\(\int x (c+a^2 c x^2)^2 \arctan (a x)^3 \, dx\) [373]

Optimal result

Integrand size = 20, antiderivative size = 242 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=-\frac {11 c^2 x}{60 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {4 i c^2 \arctan (a x)^2}{15 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {8 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^2}-\frac {4 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^2} \]

[Out]

-11/60*c^2*x/a-1/60*a*c^2*x^3+2/15*c^2*(a^2*x^2+1)*arctan(a*x)/a^2+1/20*c^2*(a^2*x^2+1)^2*arctan(a*x)/a^2-4/15
*I*c^2*arctan(a*x)^2/a^2-4/15*c^2*x*arctan(a*x)^2/a-2/15*c^2*x*(a^2*x^2+1)*arctan(a*x)^2/a-1/10*c^2*x*(a^2*x^2
+1)^2*arctan(a*x)^2/a+1/6*c^2*(a^2*x^2+1)^3*arctan(a*x)^3/a^2-8/15*c^2*arctan(a*x)*ln(2/(1+I*a*x))/a^2-4/15*I*
c^2*polylog(2,1-2/(1+I*a*x))/a^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5050, 5000, 4930, 5040, 4964, 2449, 2352, 8} \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=-\frac {c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{10 a}-\frac {2 c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^2}{15 a}+\frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^3}{6 a^2}+\frac {c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)}{20 a^2}+\frac {2 c^2 \left (a^2 x^2+1\right ) \arctan (a x)}{15 a^2}-\frac {4 i c^2 \arctan (a x)^2}{15 a^2}-\frac {8 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^2}-\frac {4 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{15 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {1}{60} a c^2 x^3-\frac {11 c^2 x}{60 a} \]

[In]

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

[Out]

(-11*c^2*x)/(60*a) - (a*c^2*x^3)/60 + (2*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a^2) + (c^2*(1 + a^2*x^2)^2*ArcTan
[a*x])/(20*a^2) - (((4*I)/15)*c^2*ArcTan[a*x]^2)/a^2 - (4*c^2*x*ArcTan[a*x]^2)/(15*a) - (2*c^2*x*(1 + a^2*x^2)
*ArcTan[a*x]^2)/(15*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(10*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/(6
*a^2) - (8*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^2) - (((4*I)/15)*c^2*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^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {\int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx}{2 a} \\ & = \frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {c \int \left (c+a^2 c x^2\right ) \, dx}{20 a}-\frac {(2 c) \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx}{5 a} \\ & = -\frac {c^2 x}{20 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {\left (2 c^2\right ) \int 1 \, dx}{15 a}-\frac {\left (4 c^2\right ) \int \arctan (a x)^2 \, dx}{15 a} \\ & = -\frac {11 c^2 x}{60 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}+\frac {1}{15} \left (8 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {11 c^2 x}{60 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {4 i c^2 \arctan (a x)^2}{15 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {\left (8 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{15 a} \\ & = -\frac {11 c^2 x}{60 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {4 i c^2 \arctan (a x)^2}{15 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {8 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^2}+\frac {\left (8 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{15 a} \\ & = -\frac {11 c^2 x}{60 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {4 i c^2 \arctan (a x)^2}{15 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {8 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^2}-\frac {\left (8 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^2} \\ & = -\frac {11 c^2 x}{60 a}-\frac {1}{60} a c^2 x^3+\frac {2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}{15 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)}{20 a^2}-\frac {4 i c^2 \arctan (a x)^2}{15 a^2}-\frac {4 c^2 x \arctan (a x)^2}{15 a}-\frac {2 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{15 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{10 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^3}{6 a^2}-\frac {8 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^2}-\frac {4 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.54 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 \left (-a x \left (11+a^2 x^2\right )-2 \left (-8 i+15 a x+10 a^3 x^3+3 a^5 x^5\right ) \arctan (a x)^2+10 \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\arctan (a x) \left (11+14 a^2 x^2+3 a^4 x^4-32 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+16 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{60 a^2} \]

[In]

Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,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 + 10*(1 + a^2*x^2)^3*Ar
cTan[a*x]^3 + 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])]))/(60*a^2)

Maple [A] (verified)

