\(\int \frac {x^3 \arctan (a x)^2}{(c+a^2 c x^2)^{3/2}} \, dx\) [339]

Optimal result

Integrand size = 24, antiderivative size = 305 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {2 x \arctan (a x)}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{a^4 c^2}+\frac {4 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {c+a^2 c x^2}} \]

[Out]

-2/a^4/c/(a^2*c*x^2+c)^(1/2)-2*x*arctan(a*x)/a^3/c/(a^2*c*x^2+c)^(1/2)+arctan(a*x)^2/a^4/c/(a^2*c*x^2+c)^(1/2)
+4*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/c/(a^2*c*x^2+c)^(1/2)-2*I*polyl
og(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/c/(a^2*c*x^2+c)^(1/2)+2*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/c/(a^2*c*x^2+c)^(1/2)+arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^4/c^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5084, 5050, 5010, 5006, 5014} \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^4 c^2}+\frac {\arctan (a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {4 i \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {2 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {2}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {2 x \arctan (a x)}{a^3 c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

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 5014

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

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 5084

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

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (-2+3 \arctan (a x)^2-2 \cos (2 \arctan (a x))+\arctan (a x)^2 \cos (2 \arctan (a x))-\frac {4 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {4 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )}{2 a^4 c^2} \]

[In]

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

[Out]

(Sqrt[c*(1 + a^2*x^2)]*(-2 + 3*ArcTan[a*x]^2 - 2*Cos[2*ArcTan[a*x]] + ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] - (4*Ar
cTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (4*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[
1 + a^2*x^2] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((4*I)*PolyLog[2, I*E^(I*ArcTan[
a*x])])/Sqrt[1 + a^2*x^2] - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(2*a^4*c^2)

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a^{4} c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) a^{4} c^{2}}+\frac {\arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{4} c^{2}}+\frac {2 \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{4} c^{2}}\)	\(294\)

[In]

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

[Out]

1/2*(arctan(a*x)^2-2+2*I*arctan(a*x))*(1+I*a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)/a^4/c^2-1/2*(c*(a*x-I)*(
I+a*x))^(1/2)*(I*a*x-1)*(arctan(a*x)^2-2-2*I*arctan(a*x))/(a^2*x^2+1)/a^4/c^2+arctan(a*x)^2*(c*(a*x-I)*(I+a*x)
)^(1/2)/a^4/c^2+2*(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^2

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^3*arctan(a*x)^2/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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