 c x^2)^2 \arctan (a x)^2 \, dx\) [269]

Optimal result

Integrand size = 19, antiderivative size = 205 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {11 c^2 x}{30}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {8 i c^2 \arctan (a x)^2}{15 a}+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a}+\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a} \]

[Out]

11/30*c^2*x+1/30*a^2*c^2*x^3-4/15*c^2*(a^2*x^2+1)*arctan(a*x)/a-1/10*c^2*(a^2*x^2+1)^2*arctan(a*x)/a+8/15*I*c^
2*arctan(a*x)^2/a+8/15*c^2*x*arctan(a*x)^2+4/15*c^2*x*(a^2*x^2+1)*arctan(a*x)^2+1/5*c^2*x*(a^2*x^2+1)^2*arctan
(a*x)^2+16/15*c^2*arctan(a*x)*ln(2/(1+I*a*x))/a+8/15*I*c^2*polylog(2,1-2/(1+I*a*x))/a

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5000, 4930, 5040, 4964, 2449, 2352, 8} \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^2+\frac {4}{15} c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}-\frac {4 c^2 \left (a^2 x^2+1\right ) \arctan (a x)}{15 a}+\frac {1}{30} a^2 c^2 x^3+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {8 i c^2 \arctan (a x)^2}{15 a}+\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a}+\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{15 a}+\frac {11 c^2 x}{30} \]

[In]

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

[Out]

(11*c^2*x)/30 + (a^2*c^2*x^3)/30 - (4*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x]
)/(10*a) + (((8*I)/15)*c^2*ArcTan[a*x]^2)/a + (8*c^2*x*ArcTan[a*x]^2)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^
2)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/5 + (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a) + (((8*I)/15)
*c^2*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^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{10} c \int \left (c+a^2 c x^2\right ) \, dx+\frac {1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx \\ & = \frac {c^2 x}{10}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{15} \left (4 c^2\right ) \int 1 \, dx+\frac {1}{15} \left (8 c^2\right ) \int \arctan (a x)^2 \, dx \\ & = \frac {11 c^2 x}{30}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2-\frac {1}{15} \left (16 a c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {11 c^2 x}{30}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {8 i c^2 \arctan (a x)^2}{15 a}+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{15} \left (16 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx \\ & = \frac {11 c^2 x}{30}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {8 i c^2 \arctan (a x)^2}{15 a}+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a}-\frac {1}{15} \left (16 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = \frac {11 c^2 x}{30}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {8 i c^2 \arctan (a x)^2}{15 a}+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a}+\frac {\left (16 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{15 a} \\ & = \frac {11 c^2 x}{30}+\frac {1}{30} a^2 c^2 x^3-\frac {4 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{10 a}+\frac {8 i c^2 \arctan (a x)^2}{15 a}+\frac {8}{15} c^2 x \arctan (a x)^2+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a}+\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.55 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^2 \left (a x \left (11+a^2 x^2\right )+2 \left (-8 i+15 a x+10 a^3 x^3+3 a^5 x^5\right ) \arctan (a x)^2-\arctan (a x) \left (11+14 a^2 x^2+3 a^4 x^4-32 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-16 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{30 a} \]

[In]

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

[Out]

(c^2*(a*x*(11 + a^2*x^2) + 2*(-8*I + 15*a*x + 10*a^3*x^3 + 3*a^5*x^5)*ArcTan[a*x]^2 - ArcTan[a*x]*(11 + 14*a^2
*x^2 + 3*a^4*x^4 - 32*Log[1 + E^((2*I)*ArcTan[a*x])]) - (16*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(30*a)

