\(\int \frac {x \arctan (a x)^{3/2}}{(c+a^2 c x^2)^3} \, dx\) [794]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 \arctan (a x)^{3/2}}{32 a^2 c^3}-\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^2 c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{64 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^2 c^3} \]

[Out]

3/32*arctan(a*x)^(3/2)/a^2/c^3-1/4*arctan(a*x)^(3/2)/a^2/c^3/(a^2*x^2+1)^2-3/1024*FresnelS(2*2^(1/2)/Pi^(1/2)*
arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2/c^3-3/64*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^2/c^3+3/32*
sin(2*arctan(a*x))*arctan(a*x)^(1/2)/a^2/c^3+3/256*sin(4*arctan(a*x))*arctan(a*x)^(1/2)/a^2/c^3

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5050, 5024, 3393, 3377, 3386, 3432} \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^2 c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{64 a^2 c^3}-\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^{3/2}}{32 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^2 c^3} \]

[In]

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

[Out]

(3*ArcTan[a*x]^(3/2))/(32*a^2*c^3) - ArcTan[a*x]^(3/2)/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*Sqrt[Pi/2]*FresnelS[2*
Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^2*c^3) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(64*a^2*c
^3) + (3*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(32*a^2*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(256*a^
2*c^3)

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3432

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

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)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a} \\ & = -\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \text {Subst}\left (\int \sqrt {x} \cos ^4(x) \, dx,x,\arctan (a x)\right )}{8 a^2 c^3} \\ & = -\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \text {Subst}\left (\int \left (\frac {3 \sqrt {x}}{8}+\frac {1}{2} \sqrt {x} \cos (2 x)+\frac {1}{8} \sqrt {x} \cos (4 x)\right ) \, dx,x,\arctan (a x)\right )}{8 a^2 c^3} \\ & = \frac {3 \arctan (a x)^{3/2}}{32 a^2 c^3}-\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \text {Subst}\left (\int \sqrt {x} \cos (4 x) \, dx,x,\arctan (a x)\right )}{64 a^2 c^3}+\frac {3 \text {Subst}\left (\int \sqrt {x} \cos (2 x) \, dx,x,\arctan (a x)\right )}{16 a^2 c^3} \\ & = \frac {3 \arctan (a x)^{3/2}}{32 a^2 c^3}-\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^2 c^3}-\frac {3 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a^2 c^3}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a^2 c^3} \\ & = \frac {3 \arctan (a x)^{3/2}}{32 a^2 c^3}-\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^2 c^3}-\frac {3 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{256 a^2 c^3}-\frac {3 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{32 a^2 c^3} \\ & = \frac {3 \arctan (a x)^{3/2}}{32 a^2 c^3}-\frac {\arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^2 c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{64 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^2 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^2 c^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.07 \[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {480 a x \arctan (a x)+288 a^3 x^3 \arctan (a x)-320 \arctan (a x)^2+384 a^2 x^2 \arctan (a x)^2+192 a^4 x^4 \arctan (a x)^2+24 \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+24 \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+3 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+6 a^2 x^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+3 a^4 x^4 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+3 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )+6 a^2 x^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )+3 a^4 x^4 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{2048 c^3 \left (a+a^3 x^2\right )^2 \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

(480*a*x*ArcTan[a*x] + 288*a^3*x^3*ArcTan[a*x] - 320*ArcTan[a*x]^2 + 384*a^2*x^2*ArcTan[a*x]^2 + 192*a^4*x^4*A
rcTan[a*x]^2 + 24*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 24*Sqrt[2]*(
1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + 3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*
ArcTan[a*x]] + 6*a^2*x^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 3*a^4*x^4*Sqrt[(-I)*ArcTan[a*
x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]] + 6*a^2*x^2*Sqrt[I*Ar
cTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]] + 3*a^4*x^4*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(2048*
c^3*(a + a^3*x^2)^2*Sqrt[ArcTan[a*x]])

Maple [A] (verified)

Time = 7.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.74

method	result	size
default	\(-\frac {3 \,\operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }+128 \arctan \left (a x \right )^{2} \cos \left (2 \arctan \left (a x \right )\right )+48 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+32 \arctan \left (a x \right )^{2} \cos \left (4 \arctan \left (a x \right )\right )-96 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-12 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{1024 c^{3} a^{2} \sqrt {\arctan \left (a x \right )}}\)	\(124\)

[In]

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

[Out]

-1/1024/c^3/a^2*(3*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)+128*arcta
n(a*x)^2*cos(2*arctan(a*x))+48*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))+32*arctan(a*x
)^2*cos(4*arctan(a*x))-96*sin(2*arctan(a*x))*arctan(a*x)-12*sin(4*arctan(a*x))*arctan(a*x))/arctan(a*x)^(1/2)

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(x*atan(a*x)**(3/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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