\(\int \frac {1}{(c+a^2 c x^2)^{3/2} \arctan (a x)^{3/2}} \, dx\) [1028]

Optimal result

Integrand size = 23, antiderivative size = 92 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {2 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5022, 5091, 5090, 3386, 3432} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=-\frac {2 \sqrt {2 \pi } \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}} \]

[In]

Int[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]

[Out]

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

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 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 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])

Rule 5091

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1
/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]), Int[x^m*(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] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(IntegerQ[q] || GtQ[d, 0])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\frac {-2+\sqrt {1+a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+\sqrt {1+a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]

[In]

Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]

[Out]

(-2 + Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[
a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])

Maple [F]

\[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}d x\]

[In]

int(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2),x)

[Out]

int(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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