3.1.81 \(\int x^2 (a+b \arctan (c x^2))^2 \, dx\) [81]

3.1.81.1 Optimal result

Integrand size = 16, antiderivative size = 1393 \[ \int x^2 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \]

output

1/12*x^3*(2*a+I*b*ln(1-I*c*x^2))^2-1/3*(-1)^(3/4)*b^2*arctanh((-1)^(3/4)*x 
*c^(1/2))^2/c^(3/2)+1/6*b^2*x^3*ln(1-I*c*x^2)*ln(1+I*c*x^2)+1/3*(-1)^(1/4) 
*b^2*polylog(2,1-2/(1-(-1)^(1/4)*x*c^(1/2)))/c^(3/2)+1/3*(-1)^(1/4)*b^2*po 
lylog(2,1-2/(1+(-1)^(1/4)*x*c^(1/2)))/c^(3/2)-1/6*(-1)^(1/4)*b^2*polylog(2 
,1-2^(1/2)*((-1)^(1/4)+x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))/c^(3/2)+1/3*(- 
1)^(3/4)*b^2*polylog(2,1-2/(1-(-1)^(3/4)*x*c^(1/2)))/c^(3/2)+1/3*(-1)^(3/4 
)*b^2*polylog(2,1-2/(1+(-1)^(3/4)*x*c^(1/2)))/c^(3/2)-1/6*(-1)^(3/4)*b^2*p 
olylog(2,1+2^(1/2)*((-1)^(3/4)+x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))/c^(3/2 
)-1/6*(-1)^(3/4)*b^2*polylog(2,1-(1+I)*(1+(-1)^(1/4)*x*c^(1/2))/(1+(-1)^(3 
/4)*x*c^(1/2)))/c^(3/2)-1/6*(-1)^(1/4)*b^2*polylog(2,1+(-1+I)*(1+(-1)^(3/4 
)*x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))/c^(3/2)-1/9*I*b*x^3*(2*a+I*b*ln(1-I 
*c*x^2))+4/3*(-1)^(3/4)*b^2*arctan((-1)^(3/4)*x*c^(1/2))/c^(3/2)+1/3*(-1)^ 
(1/4)*b^2*arctan((-1)^(3/4)*x*c^(1/2))^2/c^(3/2)-4/3*(-1)^(3/4)*b^2*arctan 
h((-1)^(3/4)*x*c^(1/2))/c^(3/2)+2/3*I*b^2*x*ln(1+I*c*x^2)/c-1/9*b^2*x^3*ln 
(1-I*c*x^2)-1/12*b^2*x^3*ln(1+I*c*x^2)^2-2/3*(-1)^(1/4)*a*b*arctanh((-1)^( 
3/4)*x*c^(1/2))/c^(3/2)-1/3*(-1)^(3/4)*b^2*arctanh((-1)^(3/4)*x*c^(1/2))*l 
n(1-I*c*x^2)/c^(3/2)-1/3*(-1)^(1/4)*b*arctan((-1)^(3/4)*x*c^(1/2))*(2*a+I* 
b*ln(1-I*c*x^2))/c^(3/2)+1/3*(-1)^(3/4)*b^2*arctan((-1)^(3/4)*x*c^(1/2))*l 
n(1+I*c*x^2)/c^(3/2)+1/3*(-1)^(3/4)*b^2*arctanh((-1)^(3/4)*x*c^(1/2))*ln(1 
+I*c*x^2)/c^(3/2)-2/3*(-1)^(3/4)*b^2*arctan((-1)^(3/4)*x*c^(1/2))*ln(2/...
 

3.1.81.2 Mathematica [F]

\[ \int x^2 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int x^2 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx \]

input

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

output

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

3.1.81.3 Rubi [A] (verified)

Time = 2.56 (sec) , antiderivative size = 1393, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5365, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx\)

