\(\int \frac {\arctan (a x)^{3/2}}{(c+a^2 c x^2)^2} \, dx\) [787]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {3 \sqrt {\arctan (a x)}}{16 a c^2}+\frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a c^2} \]

[Out]

1/2*x*arctan(a*x)^(3/2)/c^2/(a^2*x^2+1)+1/5*arctan(a*x)^(5/2)/a/c^2-3/32*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2)
)*Pi^(1/2)/a/c^2-3/16*arctan(a*x)^(1/2)/a/c^2+3/8*arctan(a*x)^(1/2)/a/c^2/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5012, 5050, 5024, 3393, 3385, 3433} \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (a^2 x^2+1\right )}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a c^2}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \sqrt {\arctan (a x)}}{16 a c^2} \]

[In]

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

[Out]

(-3*Sqrt[ArcTan[a*x]])/(16*a*c^2) + (3*Sqrt[ArcTan[a*x]])/(8*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(2*c
^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(5*a*c^2) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*
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 {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {1}{4} (3 a) \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = \frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3}{16} \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = \frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{16 a c^2} \\ & = \frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{16 a c^2} \\ & = -\frac {3 \sqrt {\arctan (a x)}}{16 a c^2}+\frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{32 a c^2} \\ & = -\frac {3 \sqrt {\arctan (a x)}}{16 a c^2}+\frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{16 a c^2} \\ & = -\frac {3 \sqrt {\arctan (a x)}}{16 a c^2}+\frac {3 \sqrt {\arctan (a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{5/2}}{5 a c^2}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a c^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.50 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\frac {16 \sqrt {\arctan (a x)} \left (15-15 a^2 x^2+40 a x \arctan (a x)+16 \left (1+a^2 x^2\right ) \arctan (a x)^2\right )}{1+a^2 x^2}+60 \left (2 \sqrt {\arctan (a x)}-\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )\right )-\frac {15 \left (8 \arctan (a x)-i \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )\right )}{\sqrt {\arctan (a x)}}}{1280 a c^2} \]

[In]

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

[Out]

((16*Sqrt[ArcTan[a*x]]*(15 - 15*a^2*x^2 + 40*a*x*ArcTan[a*x] + 16*(1 + a^2*x^2)*ArcTan[a*x]^2))/(1 + a^2*x^2)
+ 60*(2*Sqrt[ArcTan[a*x]] - Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) - (15*(8*ArcTan[a*x] - I*Sqrt[2
]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + I*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTa
n[a*x]]))/Sqrt[ArcTan[a*x]])/(1280*a*c^2)

Maple [A] (verified)

Time = 7.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {32 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }+40 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }+30 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )-15 \pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{160 c^{2} a \sqrt {\pi }}\)	\(75\)

[In]

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

[Out]

1/160/c^2/a*(32*arctan(a*x)^(5/2)*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)))/Pi^(1/2)

Fricas [F(-2)]

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

[In]

integrate(arctan(a*x)^(3/2)/(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]

\[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(atan(a*x)**(3/2)/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

Maxima [F(-2)]

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

[In]

integrate(arctan(a*x)^(3/2)/(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]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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