3.9.25 \(\int \frac {\text {ArcTan}(a x)^{3/2}}{(c+a^2 c x^2)^{3/2}} \, dx\) [825]

Optimal. Leaf size=125 \[ \frac {3 \sqrt {\text {ArcTan}(a x)}}{2 a c \sqrt {c+a^2 c x^2}}+\frac {x \text {ArcTan}(a x)^{3/2}}{c \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{2 a c \sqrt {c+a^2 c x^2}} \]

[Out]

x*arctan(a*x)^(3/2)/c/(a^2*c*x^2+c)^(1/2)-3/4*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/(a^2*c*x^2+c)^(1/2)+3/2*arctan(a*x)^(1/2)/a/c/(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5018, 5025, 5024, 3385, 3433} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \text {ArcTan}(a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\text {ArcTan}(a x)}}{2 a c \sqrt {a^2 c x^2+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 86, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {\text {ArcTan}(a x)} (3+2 a x \text {ArcTan}(a x))-3 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{4 a c \sqrt {c+a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.46, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________