\(\int \frac {x}{(c+a^2 c x^2)^3 \arctan (a x)^{5/2}} \, dx\) [1070]

Optimal result

Integrand size = 22, antiderivative size = 155 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=-\frac {2 x}{3 a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4 x^2}{c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^2 c^3}-\frac {4 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^3} \]

[Out]

-2/3*x/a/c^3/(a^2*x^2+1)^2/arctan(a*x)^(3/2)-4/3*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^2/c^3-4/3*F
resnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2/c^3-4/3/a^2/c^3/(a^2*x^2+1)^2/arctan(a*x)^(
1/2)+4*x^2/c^3/(a^2*x^2+1)^2/arctan(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5088, 5090, 4491, 3386, 3432, 5022} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^2 c^3}-\frac {4 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^3}+\frac {4 x^2}{c^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}-\frac {2 x}{3 a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

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

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 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 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 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])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.42 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {i \sqrt {2} \left (1+a^2 x^2\right )^2 (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+\sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \arctan (a x) \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+2 \left (-a x-2 \arctan (a x)+6 a^2 x^2 \arctan (a x)+i \left (1+a^2 x^2\right )^2 (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+\left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \arctan (a x) \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{3 c^3 \left (a+a^3 x^2\right )^2 \arctan (a x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71

method	result	size
default	\(-\frac {16 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+16 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+8 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+8 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+2 \sin \left (2 \arctan \left (a x \right )\right )+\sin \left (4 \arctan \left (a x \right )\right )}{12 a^{2} c^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\)	\(110\)

[In]

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

[Out]

-1/12/a^2/c^3*(16*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*arctan(a*x)^(3/2)+16*Pi^(1/2
)*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*arctan(a*x)^(3/2)+8*cos(2*arctan(a*x))*arctan(a*x)+8*cos(4*arctan(a*x
))*arctan(a*x)+2*sin(2*arctan(a*x))+sin(4*arctan(a*x)))/arctan(a*x)^(3/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {x}{a^{6} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(x/(a**6*x**6*atan(a*x)**(5/2) + 3*a**4*x**4*atan(a*x)**(5/2) + 3*a**2*x**2*atan(a*x)**(5/2) + atan(a*
x)**(5/2)), x)/c**3

Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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