\(\int \frac {x^2}{(c+a^2 c x^2)^{5/2} \arctan (a x)^2} \, dx\) [590]

Optimal result

Integrand size = 24, antiderivative size = 142 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

1/a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-1/a^3/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)-1/4*Si(arctan(a*x))*(a^2*x^2
+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)+3/4*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5084, 5022, 5091, 5090, 3380, 4491} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{4 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {a^2 x^2+1} \text {Si}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a^3 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}+\frac {1}{a^3 c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

1/(a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 1/(a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]
*SinIntegral[ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/
(4*a^3*c^2*Sqrt[c + a^2*c*x^2])

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 5022

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

Rule 5090

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(m
+ 1), Subst[Int[(a + b*x)^p*(Sin[x]^m/Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d,
e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 5091

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx}{a^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{a^2 c} \\ & = \frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx}{a}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx}{a c} \\ & = \frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {4 a^2 x^2+\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \text {Si}(\arctan (a x))-3 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \text {Si}(3 \arctan (a x))}{4 a^3 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)} \]

[In]

Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

-1/4*(4*a^2*x^2 + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*SinIntegral[ArcTan[a*x]] - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]
*SinIntegral[3*ArcTan[a*x]])/(a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 13.75 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.18

method	result	size
default	\(-\frac {i \left (\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}-3 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}-\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+2 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}-6 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-2 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right ) \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-8 i \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )-3 \,\operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+3 \,\operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) a^{3} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\)	\(309\)

[In]

int(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/8*I*(arctan(a*x)*Ei(1,-I*arctan(a*x))*a^4*x^4-3*arctan(a*x)*Ei(1,-3*I*arctan(a*x))*a^4*x^4-arctan(a*x)*Ei(1
,I*arctan(a*x))*a^4*x^4+3*arctan(a*x)*Ei(1,3*I*arctan(a*x))*a^4*x^4+2*arctan(a*x)*Ei(1,-I*arctan(a*x))*a^2*x^2
-6*arctan(a*x)*Ei(1,-3*I*arctan(a*x))*a^2*x^2-2*arctan(a*x)*Ei(1,I*arctan(a*x))*a^2*x^2+6*arctan(a*x)*Ei(1,3*I
*arctan(a*x))*a^2*x^2-8*I*(a^2*x^2+1)^(1/2)*a^2*x^2+Ei(1,-I*arctan(a*x))*arctan(a*x)-3*Ei(1,-3*I*arctan(a*x))*
arctan(a*x)-Ei(1,I*arctan(a*x))*arctan(a*x)+3*Ei(1,3*I*arctan(a*x))*arctan(a*x))/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*
(I+a*x))^(1/2)/arctan(a*x)/a^3/c^3/(a^4*x^4+2*a^2*x^2+1)

Fricas [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^2/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^2), x)

Sympy [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(x**2/(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)

[Out]

Integral(x**2/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2), x)

Maxima [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

integrate(x^2/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^2), x)

Giac [F]

\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(x^2/(atan(a*x)^2*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(x^2/(atan(a*x)^2*(c + a^2*c*x^2)^(5/2)), x)