\(\int \frac {\arctan (a x)^{3/2}}{(c+a^2 c x^2)^{5/2}} \, dx\) [834]

Optimal result

Integrand size = 23, antiderivative size = 252 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{24 a c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

1/3*x*arctan(a*x)^(3/2)/c/(a^2*c*x^2+c)^(3/2)+2/3*x*arctan(a*x)^(3/2)/c^2/(a^2*c*x^2+c)^(1/2)-1/144*FresnelC(6
^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/a/c^2/(a^2*c*x^2+c)^(1/2)-9/16*FresnelC(
2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/a/c^2/(a^2*c*x^2+c)^(1/2)+1/6*arctan(a*
x)^(1/2)/a/c/(a^2*c*x^2+c)^(3/2)+arctan(a*x)^(1/2)/a/c^2/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5020, 5018, 5025, 5024, 3385, 3433, 3393} \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {9 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{24 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

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

[Out]

Sqrt[ArcTan[a*x]]/(6*a*c*(c + a^2*c*x^2)^(3/2)) + Sqrt[ArcTan[a*x]]/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*
x]^(3/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^(3/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[Pi/2]*Sq
rt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a
^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a*c^2*Sqrt[c + a^2*c*x^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 5018

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

Rule 5020

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(d + e*x^2)^(q +
1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d*(q + 1)^2)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[b^2*p*((p - 1)/(4*(q + 1)^2)), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d*(q + 1))), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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 5025

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1/2)*(Sqrt[1
+ c^2*x^2]/Sqrt[d + e*x^2]), Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, 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])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {1}{12} \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {2 \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c} \\ & = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{2 c}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{12 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{12 a c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{12 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{2 a c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{48 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{16 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{24 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{8 a c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{24 a c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.37 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {336 \arctan (a x)+288 a^2 x^2 \arctan (a x)+288 a x \arctan (a x)^2+192 a^3 x^3 \arctan (a x)^2+81 i \left (1+a^2 x^2\right )^{3/2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-81 i \left (1+a^2 x^2\right )^{3/2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )+i \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )-i \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )-i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )}{288 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]

[In]

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

[Out]

(336*ArcTan[a*x] + 288*a^2*x^2*ArcTan[a*x] + 288*a*x*ArcTan[a*x]^2 + 192*a^3*x^3*ArcTan[a*x]^2 + (81*I)*(1 + a
^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (81*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a
*x]]*Gamma[1/2, I*ArcTan[a*x]] + I*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] +
I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - I*Sqrt[3 + 3*a^2*x^2]*S
qrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] - I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2
, (3*I)*ArcTan[a*x]])/(288*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])

Maple [F]

\[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral(atan(a*x)**(3/2)/(c*(a**2*x**2 + 1))**(5/2), x)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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