\(\int (d+e x)^3 (a+b \arctan (c x))^2 \, dx\) [9]

Optimal result

Integrand size = 18, antiderivative size = 376 \[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}+\frac {2 b d (c d-e) (c d+e) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {i b^2 d (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3} \]

[Out]

b^2*d*e^2*x/c^2-1/2*a*b*e*(6*c^2*d^2-e^2)*x/c^3+1/12*b^2*e^3*x^2/c^2-b^2*d*e^2*arctan(c*x)/c^3-1/2*b^2*e*(6*c^
2*d^2-e^2)*x*arctan(c*x)/c^3-b*d*e^2*x^2*(a+b*arctan(c*x))/c-1/6*b*e^3*x^3*(a+b*arctan(c*x))/c+I*d*(c*d-e)*(c*
d+e)*(a+b*arctan(c*x))^2/c^3-1/4*(c^4*d^4-6*c^2*d^2*e^2+e^4)*(a+b*arctan(c*x))^2/c^4/e+1/4*(e*x+d)^4*(a+b*arct
an(c*x))^2/e+2*b*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3-1/12*b^2*e^3*ln(c^2*x^2+1)/c^4+1/4*b^
2*e*(6*c^2*d^2-e^2)*ln(c^2*x^2+1)/c^4+I*b^2*d*(c*d-e)*(c*d+e)*polylog(2,1-2/(1+I*c*x))/c^3

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {4974, 4930, 266, 4946, 327, 209, 272, 45, 5104, 5004, 5040, 4964, 2449, 2352} \[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}+\frac {2 b d (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}-\frac {a b e x \left (6 c^2 d^2-e^2\right )}{2 c^3}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e x \arctan (c x) \left (6 c^2 d^2-e^2\right )}{2 c^3}+\frac {i b^2 d (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^3}+\frac {b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2}{12 c^2}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{4 c^4}-\frac {b^2 e^3 \log \left (c^2 x^2+1\right )}{12 c^4} \]

[In]

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

[Out]

(b^2*d*e^2*x)/c^2 - (a*b*e*(6*c^2*d^2 - e^2)*x)/(2*c^3) + (b^2*e^3*x^2)/(12*c^2) - (b^2*d*e^2*ArcTan[c*x])/c^3
- (b^2*e*(6*c^2*d^2 - e^2)*x*ArcTan[c*x])/(2*c^3) - (b*d*e^2*x^2*(a + b*ArcTan[c*x]))/c - (b*e^3*x^3*(a + b*A
rcTan[c*x]))/(6*c) + (I*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^2)/c^3 - ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a
+ b*ArcTan[c*x])^2)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTan[c*x])^2)/(4*e) + (2*b*d*(c*d - e)*(c*d + e)*(a + b
*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^3 - (b^2*e^3*Log[1 + c^2*x^2])/(12*c^4) + (b^2*e*(6*c^2*d^2 - e^2)*Log[1 +
c^2*x^2])/(4*c^4) + (I*b^2*d*(c*d - e)*(c*d + e)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -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 4974

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

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[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 5104

