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

Optimal. Leaf size=192 \[ \frac{(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{2 e}-\frac{b e \log \left (c^2 x^4+1\right )}{4 c}-\frac{b d^2 \tan ^{-1}\left (c x^2\right )}{2 e}-\frac{b d \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b d \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b d \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2} \sqrt{c}} \]

[Out]

-(b*d^2*ArcTan[c*x^2])/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x^2]))/(2*e) + (b*d*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])
/(Sqrt[2]*Sqrt[c]) - (b*d*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*d*Log[1 - Sqrt[2]*Sqrt[c]*x +
c*x^2])/(2*Sqrt[2]*Sqrt[c]) + (b*d*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c]) - (b*e*Log[1 + c^2*
x^4])/(4*c)

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Rubi [A]  time = 0.208994, antiderivative size = 191, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6742, 5027, 297, 1162, 617, 204, 1165, 628, 5033, 260} \[ \frac{a (d+e x)^2}{2 e}-\frac{b e \log \left (c^2 x^4+1\right )}{4 c}-\frac{b d \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b d \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+b d x \tan ^{-1}\left (c x^2\right )+\frac{b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b d \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2} \sqrt{c}}+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(d + e*x)^2)/(2*e) + b*d*x*ArcTan[c*x^2] + (b*e*x^2*ArcTan[c*x^2])/2 + (b*d*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/
(Sqrt[2]*Sqrt[c]) - (b*d*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*d*Log[1 - Sqrt[2]*Sqrt[c]*x + c
*x^2])/(2*Sqrt[2]*Sqrt[c]) + (b*d*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c]) - (b*e*Log[1 + c^2*x
^4])/(4*c)

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 5027

Int[ArcTan[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTan[c*x^n], x] - Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x],
x] /; FreeQ[{c, n}, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

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

Rule 5033

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

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]

Rubi steps

\begin{align*} \int (d+e x) \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\int \left (a (d+e x)+b (d+e x) \tan ^{-1}\left (c x^2\right )\right ) \, dx\\ &=\frac{a (d+e x)^2}{2 e}+b \int (d+e x) \tan ^{-1}\left (c x^2\right ) \, dx\\ &=\frac{a (d+e x)^2}{2 e}+b \int \left (d \tan ^{-1}\left (c x^2\right )+e x \tan ^{-1}\left (c x^2\right )\right ) \, dx\\ &=\frac{a (d+e x)^2}{2 e}+(b d) \int \tan ^{-1}\left (c x^2\right ) \, dx+(b e) \int x \tan ^{-1}\left (c x^2\right ) \, dx\\ &=\frac{a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-(2 b c d) \int \frac{x^2}{1+c^2 x^4} \, dx-(b c e) \int \frac{x^3}{1+c^2 x^4} \, dx\\ &=\frac{a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-\frac{b e \log \left (1+c^2 x^4\right )}{4 c}+(b d) \int \frac{1-c x^2}{1+c^2 x^4} \, dx-(b d) \int \frac{1+c x^2}{1+c^2 x^4} \, dx\\ &=\frac{a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-\frac{b e \log \left (1+c^2 x^4\right )}{4 c}-\frac{(b d) \int \frac{1}{\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx}{2 c}-\frac{(b d) \int \frac{1}{\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx}{2 c}-\frac{(b d) \int \frac{\frac{\sqrt{2}}{\sqrt{c}}+2 x}{-\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{2 \sqrt{2} \sqrt{c}}-\frac{(b d) \int \frac{\frac{\sqrt{2}}{\sqrt{c}}-2 x}{-\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{2 \sqrt{2} \sqrt{c}}\\ &=\frac{a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-\frac{b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}-\frac{b e \log \left (1+c^2 x^4\right )}{4 c}-\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}+\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}\\ &=\frac{a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )+\frac{b d \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b d \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b d \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b d \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}-\frac{b e \log \left (1+c^2 x^4\right )}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0953756, size = 153, normalized size = 0.8 \[ a d x+\frac{1}{2} a e x^2-\frac{b e \log \left (c^2 x^4+1\right )}{4 c}+b d x \tan ^{-1}\left (c x^2\right )-\frac{b d \left (\log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )-\log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )\right )}{2 \sqrt{2} \sqrt{c}}+\frac{1}{2} b e x^2 \tan ^{-1}\left (c x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d*x + (a*e*x^2)/2 + b*d*x*ArcTan[c*x^2] + (b*e*x^2*ArcTan[c*x^2])/2 - (b*d*(-2*ArcTan[1 - Sqrt[2]*Sqrt[c]*x]
+ 2*ArcTan[1 + Sqrt[2]*Sqrt[c]*x] + Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2] - Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2]))
/(2*Sqrt[2]*Sqrt[c]) - (b*e*Log[1 + c^2*x^4])/(4*c)