\(\Big \downarrow \) 5365

\(\displaystyle \int \left (\frac {1}{4} x^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac {1}{2} b x^2 \log \left (1+i c x^2\right ) \left (b \log \left (1-i c x^2\right )-2 i a\right )-\frac {1}{4} b^2 x^2 \log ^2\left (1+i c x^2\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{12} \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 x^3-\frac {1}{12} b^2 \log ^2\left (i c x^2+1\right ) x^3+\frac {2}{9} i a b x^3-\frac {1}{9} b^2 \log \left (1-i c x^2\right ) x^3-\frac {1}{9} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) x^3-\frac {1}{3} i a b \log \left (i c x^2+1\right ) x^3+\frac {1}{6} b^2 \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right ) x^3-\frac {2 i b^2 \log \left (1-i c x^2\right ) x}{3 c}+\frac {2 i b^2 \log \left (i c x^2+1\right ) x}{3 c}-\frac {4 a b x}{3 c}+\frac {\sqrt [4]{-1} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{3 c^{3/2}}-\frac {(-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{3 c^{3/2}}+\frac {4 (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{3 c^{3/2}}-\frac {4 (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{3 c^{3/2}}-\frac {2 \sqrt [4]{-1} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{3 c^{3/2}}-\frac {2 (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{3 c^{3/2}}+\frac {2 (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{3 c^{3/2}}-\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{3 c^{3/2}}+\frac {2 (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{3 c^{3/2}}-\frac {2 (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right )}{3 c^{3/2}}+\frac {(-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}\right )}{3 c^{3/2}}+\frac {(-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\right )}{3 c^{3/2}}-\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{3 c^{3/2}}-\frac {(-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac {\sqrt [4]{-1} b \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{3 c^{3/2}}+\frac {(-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{3 c^{3/2}}+\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{3 c^{3/2}}+\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{3 c^{3/2}}-\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{6 c^{3/2}}+\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{3 c^{3/2}}+\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right )}{3 c^{3/2}}-\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}+1\right )}{6 c^{3/2}}-\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\right )}{6 c^{3/2}}-\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{6 c^{3/2}}\)

input

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

output

(-4*a*b*x)/(3*c) + ((2*I)/9)*a*b*x^3 + (4*(-1)^(3/4)*b^2*ArcTan[(-1)^(3/4) 
*Sqrt[c]*x])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2) 
/(3*c^(3/2)) - (2*(-1)^(1/4)*a*b*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2) 
) - (4*(-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) - ((-1)^( 
3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/(3*c^(3/2)) - (2*(-1)^(3/4)*b^2* 
ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2) 
) + (2*(-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*S 
qrt[c]*x)])/(3*c^(3/2)) - ((-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log 
[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2 
)) + (2*(-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4) 
*Sqrt[c]*x)])/(3*c^(3/2)) - (2*(-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x 
]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*ArcTanh 
[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^ 
(3/4)*Sqrt[c]*x))])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[ 
c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] 
)/(3*c^(3/2)) - ((-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)* 
(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((( 
2*I)/3)*b^2*x*Log[1 - I*c*x^2])/c - (b^2*x^3*Log[1 - I*c*x^2])/9 - ((-1)^( 
3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/(3*c^(3/2)) - (I/ 
9)*b*x^3*(2*a + I*b*Log[1 - I*c*x^2]) - ((-1)^(1/4)*b*ArcTan[(-1)^(3/4)...
 

3.1.81.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5365

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> I 
nt[ExpandIntegrand[x^m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n 
])/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && Integ 
erQ[m]
 

3.1.81.4 Maple [F]

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

input

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

output

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

3.1.81.5 Fricas [F]

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

input

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

output

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

3.1.81.6 Sympy [F]

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

input

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

output

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

3.1.81.7 Maxima [F]

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

input

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

output

1/3*a^2*x^3 + 1/6*(4*x^3*arctan(c*x^2) - c*(8*x/c^2 - (2*sqrt(2)*arctan(1/ 
2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/sqrt(c) + 2*sqrt(2)*arctan(1/ 
2*sqrt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/sqrt(c) + sqrt(2)*log(c*x^2 + 
 sqrt(2)*sqrt(c)*x + 1)/sqrt(c) - sqrt(2)*log(c*x^2 - sqrt(2)*sqrt(c)*x + 
1)/sqrt(c))/c^2))*a*b + 1/48*(4*x^3*arctan(c*x^2)^2 - x^3*log(c^2*x^4 + 1) 
^2 + 48*integrate(1/48*(8*c^2*x^6*log(c^2*x^4 + 1) - 16*c*x^4*arctan(c*x^2 
) + 36*(c^2*x^6 + x^2)*arctan(c*x^2)^2 + 3*(c^2*x^6 + x^2)*log(c^2*x^4 + 1 
)^2)/(c^2*x^4 + 1), x))*b^2
 

3.1.81.8 Giac [F]

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

input

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

output

integrate((b*arctan(c*x^2) + a)^2*x^2, x)
 

3.1.81.9 Mupad [F(-1)]

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

input

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

output

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