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {(b c) \int \left (\frac {e^2 \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))}{c^4}+\frac {4 d e^3 x (a+b \arctan (c x))}{c^2}+\frac {e^4 x^2 (a+b \arctan (c x))}{c^2}+\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x\right ) (a+b \arctan (c x))}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{2 e} \\ & = \frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {b \int \frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 c^3 e}-\frac {\left (2 b d e^2\right ) \int x (a+b \arctan (c x)) \, dx}{c}-\frac {\left (b e^3\right ) \int x^2 (a+b \arctan (c x)) \, dx}{2 c}-\frac {\left (b e \left (6 c^2 d^2-e^2\right )\right ) \int (a+b \arctan (c x)) \, dx}{2 c^3} \\ & = -\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {b \int \left (\frac {c^4 d^4 \left (1+\frac {-6 c^2 d^2 e^2+e^4}{c^4 d^4}\right ) (a+b \arctan (c x))}{1+c^2 x^2}+\frac {4 c^2 d (c d-e) e (c d+e) x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{2 c^3 e}+\left (b^2 d e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {1}{6} \left (b^2 e^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {\left (b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \arctan (c x) \, dx}{2 c^3} \\ & = \frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {\left (b^2 d e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c^2}+\frac {1}{12} \left (b^2 e^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {(2 b d (c d-e) (c d+e)) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c}+\frac {\left (b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right )\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 c^3 e} \\ & = \frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{12} \left (b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {(2 b d (c d-e) (c d+e)) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{c^2} \\ & = \frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}+\frac {2 b d (c d-e) (c d+e) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}-\frac {\left (2 b^2 d (c d-e) (c d+e)\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2} \\ & = \frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}+\frac {2 b d (c d-e) (c d+e) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {\left (2 i b^2 d (c d-e) (c d+e)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^3} \\ & = \frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}+\frac {2 b d (c d-e) (c d+e) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {i b^2 d (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.26 \[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\frac {b^2 e^3+12 a^2 c^4 d^3 x-36 a b c^3 d^2 e x+12 b^2 c^2 d e^2 x+6 a b c e^3 x+18 a^2 c^4 d^2 e x^2-12 a b c^3 d e^2 x^2+b^2 c^2 e^3 x^2+12 a^2 c^4 d e^2 x^3-2 a b c^3 e^3 x^3+3 a^2 c^4 e^3 x^4+3 b^2 \left (-4 i c^3 d^3+6 c^2 d^2 e+4 i c d e^2-e^3+c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \arctan (c x)^2+2 b \arctan (c x) \left (-b c e \left (18 c^2 d^2 x+e^2 x \left (-3+c^2 x^2\right )+6 d \left (e+c^2 e x^2\right )\right )+3 a \left (6 c^2 d^2 e-e^3+c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+12 b c d \left (c^2 d^2-e^2\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )-12 a b c^3 d^3 \log \left (1+c^2 x^2\right )+18 b^2 c^2 d^2 e \log \left (1+c^2 x^2\right )+12 a b c d e^2 \log \left (1+c^2 x^2\right )-4 b^2 e^3 \log \left (1+c^2 x^2\right )-12 i b^2 c d \left (c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{12 c^4} \]

[In]

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

[Out]

(b^2*e^3 + 12*a^2*c^4*d^3*x - 36*a*b*c^3*d^2*e*x + 12*b^2*c^2*d*e^2*x + 6*a*b*c*e^3*x + 18*a^2*c^4*d^2*e*x^2 -
12*a*b*c^3*d*e^2*x^2 + b^2*c^2*e^3*x^2 + 12*a^2*c^4*d*e^2*x^3 - 2*a*b*c^3*e^3*x^3 + 3*a^2*c^4*e^3*x^4 + 3*b^2
*((-4*I)*c^3*d^3 + 6*c^2*d^2*e + (4*I)*c*d*e^2 - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcT
an[c*x]^2 + 2*b*ArcTan[c*x]*(-(b*c*e*(18*c^2*d^2*x + e^2*x*(-3 + c^2*x^2) + 6*d*(e + c^2*e*x^2))) + 3*a*(6*c^2
*d^2*e - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + 12*b*c*d*(c^2*d^2 - e^2)*Log[1 + E^((2*I)*
ArcTan[c*x])]) - 12*a*b*c^3*d^3*Log[1 + c^2*x^2] + 18*b^2*c^2*d^2*e*Log[1 + c^2*x^2] + 12*a*b*c*d*e^2*Log[1 +
c^2*x^2] - 4*b^2*e^3*Log[1 + c^2*x^2] - (12*I)*b^2*c*d*(c^2*d^2 - e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(12
*c^4)

Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.77

method	result	size
parts	\(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (\frac {c \,e^{3} \arctan \left (c x \right )^{2} x^{4}}{4}+c \,e^{2} \arctan \left (c x \right )^{2} x^{3} d +\frac {3 c e \arctan \left (c x \right )^{2} x^{2} d^{2}}{2}+\arctan \left (c x \right )^{2} c x \,d^{3}+\frac {c \arctan \left (c x \right )^{2} d^{4}}{4 e}-\frac {6 \arctan \left (c x \right ) c^{3} d^{2} e^{2} x +2 \arctan \left (c x \right ) e^{3} c^{3} d \,x^{2}+\frac {\arctan \left (c x \right ) e^{4} c^{3} x^{3}}{3}-\arctan \left (c x \right ) e^{4} c x +2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{3} d^{3} e -2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d \,e^{3}+\arctan \left (c x \right )^{2} c^{4} d^{4}-6 \arctan \left (c x \right )^{2} c^{2} d^{2} e^{2}+\arctan \left (c x \right )^{2} e^{4}-\frac {\left (6 c^{4} d^{4}-36 c^{2} d^{2} e^{2}+6 e^{4}\right ) \arctan \left (c x \right )^{2}}{12}-\frac {e^{2} \left (6 c^{2} d e x +\frac {c^{2} e^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}-4 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-6 e \arctan \left (c x \right ) c d \right )}{3}-2 c d e \left (c^{2} d^{2}-e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2 c^{3} e}\right )}{c}+\frac {2 a b \left (\frac {c \,e^{3} \arctan \left (c x \right ) x^{4}}{4}+c \,e^{2} \arctan \left (c x \right ) x^{3} d +\frac {3 c e \arctan \left (c x \right ) x^{2} d^{2}}{2}+\arctan \left (c x \right ) c x \,d^{3}+\frac {c \arctan \left (c x \right ) d^{4}}{4 e}-\frac {6 c^{3} d^{2} e^{2} x +2 e^{3} c^{3} d \,x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-c \,e^{4} x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 c^{3} e}\right )}{c}\)	\(667\)
derivativedivides	\(\frac {\frac {a^{2} \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{4} d^{4}}{4 e}+\arctan \left (c x \right )^{2} c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right )^{2} c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right )^{2} c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {6 \arctan \left (c x \right ) c^{3} d^{2} e^{2} x +2 \arctan \left (c x \right ) e^{3} c^{3} d \,x^{2}+\frac {\arctan \left (c x \right ) e^{4} c^{3} x^{3}}{3}-\arctan \left (c x \right ) e^{4} c x +2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{3} d^{3} e -2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d \,e^{3}+\arctan \left (c x \right )^{2} c^{4} d^{4}-6 \arctan \left (c x \right )^{2} c^{2} d^{2} e^{2}+\arctan \left (c x \right )^{2} e^{4}-\frac {\left (6 c^{4} d^{4}-36 c^{2} d^{2} e^{2}+6 e^{4}\right ) \arctan \left (c x \right )^{2}}{12}-\frac {e^{2} \left (6 c^{2} d e x +\frac {c^{2} e^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}-4 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-6 e \arctan \left (c x \right ) c d \right )}{3}-2 c d e \left (c^{2} d^{2}-e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2 e}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{4} d^{4}}{4 e}+\arctan \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {6 c^{3} d^{2} e^{2} x +2 e^{3} c^{3} d \,x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-c \,e^{4} x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 e}\right )}{c^{3}}}{c}\)	\(691\)
default	\(\frac {\frac {a^{2} \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{4} d^{4}}{4 e}+\arctan \left (c x \right )^{2} c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right )^{2} c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right )^{2} c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {6 \arctan \left (c x \right ) c^{3} d^{2} e^{2} x +2 \arctan \left (c x \right ) e^{3} c^{3} d \,x^{2}+\frac {\arctan \left (c x \right ) e^{4} c^{3} x^{3}}{3}-\arctan \left (c x \right ) e^{4} c x +2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{3} d^{3} e -2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d \,e^{3}+\arctan \left (c x \right )^{2} c^{4} d^{4}-6 \arctan \left (c x \right )^{2} c^{2} d^{2} e^{2}+\arctan \left (c x \right )^{2} e^{4}-\frac {\left (6 c^{4} d^{4}-36 c^{2} d^{2} e^{2}+6 e^{4}\right ) \arctan \left (c x \right )^{2}}{12}-\frac {e^{2} \left (6 c^{2} d e x +\frac {c^{2} e^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}-4 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-6 e \arctan \left (c x \right ) c d \right )}{3}-2 c d e \left (c^{2} d^{2}-e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2 e}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{4} d^{4}}{4 e}+\arctan \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {6 c^{3} d^{2} e^{2} x +2 e^{3} c^{3} d \,x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-c \,e^{4} x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 e}\right )}{c^{3}}}{c}\)	\(691\)
risch	\(\text {Expression too large to display}\)	\(1348\)

[In]

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

[Out]