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Maple [A]  time = 0.028, size = 167, normalized size = 0.9 \begin{align*}{\frac{a{x}^{2}e}{2}}+adx+{\frac{b\arctan \left ( c{x}^{2} \right ){x}^{2}e}{2}}+b\arctan \left ( c{x}^{2} \right ) dx-{\frac{bd\sqrt{2}}{4\,c}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) \left ({x}^{2}+\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{bd\sqrt{2}}{2\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{bd\sqrt{2}}{2\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{be\ln \left ({c}^{2}{x}^{4}+1 \right ) }{4\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2*e+a*d*x+1/2*b*arctan(c*x^2)*x^2*e+b*arctan(c*x^2)*d*x-1/4*b*d/c/(1/c^2)^(1/4)*2^(1/2)*ln((x^2-(1/c^2
)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2))/(x^2+(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)))-1/2*b*d/c/(1/c^2)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(1/c^2)^(1/4)*x+1)-1/2*b*d/c/(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x-1)-1/4*b*e*ln
(c^2*x^4+1)/c

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Maxima [A]  time = 1.50104, size = 404, normalized size = 2.1 \begin{align*} \frac{1}{2} \, a e x^{2} + \frac{1}{4} \,{\left (c{\left (\frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{c^{2}} \sqrt{-\sqrt{c^{2}}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{c^{2}} \sqrt{-\sqrt{c^{2}}}}\right )} + 4 \, x \arctan \left (c x^{2}\right )\right )} b d + a d x + \frac{{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b e}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e*x^2 + 1/4*(c*(sqrt(2)*log(sqrt(c^2)*x^2 + sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(3/4) - sqrt(2)*log(sqrt(c^
2)*x^2 - sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(3/4) - sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) + sqrt
(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^(1/4)))/(sqrt(c^2)*sqrt(-sqrt(c^2))
) - sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt
(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4)))/(sqrt(c^2)*sqrt(-sqrt(c^2)))) + 4*x*arctan(c*x^2))*b*d + a*d*x + 1/4*(2*c
*x^2*arctan(c*x^2) - log(c^2*x^4 + 1))*b*e/c