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {2 a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}+a \,c^{2} x \arctan \left (a x \right )^{2}-\frac {2 c^{2} \left (\frac {3 \arctan \left (a x \right ) a^{4} x^{4}}{4}+\frac {7 a^{2} \arctan \left (a x \right ) x^{2}}{2}+4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{4}-\frac {11 a x}{4}+\frac {11 \arctan \left (a x \right )}{4}+2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{15}}{a}\)	\(244\)
default	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {2 a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}+a \,c^{2} x \arctan \left (a x \right )^{2}-\frac {2 c^{2} \left (\frac {3 \arctan \left (a x \right ) a^{4} x^{4}}{4}+\frac {7 a^{2} \arctan \left (a x \right ) x^{2}}{2}+4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{4}-\frac {11 a x}{4}+\frac {11 \arctan \left (a x \right )}{4}+2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{15}}{a}\)	\(244\)
parts	\(\frac {a^{4} c^{2} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {2 a^{2} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}+c^{2} x \arctan \left (a x \right )^{2}-\frac {2 c^{2} \left (\frac {3 a^{3} \arctan \left (a x \right ) x^{4}}{4}+\frac {7 a \arctan \left (a x \right ) x^{2}}{2}+\frac {4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {a^{3} x^{3}+11 a x -11 \arctan \left (a x \right )-8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{4 a}\right )}{15}\)	\(245\)
risch	\(\frac {7 i c^{2} a \ln \left (i a x +1\right ) x^{2}}{30}+\frac {c^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}+\frac {a^{2} c^{2} x^{3}}{30}+\frac {11 c^{2} x}{30}-\frac {8 i c^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{15 a}+\frac {i c^{2} a^{3} \ln \left (i a x +1\right ) x^{4}}{20}+\frac {2 i c^{2} \ln \left (i a x +1\right )^{2}}{15 a}-\frac {c^{2} a^{4} \ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {c^{2} a^{2} \ln \left (i a x +1\right )^{2} x^{3}}{6}-\frac {c^{2} a^{4} \ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {c^{2} a^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{6}+\frac {8 i c^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{15 a}+\frac {4 i c^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{15 a}+\frac {8 i c^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{15 a}-\frac {c^{2} \ln \left (i a x +1\right )^{2} x}{4}-\frac {c^{2} \ln \left (-i a x +1\right )^{2} x}{4}-\frac {11 c^{2} \arctan \left (a x \right )}{30 a}+\frac {3739 i c^{2}}{6750 a}-\frac {2 i c^{2} \ln \left (-i a x +1\right )^{2}}{15 a}-\frac {7 i c^{2} a \ln \left (-i a x +1\right ) x^{2}}{30}-\frac {i c^{2} a^{3} \ln \left (-i a x +1\right ) x^{4}}{20}+\frac {c^{2} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{3}+\frac {c^{2} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}\)	\(438\)

[In]

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

[Out]

1/a*(1/5*c^2*arctan(a*x)^2*a^5*x^5+2/3*a^3*c^2*x^3*arctan(a*x)^2+a*c^2*x*arctan(a*x)^2-2/15*c^2*(3/4*arctan(a*
x)*a^4*x^4+7/2*a^2*arctan(a*x)*x^2+4*arctan(a*x)*ln(a^2*x^2+1)-1/4*a^3*x^3-11/4*a*x+11/4*arctan(a*x)+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)-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]

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

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^2, x)

Sympy [F]

\[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=c^{2} \left (\int 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(2*a**2*x**2*atan(a*x)**2, x) + Integral(a**4*x**4*atan(a*x)**2, x) + Integral(atan(a*x)**2, x))

Maxima [F]

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

[In]

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

[Out]

180*a^6*c^2*integrate(1/240*x^6*arctan(a*x)^2/(a^2*x^2 + 1), x) + 15*a^6*c^2*integrate(1/240*x^6*log(a^2*x^2 +
1)^2/(a^2*x^2 + 1), x) + 12*a^6*c^2*integrate(1/240*x^6*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 24*a^5*c^2*integ
rate(1/240*x^5*arctan(a*x)/(a^2*x^2 + 1), x) + 540*a^4*c^2*integrate(1/240*x^4*arctan(a*x)^2/(a^2*x^2 + 1), x)
+ 45*a^4*c^2*integrate(1/240*x^4*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 40*a^4*c^2*integrate(1/240*x^4*log(a^
2*x^2 + 1)/(a^2*x^2 + 1), x) - 80*a^3*c^2*integrate(1/240*x^3*arctan(a*x)/(a^2*x^2 + 1), x) + 540*a^2*c^2*inte
grate(1/240*x^2*arctan(a*x)^2/(a^2*x^2 + 1), x) + 45*a^2*c^2*integrate(1/240*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 +
1), x) + 60*a^2*c^2*integrate(1/240*x^2*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) + 1/4*c^2*arctan(a*x)^3/a - 120*a*
c^2*integrate(1/240*x*arctan(a*x)/(a^2*x^2 + 1), x) + 1/60*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arctan(
a*x)^2 + 15*c^2*integrate(1/240*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) - 1/240*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 +
15*c^2*x)*log(a^2*x^2 + 1)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(atan(a*x)^2*(c + a^2*c*x^2)^2,x)

[Out]

int(atan(a*x)^2*(c + a^2*c*x^2)^2, x)