3.10 \(\int (d+e x)^2 (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=270 \[ -\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {2 b \left (3 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}-\frac {2 a b d e x}{c}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}+\frac {b^2 d e \log \left (c^2 x^2+1\right )}{c^2}+\frac {b^2 e^2 x}{3 c^2}+\frac {i b^2 \left (3 c^2 d^2-e^2\right ) \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c} \]

[Out]

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

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Rubi [A]  time = 0.40, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4864, 4846, 260, 4852, 321, 203, 4984, 4884, 4920, 4854, 2402, 2315} \[ \frac {i b^2 \left (3 c^2 d^2-e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {2 b \left (3 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}-\frac {2 a b d e x}{c}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {b^2 d e \log \left (c^2 x^2+1\right )}{c^2}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*d*e*x)/c + (b^2*e^2*x)/(3*c^2) - (b^2*e^2*ArcTan[c*x])/(3*c^3) - (2*b^2*d*e*x*ArcTan[c*x])/c - (b*e^2*
x^2*(a + b*ArcTan[c*x]))/(3*c) + ((I/3)*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^3 - (d*(d^2 - (3*e^2)/c^2)*
(a + b*ArcTan[c*x])^2)/(3*e) + ((d + e*x)^3*(a + b*ArcTan[c*x])^2)/(3*e) + (2*b*(3*c^2*d^2 - e^2)*(a + b*ArcTa
n[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (b^2*d*e*Log[1 + c^2*x^2])/c^2 + ((I/3)*b^2*(3*c^2*d^2 - e^2)*PolyLog[2,
1 - 2/(1 + I*c*x)])/c^3

Rule 203

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

Rule 260

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 321

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2402

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 4846

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

Rule 4852

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

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 4864

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 4884

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 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4984

