\(\int \frac {1}{(c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [671]

Optimal result

Integrand size = 21, antiderivative size = 145 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

-1/2/a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2+3/2*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-3/8*Ci(arctan(a*x))*(a^2*x^
2+1)^(1/2)/a/c^2/(a^2*c*x^2+c)^(1/2)-9/8*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a/c^2/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5022, 5088, 5091, 5090, 4491, 3383, 5025, 5024, 3393} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {3 \sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {a^2 c x^2+c}}-\frac {9 \sqrt {a^2 x^2+1} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {a^2 c x^2+c}}+\frac {3 x}{2 c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

-1/2*1/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + (3*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - (3*Sqrt[1 +
a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[
a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2])

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 5024

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

Rule 5025

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1/2)*(Sqrt[1
+ c^2*x^2]/Sqrt[d + e*x^2]), Int[(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] && ILtQ[2*(q + 1), 0] &&  !(IntegerQ[q] || GtQ[d, 0])

Rule 5088

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

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 {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {1}{2} (3 a) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3}{2} \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx+\left (3 a^2\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{x} \, dx,x,\arctan (a x)\right )}{2 a 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 (x) \sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 x}+\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{2 a 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 {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arctan (a x)\right )}{8 a 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 (x)}{x} \, dx,x,\arctan (a x)\right )}{4 a 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 (3 x)}{x} \, dx,x,\arctan (a x)\right )}{4 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (9 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{8 a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {-4+12 a x \arctan (a x)-3 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \operatorname {CosIntegral}(\arctan (a x))-9 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \operatorname {CosIntegral}(3 \arctan (a x))}{8 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)^2} \]

[In]

Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

(-4 + 12*a*x*ArcTan[a*x] - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[ArcTan[a*x]] - 9*(1 + a^2*x^2)^(3/2
)*ArcTan[a*x]^2*CosIntegral[3*ArcTan[a*x]])/(8*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 18.91 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.39

method	result	size
default	\(\frac {\left (9 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+9 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+18 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+18 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+24 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +9 \,\operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 \,\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 \,\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+9 \,\operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-8 \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )^{2} a \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\)	\(346\)

[In]

int(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/16*(9*arctan(a*x)^2*Ei(1,3*I*arctan(a*x))*a^4*x^4+3*arctan(a*x)^2*Ei(1,-I*arctan(a*x))*a^4*x^4+3*arctan(a*x)
^2*Ei(1,I*arctan(a*x))*a^4*x^4+9*arctan(a*x)^2*Ei(1,-3*I*arctan(a*x))*a^4*x^4+18*arctan(a*x)^2*Ei(1,3*I*arctan
(a*x))*a^2*x^2+6*arctan(a*x)^2*Ei(1,-I*arctan(a*x))*a^2*x^2+6*arctan(a*x)^2*Ei(1,I*arctan(a*x))*a^2*x^2+18*arc
tan(a*x)^2*Ei(1,-3*I*arctan(a*x))*a^2*x^2+24*arctan(a*x)*(a^2*x^2+1)^(1/2)*a*x+9*Ei(1,3*I*arctan(a*x))*arctan(
a*x)^2+3*Ei(1,-I*arctan(a*x))*arctan(a*x)^2+3*Ei(1,I*arctan(a*x))*arctan(a*x)^2+9*Ei(1,-3*I*arctan(a*x))*arcta
n(a*x)^2-8*(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/arctan(a*x)^2/a/c^3/(a^4*x^4+2*a^2*x
^2+1)

Fricas [F]

\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(1/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)

[Out]

Integral(1/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), x)

Maxima [F]

\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate(1/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3), x)

Giac [F]

\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(1/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)