\(\int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\) [331]

Optimal result

Integrand size = 24, antiderivative size = 315 \[ \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2}}{3 a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2 c}-\frac {10 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}} \]

[Out]

-10/3*I*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+5/3*I*po
lylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-5/3*I*polylog(2,I*(1+I*a
*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+1/3*(a^2*c*x^2+c)^(1/2)/a^4/c-1/3*x*arcta
n(a*x)*(a^2*c*x^2+c)^(1/2)/a^3/c-2/3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^4/c+1/3*x^2*arctan(a*x)^2*(a^2*c*x^2+
c)^(1/2)/a^2/c

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5072, 267, 5010, 5006, 5050} \[ \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^4 c}-\frac {10 i \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{3 a^4 \sqrt {a^2 c x^2+c}}+\frac {5 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {a^2 c x^2+c}}-\frac {5 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c}}{3 a^4 c}-\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^3 c} \]

[In]

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

[Out]

Sqrt[c + a^2*c*x^2]/(3*a^4*c) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[
a*x]^2)/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^2*c) - (((10*I)/3)*Sqrt[1 + a^2*x^2]*ArcTan[a
*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (((5*I)/3)*Sqrt[1 + a^2*x^2]*PolyLog[
2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) - (((5*I)/3)*Sqrt[1 + a^2*x^2]*PolyLog[2
, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2])

Rule 267

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

Rule 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 5050

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(d + e*x^2)^(
q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Dist[b*(p/(2*c*(q + 1))), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 5072

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + (-Dist[b*f*(p/(c*m)), Int[(f*x)^(m - 1
)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[f^2*((m - 1)/(c^2*m)), Int[(f*x)^(m - 2)*((a +
b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && Gt
Q[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a} \\ & = -\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2 c}+\frac {\int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {4 \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2} \\ & = \frac {\sqrt {c+a^2 c x^2}}{3 a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2 c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2}}{3 a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2 c}-\frac {10 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\left (1+a^2 x^2\right ) \sqrt {c \left (1+a^2 x^2\right )} \left (2-2 \arctan (a x)^2+2 \cos (2 \arctan (a x))-6 \arctan (a x)^2 \cos (2 \arctan (a x))+\frac {15 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+5 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-\frac {15 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-5 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+\frac {20 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-\frac {20 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )}{12 a^4 c} \]

[In]

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

[Out]

((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 - 2*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - 6*ArcTan[a*x]^2*Cos[2*ArcTa
n[a*x]] + (15*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 5*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*L
og[1 - I*E^(I*ArcTan[a*x])] - (15*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 5*ArcTan[a*x]*
Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + ((20*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3
/2) - ((20*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(12*a^
4*c)

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {\left (x^{2} \arctan \left (a x \right )^{2} a^{2}-x \arctan \left (a x \right ) a -2 \arctan \left (a x \right )^{2}+1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 a^{4} c}-\frac {5 \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 \sqrt {a^{2} x^{2}+1}\, a^{4} c}\)	\(206\)

[In]

int(x^3*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(x^2*arctan(a*x)^2*a^2-x*arctan(a*x)*a-2*arctan(a*x)^2+1)*(c*(a*x-I)*(I+a*x))^(1/2)/a^4/c-5/3*(arctan(a*x)
*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^
2*x^2+1)^(1/2))+I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)/a^4/c

Fricas [F]

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

[In]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(x^3*arctan(a*x)^2/sqrt(a^2*c*x^2 + c), x)

Sympy [F]

\[ \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(x**3*atan(a*x)**2/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(x**3*atan(a*x)**2/sqrt(c*(a**2*x**2 + 1)), x)

Maxima [F]

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

[In]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^3*arctan(a*x)^2/sqrt(a^2*c*x^2 + c), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int((x^3*atan(a*x)^2)/(c + a^2*c*x^2)^(1/2),x)

[Out]

int((x^3*atan(a*x)^2)/(c + a^2*c*x^2)^(1/2), x)