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*} \int (d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {(2 b c) \int \left (\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac {e^3 x \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac {\left (c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 e}\\ &=\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {(2 b) \int \frac {\left (c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c e}-\frac {(2 b d e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}-\frac {\left (2 b e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}\\ &=-\frac {2 a b d e x}{c}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac {(2 b) \int \left (\frac {c^2 d^3 \left (1-\frac {3 e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}-\frac {e \left (-3 c^2 d^2+e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{3 c e}-\frac {\left (2 b^2 d e\right ) \int \tan ^{-1}(c x) \, dx}{c}+\frac {1}{3} \left (b^2 e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx\\ &=-\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\left (2 b^2 d e\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {\left (b^2 e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}-\frac {1}{3} \left (2 b d \left (\frac {c d^2}{e}-\frac {3 e}{c}\right )\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\frac {\left (2 b \left (3 c^2 d^2-e^2\right )\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac {\left (2 b \left (3 c^2 d^2-e^2\right )\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}\\ &=-\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {2 b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {b^2 d e \log \left (1+c^2 x^2\right )}{c^2}-\frac {\left (2 b^2 \left (3 c^2 d^2-e^2\right )\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=-\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {2 b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac {\left (2 i b^2 \left (3 c^2 d^2-e^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3}\\ &=-\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac {2 b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac {i b^2 \left (3 c^2 d^2-e^2\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 312, normalized size = 1.16 \[ \frac {3 a^2 c^3 d^2 x+3 a^2 c^3 d e x^2+a^2 c^3 e^2 x^3-3 a b c^2 d^2 \log \left (c^2 x^2+1\right )-6 a b c^2 d e x-a b c^2 e^2 x^2+a b e^2 \log \left (c^2 x^2+1\right )+b \tan ^{-1}(c x) \left (2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+6 a c d e+2 b \left (3 c^2 d^2-e^2\right ) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-b e \left (6 c^2 d x+c^2 e x^2+e\right )\right )-i b^2 \left (3 c^2 d^2-e^2\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+3 b^2 c d e \log \left (c^2 x^2+1\right )+b^2 \tan ^{-1}(c x)^2 \left (c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-3 i c^2 d^2+3 c d e+i e^2\right )+b^2 c e^2 x}{3 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} e^{2} x^{2} + 2 \, a^{2} d e x + a^{2} d^{2} + {\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{2} + 2 \, a b d e x + a b d^{2}\right )} \arctan \left (c x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [B]  time = 0.11, size = 750, normalized size = 2.78 \[ -\frac {b^{2} e^{2} \arctan \left (c x \right )}{3 c^{3}}+\frac {b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{c^{2}}-\frac {2 a b d e x}{c}-\frac {2 b^{2} d e x \arctan \left (c x \right )}{c}+2 a b e \arctan \left (c x \right ) x^{2} d +\frac {2 a b e \arctan \left (c x \right ) d}{c^{2}}+\frac {a^{2} e^{2} x^{3}}{3}+a^{2} x \,d^{2}+\frac {i b^{2} \ln \left (c x -i\right )^{2} d^{2}}{4 c}+\frac {i b^{2} \ln \left (c x +i\right )^{2} e^{2}}{12 c^{3}}-\frac {i b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right ) d^{2}}{2 c}+\frac {i b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right ) e^{2}}{6 c^{3}}-\frac {i b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right ) e^{2}}{6 c^{3}}+\frac {i b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right ) d^{2}}{2 c}+\frac {a b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}-\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{2}}{3 c}-\frac {b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) d^{2}}{c}+2 a b \arctan \left (c x \right ) x \,d^{2}+\frac {2 a b \,e^{2} \arctan \left (c x \right ) x^{3}}{3}+\frac {b^{2} e^{2} x}{3 c^{2}}+\frac {b^{2} e \arctan \left (c x \right )^{2} d}{c^{2}}-\frac {a b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}-\frac {i b^{2} \ln \left (c x +i\right )^{2} d^{2}}{4 c}-\frac {i b^{2} \ln \left (c x -i\right )^{2} e^{2}}{12 c^{3}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2} x^{3}}{3}+b^{2} \arctan \left (c x \right )^{2} x \,d^{2}+a^{2} e \,x^{2} d -\frac {i b^{2} \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x +i\right ) e^{2}}{6 c^{3}}+\frac {i b^{2} \ln \left (\frac {i \left (c x -i\right )}{2}\right ) \ln \left (c x +i\right ) e^{2}}{6 c^{3}}-\frac {i b^{2} \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x -i\right ) d^{2}}{2 c}+\frac {i b^{2} \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right ) d^{2}}{2 c}+\frac {i b^{2} \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x +i\right ) d^{2}}{2 c}-\frac {i b^{2} \ln \left (\frac {i \left (c x -i\right )}{2}\right ) \ln \left (c x +i\right ) d^{2}}{2 c}+\frac {i b^{2} \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x -i\right ) e^{2}}{6 c^{3}}-\frac {i b^{2} \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right ) e^{2}}{6 c^{3}}+\frac {a^{2} d^{3}}{3 e}-\frac {a b \,x^{2} e^{2}}{3 c}+b^{2} e \arctan \left (c x \right )^{2} x^{2} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2*e^2*arctan(c*x)/c^3+b^2*d*e*ln(c^2*x^2+1)/c^2-2*a*b*d*e*x/c-2*b^2*d*e*x*arctan(c*x)/c+1/2*I/c*b^2*ln(
-1/2*I*(I+c*x))*ln(c*x-I)*d^2+2*a*b*e*arctan(c*x)*x^2*d+1/2*I/c*b^2*ln(c^2*x^2+1)*ln(I+c*x)*d^2-1/2*I/c*b^2*ln
(1/2*I*(c*x-I))*ln(I+c*x)*d^2+1/6*I/c^3*b^2*ln(c^2*x^2+1)*ln(c*x-I)*e^2-1/6*I/c^3*b^2*ln(-1/2*I*(I+c*x))*ln(c*
x-I)*e^2-1/6*I/c^3*b^2*ln(c^2*x^2+1)*ln(I+c*x)*e^2+2/c^2*a*b*e*arctan(c*x)*d+1/3*a^2*e^2*x^3+a^2*x*d^2+1/6*I/c
^3*b^2*ln(1/2*I*(c*x-I))*ln(I+c*x)*e^2-1/2*I/c*b^2*ln(c^2*x^2+1)*ln(c*x-I)*d^2+1/3/c^3*a*b*e^2*ln(c^2*x^2+1)-1
/2*I/c*b^2*dilog(1/2*I*(c*x-I))*d^2-1/3/c*b^2*e^2*arctan(c*x)*x^2-1/c*b^2*arctan(c*x)*ln(c^2*x^2+1)*d^2+2*a*b*
arctan(c*x)*x*d^2+2/3*a*b*e^2*arctan(c*x)*x^3+1/3*b^2*e^2*x/c^2+1/c^2*b^2*e*arctan(c*x)^2*d+1/6*I/c^3*b^2*dilo
g(1/2*I*(c*x-I))*e^2-1/12*I/c^3*b^2*ln(c*x-I)^2*e^2-1/6*I/c^3*b^2*dilog(-1/2*I*(I+c*x))*e^2-1/c*a*b*ln(c^2*x^2
+1)*d^2+1/3/c^3*b^2*e^2*arctan(c*x)*ln(c^2*x^2+1)+1/2*I/c*b^2*dilog(-1/2*I*(I+c*x))*d^2-1/4*I/c*b^2*ln(I+c*x)^
2*d^2+1/12*I/c^3*b^2*ln(I+c*x)^2*e^2+1/3*b^2*e^2*arctan(c*x)^2*x^3+b^2*arctan(c*x)^2*x*d^2+a^2*e*x^2*d+1/3*a^2
/e*d^3+1/4*I/c*b^2*ln(c*x-I)^2*d^2-1/3/c*a*b*x^2*e^2+b^2*e*arctan(c*x)^2*x^2*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} e^{2} x^{3} + 36 \, b^{2} c^{2} e^{2} \int \frac {x^{4} \arctan \left (c x\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 3 \, b^{2} c^{2} e^{2} \int \frac {x^{4} \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 72 \, b^{2} c^{2} d e \int \frac {x^{3} \arctan \left (c x\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 4 \, b^{2} c^{2} e^{2} \int \frac {x^{4} \log \left (c^{2} x^{2} + 1\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 6 \, b^{2} c^{2} d e \int \frac {x^{3} \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 36 \, b^{2} c^{2} d^{2} \int \frac {x^{2} \arctan \left (c x\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 12 \, b^{2} c^{2} d e \int \frac {x^{3} \log \left (c^{2} x^{2} + 1\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 3 \, b^{2} c^{2} d^{2} \int \frac {x^{2} \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 12 \, b^{2} c^{2} d^{2} \int \frac {x^{2} \log \left (c^{2} x^{2} + 1\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + a^{2} d e x^{2} + \frac {b^{2} d^{2} \arctan \left (c x\right )^{3}}{4 \, c} - 8 \, b^{2} c e^{2} \int \frac {x^{3} \arctan \left (c x\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} - 24 \, b^{2} c d e \int \frac {x^{2} \arctan \left (c x\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} - 24 \, b^{2} c d^{2} \int \frac {x \arctan \left (c x\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 2 \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d e + \frac {1}{3} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b e^{2} + a^{2} d^{2} x + 36 \, b^{2} e^{2} \int \frac {x^{2} \arctan \left (c x\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 3 \, b^{2} e^{2} \int \frac {x^{2} \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 72 \, b^{2} d e \int \frac {x \arctan \left (c x\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 6 \, b^{2} d e \int \frac {x \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 3 \, b^{2} d^{2} \int \frac {\log \left (c^{2} x^{2} + 1\right )^{2}}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} a b d^{2}}{c} + \frac {1}{12} \, {\left (b^{2} e^{2} x^{3} + 3 \, b^{2} d e x^{2} + 3 \, b^{2} d^{2} x\right )} \arctan \left (c x\right )^{2} - \frac {1}{48} \, {\left (b^{2} e^{2} x^{3} + 3 \, b^{2} d e x^{2} + 3 \, b^{2} d^{2} x\right )} \log \left (c^{2} x^{2} + 1\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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