Time = 3.73 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.22

method	result	size
parts	\(\frac {c^{2} \arctan \left (a x \right )^{3} a^{4} x^{6}}{6}+\frac {c^{2} \arctan \left (a x \right )^{3} a^{2} x^{4}}{2}+\frac {c^{2} \arctan \left (a x \right )^{3} x^{2}}{2}+\frac {c^{2} \arctan \left (a x \right )^{3}}{6 a^{2}}-\frac {c^{2} \left (\frac {a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {2 a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{4} x^{4}}{10}-\frac {7 a^{2} \arctan \left (a x \right ) x^{2}}{15}-\frac {8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{30}+\frac {11 a x}{30}-\frac {11 \arctan \left (a x \right )}{30}-\frac {4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{15}+\frac {4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{15}\right )}{2 a^{2}}\)	\(296\)
derivativedivides	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{3} a^{6} x^{6}}{6}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{3}}{2}+\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c^{2} \arctan \left (a x \right )^{3}}{6}-\frac {c^{2} \left (\frac {a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {2 a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{4} x^{4}}{10}-\frac {7 a^{2} \arctan \left (a x \right ) x^{2}}{15}-\frac {8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{30}+\frac {11 a x}{30}-\frac {11 \arctan \left (a x \right )}{30}-\frac {4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{15}+\frac {4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{15}\right )}{2}}{a^{2}}\)	\(297\)
default	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{3} a^{6} x^{6}}{6}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{3}}{2}+\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c^{2} \arctan \left (a x \right )^{3}}{6}-\frac {c^{2} \left (\frac {a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {2 a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}+a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{4} x^{4}}{10}-\frac {7 a^{2} \arctan \left (a x \right ) x^{2}}{15}-\frac {8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{30}+\frac {11 a x}{30}-\frac {11 \arctan \left (a x \right )}{30}-\frac {4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{15}+\frac {4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{15}\right )}{2}}{a^{2}}\)	\(297\)

[In]

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

[Out]

1/6*c^2*arctan(a*x)^3*a^4*x^6+1/2*c^2*arctan(a*x)^3*a^2*x^4+1/2*c^2*arctan(a*x)^3*x^2+1/6*c^2*arctan(a*x)^3/a^
2-1/2/a^2*c^2*(1/5*a^5*arctan(a*x)^2*x^5+2/3*a^3*arctan(a*x)^2*x^3+a*arctan(a*x)^2*x-1/10*arctan(a*x)*a^4*x^4-
7/15*a^2*arctan(a*x)*x^2-8/15*arctan(a*x)*ln(a^2*x^2+1)+1/30*a^3*x^3+11/30*a*x-11/30*arctan(a*x)-4/15*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)+4/15*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]

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

[In]

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

[Out]

integral((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*arctan(a*x)^3, x)

Sympy [F]

\[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=c^{2} \left (\int x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(x*atan(a*x)**3, x) + Integral(2*a**2*x**3*atan(a*x)**3, x) + Integral(a**4*x**5*atan(a*x)**3, x
))

Maxima [F]

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

[In]

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

[Out]

1/480*(40*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*arctan(a*x)^3 + 20*(5760*a^7*c^2*integrate(1/480
*x^7*arctan(a*x)^3/(a^3*x^2 + a), x) - 1440*a^6*c^2*integrate(1/480*x^6*arctan(a*x)^2/(a^3*x^2 + a), x) - 360*
a^6*c^2*integrate(1/480*x^6*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 288*a^6*c^2*integrate(1/480*x^6*log(a^2*x^2
+ 1)/(a^3*x^2 + a), x) + 17280*a^5*c^2*integrate(1/480*x^5*arctan(a*x)^3/(a^3*x^2 + a), x) + 576*a^5*c^2*inte
grate(1/480*x^5*arctan(a*x)/(a^3*x^2 + a), x) - 4320*a^4*c^2*integrate(1/480*x^4*arctan(a*x)^2/(a^3*x^2 + a),
x) - 1080*a^4*c^2*integrate(1/480*x^4*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 960*a^4*c^2*integrate(1/480*x^4*l
og(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 17280*a^3*c^2*integrate(1/480*x^3*arctan(a*x)^3/(a^3*x^2 + a), x) + 1920*a
^3*c^2*integrate(1/480*x^3*arctan(a*x)/(a^3*x^2 + a), x) - 4320*a^2*c^2*integrate(1/480*x^2*arctan(a*x)^2/(a^3
*x^2 + a), x) - 1080*a^2*c^2*integrate(1/480*x^2*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 1440*a^2*c^2*integrate
(1/480*x^2*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 5760*a*c^2*integrate(1/480*x*arctan(a*x)^3/(a^3*x^2 + a), x) +
2880*a*c^2*integrate(1/480*x*arctan(a*x)/(a^3*x^2 + a), x) - c^2*arctan(a*x)^3/a^2 - 360*c^2*integrate(1/480*
log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x))*a^2 - 4*(3*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*arctan(a*x)^2 + (3
*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*log(a^2*x^2 + 1)^2)/a^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(x*atan(a*x)^3*(c + a^2*c*x^2)^2,x)

[Out]

int(x*atan(a*x)^3*(c + a^2*c*x^2)^2, x)