\(\int \frac {x^3 \arctan (a x)^{3/2}}{(c+a^2 c x^2)^3} \, dx\) [792]

Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \arctan (a x)^{3/2}}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^4 c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{64 a^4 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^4 c^3}-\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^4 c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5064, 5090, 3393, 3377, 3386, 3432} \[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^4 c^3}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{64 a^4 c^3}-\frac {3 \arctan (a x)^{3/2}}{32 a^4 c^3}+\frac {3 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{32 a^4 c^3}-\frac {3 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{256 a^4 c^3}+\frac {x^4 \arctan (a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2} \]

[In]

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

[Out]

(-3*ArcTan[a*x]^(3/2))/(32*a^4*c^3) + (x^4*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*Sqrt[Pi/2]*FresnelS
[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^4*c^3) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(64*a^
4*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(32*a^4*c^3) - (3*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(256
*a^4*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 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -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])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.08 \[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\sqrt {\arctan (a x)} \left (\frac {3 x \left (3+5 a^2 x^2\right )}{64 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\left (-3-6 a^2 x^2+5 a^4 x^4\right ) \arctan (a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )^2}\right )-\frac {9 \left (-2 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-2 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{4096 a^4 c^3 \sqrt {\arctan (a x)}}-\frac {15 \left (-2 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-2 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{4096 a^4 c^3 \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

Sqrt[ArcTan[a*x]]*((3*x*(3 + 5*a^2*x^2))/(64*a^3*c^3*(1 + a^2*x^2)^2) + ((-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a
*x])/(32*a^4*c^3*(1 + a^2*x^2)^2)) - (9*(-2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*
Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*
x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(4096*a^4*c^3*Sqrt[ArcTan[a*x]]) - (15*(-2*Sqrt[2]*S
qrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a
*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x
]]))/(4096*a^4*c^3*Sqrt[ArcTan[a*x]])

Maple [A] (verified)

Time = 7.56 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.74

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

[In]

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

[Out]

-1/1024/c^3/a^4*(-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*arct
an(a*x)^2*cos(2*arctan(a*x))-32*arctan(a*x)^2*cos(4*arctan(a*x))+48*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arct
an(a*x)^(1/2)/Pi^(1/2))-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. \[ \int \frac {x^3 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*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]

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

[In]

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

[Out]

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

[In]

integrate(x^3*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]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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