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Fricas [B]  time = 2.81062, size = 1166, normalized size = 6.07 \begin{align*} \frac{2 \, a b^{4} c d^{4} e x^{2} + 4 \, a b^{4} c d^{5} x + 4 \, \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{4} c d^{4} \arctan \left (-\frac{b^{8} d^{8} + \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{5}{4}} b^{3} c^{3} d^{3} x - \sqrt{2} \sqrt{b^{6} d^{6} x^{2} + \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c d^{3} x + \sqrt{\frac{b^{4} d^{4}}{c^{2}}} b^{4} d^{4}} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{5}{4}} c^{3}}{b^{8} d^{8}}\right ) + 4 \, \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{4} c d^{4} \arctan \left (\frac{b^{8} d^{8} - \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{5}{4}} b^{3} c^{3} d^{3} x + \sqrt{2} \sqrt{b^{6} d^{6} x^{2} - \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c d^{3} x + \sqrt{\frac{b^{4} d^{4}}{c^{2}}} b^{4} d^{4}} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{5}{4}} c^{3}}{b^{8} d^{8}}\right ) + 2 \,{\left (b^{5} c d^{4} e x^{2} + 2 \, b^{5} c d^{5} x\right )} \arctan \left (c x^{2}\right ) -{\left (b^{5} d^{4} e - \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{4} c d^{4}\right )} \log \left (b^{6} d^{6} x^{2} + \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c d^{3} x + \sqrt{\frac{b^{4} d^{4}}{c^{2}}} b^{4} d^{4}\right ) -{\left (b^{5} d^{4} e + \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{4} c d^{4}\right )} \log \left (b^{6} d^{6} x^{2} - \sqrt{2} \left (\frac{b^{4} d^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c d^{3} x + \sqrt{\frac{b^{4} d^{4}}{c^{2}}} b^{4} d^{4}\right )}{4 \, b^{4} c d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a*b^4*c*d^4*e*x^2 + 4*a*b^4*c*d^5*x + 4*sqrt(2)*(b^4*d^4/c^2)^(1/4)*b^4*c*d^4*arctan(-(b^8*d^8 + sqrt(2
)*(b^4*d^4/c^2)^(5/4)*b^3*c^3*d^3*x - sqrt(2)*sqrt(b^6*d^6*x^2 + sqrt(2)*(b^4*d^4/c^2)^(3/4)*b^3*c*d^3*x + sqr
t(b^4*d^4/c^2)*b^4*d^4)*(b^4*d^4/c^2)^(5/4)*c^3)/(b^8*d^8)) + 4*sqrt(2)*(b^4*d^4/c^2)^(1/4)*b^4*c*d^4*arctan((
b^8*d^8 - sqrt(2)*(b^4*d^4/c^2)^(5/4)*b^3*c^3*d^3*x + sqrt(2)*sqrt(b^6*d^6*x^2 - sqrt(2)*(b^4*d^4/c^2)^(3/4)*b
^3*c*d^3*x + sqrt(b^4*d^4/c^2)*b^4*d^4)*(b^4*d^4/c^2)^(5/4)*c^3)/(b^8*d^8)) + 2*(b^5*c*d^4*e*x^2 + 2*b^5*c*d^5
*x)*arctan(c*x^2) - (b^5*d^4*e - sqrt(2)*(b^4*d^4/c^2)^(1/4)*b^4*c*d^4)*log(b^6*d^6*x^2 + sqrt(2)*(b^4*d^4/c^2
)^(3/4)*b^3*c*d^3*x + sqrt(b^4*d^4/c^2)*b^4*d^4) - (b^5*d^4*e + sqrt(2)*(b^4*d^4/c^2)^(1/4)*b^4*c*d^4)*log(b^6
*d^6*x^2 - sqrt(2)*(b^4*d^4/c^2)^(3/4)*b^3*c*d^3*x + sqrt(b^4*d^4/c^2)*b^4*d^4))/(b^4*c*d^4)

