\(\int \frac {x \arctan (a x)^{5/2}}{(c+a^2 c x^2)^2} \, dx\) [867]

Optimal result

Integrand size = 22, antiderivative size = 156 \[ \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {15 \sqrt {\arctan (a x)}}{64 a^2 c^2}+\frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a^2 c^2} \]

[Out]

5/8*x*arctan(a*x)^(3/2)/a/c^2/(a^2*x^2+1)+1/4*arctan(a*x)^(5/2)/a^2/c^2-1/2*arctan(a*x)^(5/2)/a^2/c^2/(a^2*x^2
+1)-15/128*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^2/c^2-15/64*arctan(a*x)^(1/2)/a^2/c^2+15/32*arcta
n(a*x)^(1/2)/a^2/c^2/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5050, 5012, 5024, 3393, 3385, 3433} \[ \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (a^2 x^2+1\right )}+\frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {15 \sqrt {\arctan (a x)}}{64 a^2 c^2} \]

[In]

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

[Out]

(-15*Sqrt[ArcTan[a*x]])/(64*a^2*c^2) + (15*Sqrt[ArcTan[a*x]])/(32*a^2*c^2*(1 + a^2*x^2)) + (5*x*ArcTan[a*x]^(3
/2))/(8*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(4*a^2*c^2) - ArcTan[a*x]^(5/2)/(2*a^2*c^2*(1 + a^2*x^2)) - (
15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^2*c^2)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - 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 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 5012

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

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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a} \\ & = \frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15}{16} \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = \frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{64 a} \\ & = \frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a^2 c^2} \\ & = \frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{64 a^2 c^2} \\ & = -\frac {15 \sqrt {\arctan (a x)}}{64 a^2 c^2}+\frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{128 a^2 c^2} \\ & = -\frac {15 \sqrt {\arctan (a x)}}{64 a^2 c^2}+\frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{64 a^2 c^2} \\ & = -\frac {15 \sqrt {\arctan (a x)}}{64 a^2 c^2}+\frac {15 \sqrt {\arctan (a x)}}{32 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {5 x \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{4 a^2 c^2}-\frac {\arctan (a x)^{5/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a^2 c^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.50 \[ \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {240 \arctan (a x)-240 a^2 x^2 \arctan (a x)+640 a x \arctan (a x)^2-256 \arctan (a x)^3+256 a^2 x^2 \arctan (a x)^3-60 \sqrt {\pi } \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+15 i \sqrt {2} \left (1+a^2 x^2\right ) \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-15 i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-15 i \sqrt {2} a^2 x^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )}{1024 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

(240*ArcTan[a*x] - 240*a^2*x^2*ArcTan[a*x] + 640*a*x*ArcTan[a*x]^2 - 256*ArcTan[a*x]^3 + 256*a^2*x^2*ArcTan[a*
x]^3 - 60*Sqrt[Pi]*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + (15*I)*Sqrt[2]*(
1 + a^2*x^2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - (15*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[
1/2, (2*I)*ArcTan[a*x]] - (15*I)*Sqrt[2]*a^2*x^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]])/(1024*a^2*
c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])

Maple [A] (verified)

Time = 25.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.53

method	result	size
default	\(-\frac {32 \arctan \left (a x \right )^{\frac {5}{2}} \cos \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }-40 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }-30 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )+15 \pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{128 c^{2} a^{2} \sqrt {\pi }}\)	\(82\)

[In]

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

[Out]

-1/128/c^2/a^2/Pi^(1/2)*(32*arctan(a*x)^(5/2)*cos(2*arctan(a*x))*Pi^(1/2)-40*arctan(a*x)^(3/2)*sin(2*arctan(a*
x))*Pi^(1/2)-30*arctan(a*x)^(1/2)*Pi^(1/2)*cos(2*arctan(a*x))+15*Pi*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2)))

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(x*atan(a*x)**(5/2)/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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