3.3.77 \(\int (c+a^2 c x^2)^3 \text {ArcTan}(a x)^2 \, dx\) [277]

Optimal. Leaf size=268 \[ \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)}{21 a}+\frac {16 i c^3 \text {ArcTan}(a x)^2}{35 a}+\frac {16}{35} c^3 x \text {ArcTan}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \text {ArcTan}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)^2+\frac {32 c^3 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a} \]

[Out]

38/105*c^3*x+19/315*a^2*c^3*x^3+1/105*a^4*c^3*x^5-8/35*c^3*(a^2*x^2+1)*arctan(a*x)/a-3/35*c^3*(a^2*x^2+1)^2*ar
ctan(a*x)/a-1/21*c^3*(a^2*x^2+1)^3*arctan(a*x)/a+16/35*I*c^3*arctan(a*x)^2/a+16/35*c^3*x*arctan(a*x)^2+8/35*c^
3*x*(a^2*x^2+1)*arctan(a*x)^2+6/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^2+1/7*c^3*x*(a^2*x^2+1)^3*arctan(a*x)^2+32/
35*c^3*arctan(a*x)*ln(2/(1+I*a*x))/a+16/35*I*c^3*polylog(2,1-2/(1+I*a*x))/a

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Rubi [A]
time = 0.14, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5000, 4930, 5040, 4964, 2449, 2352, 8, 200} \begin {gather*} \frac {1}{105} a^4 c^3 x^5+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \text {ArcTan}(a x)^2+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)^2+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \text {ArcTan}(a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \text {ArcTan}(a x)}{21 a}-\frac {3 c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)}{35 a}-\frac {8 c^3 \left (a^2 x^2+1\right ) \text {ArcTan}(a x)}{35 a}+\frac {19}{315} a^2 c^3 x^3+\frac {16}{35} c^3 x \text {ArcTan}(a x)^2+\frac {16 i c^3 \text {ArcTan}(a x)^2}{35 a}+\frac {32 c^3 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{35 a}+\frac {38 c^3 x}{105} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(38*c^3*x)/105 + (19*a^2*c^3*x^3)/315 + (a^4*c^3*x^5)/105 - (8*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*
(1 + a^2*x^2)^2*ArcTan[a*x])/(35*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(21*a) + (((16*I)/35)*c^3*ArcTan[a*x]^
2)/a + (16*c^3*x*ArcTan[a*x]^2)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTa
n[a*x]^2)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/7 + (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a) + (((1
6*I)/35)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a

Rule 8

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx &=-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{21} c \int \left (c+a^2 c x^2\right )^2 \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{21} c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx+\frac {1}{35} \left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx\\ &=\frac {2 c^3 x}{15}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{35} \left (8 c^3\right ) \int 1 \, dx+\frac {1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2-\frac {1}{35} \left (32 a c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{35} \left (32 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {1}{35} \left (32 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {\left (32 i c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{35 a}\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{35 a}\\ \end {align*}

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Mathematica [A]
time = 0.87, size = 137, normalized size = 0.51 \begin {gather*} \frac {c^3 \left (a x \left (114+19 a^2 x^2+3 a^4 x^4\right )+9 \left (-16 i+35 a x+35 a^3 x^3+21 a^5 x^5+5 a^7 x^7\right ) \text {ArcTan}(a x)^2-3 \text {ArcTan}(a x) \left (38+57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-96 \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )\right )-144 i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )\right )}{315 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(a*x*(114 + 19*a^2*x^2 + 3*a^4*x^4) + 9*(-16*I + 35*a*x + 35*a^3*x^3 + 21*a^5*x^5 + 5*a^7*x^7)*ArcTan[a*x
]^2 - 3*ArcTan[a*x]*(38 + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 96*Log[1 + E^((2*I)*ArcTan[a*x])]) - (144*I)*P
olyLog[2, -E^((2*I)*ArcTan[a*x])]))/(315*a)