1/4*a^2*(e*x+d)^4/e+b^2/c*(1/4*c*e^3*arctan(c*x)^2*x^4+c*e^2*arctan(c*x)^2*x^3*d+3/2*c*e*arctan(c*x)^2*x^2*d^2
+arctan(c*x)^2*c*x*d^3+1/4*c/e*arctan(c*x)^2*d^4-1/2/c^3/e*(6*arctan(c*x)*c^3*d^2*e^2*x+2*arctan(c*x)*e^3*c^3*
d*x^2+1/3*arctan(c*x)*e^4*c^3*x^3-arctan(c*x)*e^4*c*x+2*arctan(c*x)*ln(c^2*x^2+1)*c^3*d^3*e-2*arctan(c*x)*ln(c
^2*x^2+1)*c*d*e^3+arctan(c*x)^2*c^4*d^4-6*arctan(c*x)^2*c^2*d^2*e^2+arctan(c*x)^2*e^4-1/12*(6*c^4*d^4-36*c^2*d
^2*e^2+6*e^4)*arctan(c*x)^2-1/3*e^2*(6*c^2*d*e*x+1/2*c^2*e^2*x^2+1/2*(18*c^2*d^2-4*e^2)*ln(c^2*x^2+1)-6*e*arct
an(c*x)*c*d)-2*c*d*e*(c^2*d^2-e^2)*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-dilog(-1/2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*
(c*x+I))-1/2*ln(c*x-I)^2)+1/2*I*(ln(c*x+I)*ln(c^2*x^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*
ln(c*x+I)^2))))+2*a*b/c*(1/4*c*e^3*arctan(c*x)*x^4+c*e^2*arctan(c*x)*x^3*d+3/2*c*e*arctan(c*x)*x^2*d^2+arctan(
c*x)*c*x*d^3+1/4*c/e*arctan(c*x)*d^4-1/4/c^3/e*(6*c^3*d^2*e^2*x+2*e^3*c^3*d*x^2+1/3*e^4*c^3*x^3-c*e^4*x+1/2*(4
*c^3*d^3*e-4*c*d*e^3)*ln(c^2*x^2+1)+(c^4*d^4-6*c^2*d^2*e^2+e^4)*arctan(c*x)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + 12*b^2*c^2*e^3*integrate(1/16*x^5*arctan(c*x)^2/(c^2*x^2 + 1), x) + b^2*c^2*
e^3*integrate(1/16*x^5*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 36*b^2*c^2*d*e^2*integrate(1/16*x^4*arctan(c*x)^
2/(c^2*x^2 + 1), x) + b^2*c^2*e^3*integrate(1/16*x^5*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*b^2*c^2*d*e^2*inte
grate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 36*b^2*c^2*d^2*e*integrate(1/16*x^3*arctan(c*x)^2/(c^2*x
^2 + 1), x) + 4*b^2*c^2*d*e^2*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*b^2*c^2*d^2*e*integrat
e(1/16*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 12*b^2*c^2*d^3*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^2 + 1
), x) + 6*b^2*c^2*d^2*e*integrate(1/16*x^3*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + b^2*c^2*d^3*integrate(1/16*x^2
*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 4*b^2*c^2*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) +
3/2*a^2*d^2*e*x^2 + 1/4*b^2*d^3*arctan(c*x)^3/c - 2*b^2*c*e^3*integrate(1/16*x^4*arctan(c*x)/(c^2*x^2 + 1), x)
- 8*b^2*c*d*e^2*integrate(1/16*x^3*arctan(c*x)/(c^2*x^2 + 1), x) - 12*b^2*c*d^2*e*integrate(1/16*x^2*arctan(c
*x)/(c^2*x^2 + 1), x) - 8*b^2*c*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^2 + 1), x) + 3*(x^2*arctan(c*x) - c*(x
/c^2 - arctan(c*x)/c^3))*a*b*d^2*e + (2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d*e^2 + 1/6*
(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*e^3 + a^2*d^3*x + 12*b^2*e^3*integrate(1
/16*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + b^2*e^3*integrate(1/16*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 36
*b^2*d*e^2*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^2*d*e^2*integrate(1/16*x^2*log(c^2*x^2 + 1
)^2/(c^2*x^2 + 1), x) + 36*b^2*d^2*e*integrate(1/16*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^2*d^2*e*integrate(
1/16*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + b^2*d^3*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + (2
*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d^3/c + 1/16*(b^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2 + 4*b^2
*d^3*x)*arctan(c*x)^2 - 1/64*(b^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2 + 4*b^2*d^3*x)*log(c^2*x^2 + 1)^
2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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