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Sympy [A]  time = 31.7067, size = 1515, normalized size = 7.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*(d*x + e*x**2/2), Eq(c, 0)), ((a - oo*I*b)*(d*x + e*x**2/2), Eq(c, -I/x**2)), ((a + oo*I*b)*(d*x
+ e*x**2/2), Eq(c, I/x**2)), (-2*I*a*c**11*d*x**5*(c**(-2))**(11/2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c
**17*(c**(-2))**(19/2)) - I*a*c**11*e*x**6*(c**(-2))**(11/2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c
**(-2))**(19/2)) - 2*I*a*c**9*d*x*(c**(-2))**(11/2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**
(19/2)) - I*a*c**9*e*x**2*(c**(-2))**(11/2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2))
- 2*(-1)**(1/4)*b*c**18*d*x**4*(c**(-2))**(37/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(-2*I*c**19*x**4*(c**(-2
))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - 2*(-1)**(1/4)*b*c**16*d*(c**(-2))**(37/4)*atan((-1)**(3/4)*x/(c**(
-2))**(1/4))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) + 2*(-1)**(1/4)*b*c**14*d*x**4*
(c**(-2))**(29/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2
))**(19/2)) + 2*(-1)**(1/4)*b*c**12*d*(c**(-2))**(29/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(-2*I*c**19*x**4
*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - 2*I*b*c**11*d*x**5*(c**(-2))**(11/2)*atan(c*x**2)/(-2*I*c*
*19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - I*b*c**11*e*x**6*(c**(-2))**(11/2)*atan(c*x**2)/(-
2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - 2*I*b*c**9*d*x*(c**(-2))**(11/2)*atan(c*x**2
)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - I*b*c**9*e*x**2*(c**(-2))**(11/2)*atan(c
*x**2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - 2*(-1)**(3/4)*b*c**5*d*x**4*(c**(-2
))**(11/4)*atan(c*x**2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - 2*(-1)**(1/4)*b*c*
*4*d*x**4*(c**(-2))**(9/4)*log(x**2 + I*sqrt(c**(-2)))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2)
)**(19/2)) - 2*(-1)**(3/4)*b*c**3*d*(c**(-2))**(11/4)*atan(c*x**2)/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c*
*17*(c**(-2))**(19/2)) + (-1)**(1/4)*b*c**2*d*x**4*(c**(-2))**(5/4)*log(x**2 + I*sqrt(c**(-2)))/(-2*I*c**19*x*
*4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) - 2*(-1)**(1/4)*b*c**2*d*(c**(-2))**(9/4)*log(x**2 + I*sqr
t(c**(-2)))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) + I*b*c**2*e*x**4*(c**(-2))**(3/
2)*log(x**2 + I*sqrt(c**(-2)))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)) + (-1)**(1/4)
*b*d*(c**(-2))**(5/4)*log(x**2 + I*sqrt(c**(-2)))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(1
9/2)) - b*e*x**4*atan(c*x**2)/(-2*I*c**20*x**4*(c**(-2))**(19/2) - 2*I*c**18*(c**(-2))**(19/2)) - b*e*atan(c*x
**2)/(-2*I*c**22*x**4*(c**(-2))**(19/2) - 2*I*c**20*(c**(-2))**(19/2)) + I*b*e*(c**(-2))**(3/2)*log(x**2 + I*s
qrt(c**(-2)))/(-2*I*c**19*x**4*(c**(-2))**(19/2) - 2*I*c**17*(c**(-2))**(19/2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.26275, size = 271, normalized size = 1.41 \begin{align*} -\frac{1}{4} \, b c^{3} d{\left (\frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{4}} + \frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{4}} - \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{4}} + \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{4}}\right )} + \frac{2 \, b c x^{2} \arctan \left (c x^{2}\right ) e + 4 \, b c d x \arctan \left (c x^{2}\right ) + 2 \, a c x^{2} e + 4 \, a c d x - b e \log \left (c^{2} x^{4} + 1\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*c^3*d*(2*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/c^4 + 2*sqr
t(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/c^4 - sqrt(2)*sqrt(abs(c))*log
(x^2 + sqrt(2)*x/sqrt(abs(c)) + 1/abs(c))/c^4 + sqrt(2)*sqrt(abs(c))*log(x^2 - sqrt(2)*x/sqrt(abs(c)) + 1/abs(
c))/c^4) + 1/4*(2*b*c*x^2*arctan(c*x^2)*e + 4*b*c*d*x*arctan(c*x^2) + 2*a*c*x^2*e + 4*a*c*d*x - b*e*log(c^2*x^
4 + 1))/c