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Maple [A]
time = 0.21, size = 282, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 \arctan \left (a x \right ) a^{6} x^{6}}{6}+4 \arctan \left (a x \right ) a^{4} x^{4}+\frac {19 \arctan \left (a x \right ) a^{2} x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}-4 i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-2 i \ln \left (a x -i\right )^{2}-4 i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )+4 i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )+2 i \ln \left (a x +i\right )^{2}+4 i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )+4 i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-4 i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\)	\(282\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 \arctan \left (a x \right ) a^{6} x^{6}}{6}+4 \arctan \left (a x \right ) a^{4} x^{4}+\frac {19 \arctan \left (a x \right ) a^{2} x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}-4 i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-2 i \ln \left (a x -i\right )^{2}-4 i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )+4 i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )+2 i \ln \left (a x +i\right )^{2}+4 i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )+4 i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-4 i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\)	\(282\)
risch	\(\frac {c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}+\frac {38 c^{3} x}{105}+\frac {19 i c^{3} a \ln \left (i a x +1\right ) x^{2}}{70}+\frac {20469 i c^{3}}{42875 a}-\frac {c^{3} \ln \left (-i a x +1\right )^{2} x}{4}-\frac {c^{3} \ln \left (i a x +1\right )^{2} x}{4}+\frac {19 a^{2} c^{3} x^{3}}{315}+\frac {a^{4} c^{3} x^{5}}{105}-\frac {38 c^{3} \arctan \left (a x \right )}{105 a}+\frac {c^{3} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{2}+\frac {c^{3} a^{6} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}+\frac {3 c^{3} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {8 i c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{35 a}-\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{35 a}+\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}-\frac {i c^{3} a^{5} \ln \left (-i a x +1\right ) x^{6}}{42}+\frac {i c^{3} a^{5} \ln \left (i a x +1\right ) x^{6}}{42}-\frac {19 i c^{3} a \ln \left (-i a x +1\right ) x^{2}}{70}-\frac {4 i c^{3} a^{3} \ln \left (-i a x +1\right ) x^{4}}{35}+\frac {4 i c^{3} a^{3} \ln \left (i a x +1\right ) x^{4}}{35}+\frac {16 i c^{3} \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}-\frac {4 i c^{3} \ln \left (-i a x +1\right )^{2}}{35 a}+\frac {4 i c^{3} \ln \left (i a x +1\right )^{2}}{35 a}-\frac {c^{3} a^{6} \ln \left (-i a x +1\right )^{2} x^{7}}{28}-\frac {c^{3} a^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{4}-\frac {3 c^{3} a^{4} \ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {3 c^{3} a^{4} \ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {c^{3} a^{2} \ln \left (i a x +1\right )^{2} x^{3}}{4}-\frac {c^{3} a^{6} \ln \left (i a x +1\right )^{2} x^{7}}{28}\)	\(558\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(1/7*c^3*arctan(a*x)^2*a^7*x^7+3/5*a^5*c^3*x^5*arctan(a*x)^2+a^3*c^3*x^3*arctan(a*x)^2+a*c^3*x*arctan(a*x)
^2-2/35*c^3*(5/6*arctan(a*x)*a^6*x^6+4*arctan(a*x)*a^4*x^4+19/2*arctan(a*x)*a^2*x^2+8*arctan(a*x)*ln(a^2*x^2+1
)-1/6*a^5*x^5-19/18*a^3*x^3-19/3*a*x+19/3*arctan(a*x)-4*I*ln(a*x-I)*ln(-1/2*I*(I+a*x))-2*I*ln(a*x-I)^2-4*I*dil
og(-1/2*I*(I+a*x))+2*I*ln(I+a*x)^2+4*I*ln(a*x-I)*ln(a^2*x^2+1)+4*I*ln(I+a*x)*ln(1/2*I*(a*x-I))+4*I*dilog(1/2*I
*(a*x-I))-4*I*ln(I+a*x)*ln(a^2*x^2+1)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

420*a^8*c^3*integrate(1/560*x^8*arctan(a*x)^2/(a^2*x^2 + 1), x) + 35*a^8*c^3*integrate(1/560*x^8*log(a^2*x^2 +
1)^2/(a^2*x^2 + 1), x) + 20*a^8*c^3*integrate(1/560*x^8*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 40*a^7*c^3*integ
rate(1/560*x^7*arctan(a*x)/(a^2*x^2 + 1), x) + 1680*a^6*c^3*integrate(1/560*x^6*arctan(a*x)^2/(a^2*x^2 + 1), x
) + 140*a^6*c^3*integrate(1/560*x^6*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 84*a^6*c^3*integrate(1/560*x^6*log(
a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 168*a^5*c^3*integrate(1/560*x^5*arctan(a*x)/(a^2*x^2 + 1), x) + 2520*a^4*c^3*
integrate(1/560*x^4*arctan(a*x)^2/(a^2*x^2 + 1), x) + 210*a^4*c^3*integrate(1/560*x^4*log(a^2*x^2 + 1)^2/(a^2*
x^2 + 1), x) + 140*a^4*c^3*integrate(1/560*x^4*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 280*a^3*c^3*integrate(1/56
0*x^3*arctan(a*x)/(a^2*x^2 + 1), x) + 1680*a^2*c^3*integrate(1/560*x^2*arctan(a*x)^2/(a^2*x^2 + 1), x) + 140*a
^2*c^3*integrate(1/560*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 140*a^2*c^3*integrate(1/560*x^2*log(a^2*x^2
+ 1)/(a^2*x^2 + 1), x) + 1/4*c^3*arctan(a*x)^3/a - 280*a*c^3*integrate(1/560*x*arctan(a*x)/(a^2*x^2 + 1), x) +
35*c^3*integrate(1/560*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 1/140*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*
c^3*x^3 + 35*c^3*x)*arctan(a*x)^2 - 1/560*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*log(a^2
*x^2 + 1)^2

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(3*a**2*x**2*atan(a*x)**2, x) + Integral(3*a**4*x**4*atan(a*x)**2, x) + Integral(a**6*x**6*atan(
a*x)**2, x) + Integral(atan(a*x)**2, x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^2*(c + a^2*c*x^2)^3,x)

[Out]

int(atan(a*x)^2*(c + a^2*c*x^2)^3, x)

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