\(\int \frac {1}{x (c+a^2 c x^2)^3 \arctan (a x)^{3/2}} \, dx\) [1003]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {15 \sqrt {\arctan (a x)}}{2 c^3}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4 c^3}-\frac {5 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{c^3}-\frac {2 \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}},x\right )}{a} \]

[Out]

-5/8*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/c^3-5*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/
2))*Pi^(1/2)/c^3-2/a/c^3/x/(a^2*x^2+1)^2/arctan(a*x)^(1/2)-15/2*arctan(a*x)^(1/2)/c^3-2*Unintegrable(1/x^2/(a^
2*c*x^2+c)^3/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

-2/(a*c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (15*Sqrt[ArcTan[a*x]])/(2*c^3) - (5*Sqrt[Pi/2]*FresnelC[2*Sqr
t[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*c^3) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 - (2*Defer[Int
][1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a}-(10 a) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a}-\frac {10 \text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^3} \\ & = -\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a}-\frac {10 \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{c^3} \\ & = -\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {15 \sqrt {\arctan (a x)}}{2 c^3}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a}-\frac {5 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 c^3}-\frac {5 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^3} \\ & = -\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {15 \sqrt {\arctan (a x)}}{2 c^3}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a}-\frac {5 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2 c^3}-\frac {10 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^3} \\ & = -\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}-\frac {15 \sqrt {\arctan (a x)}}{2 c^3}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4 c^3}-\frac {5 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{c^3}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 13.71 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(1/(a**6*x**7*atan(a*x)**(3/2) + 3*a**4*x**5*atan(a*x)**(3/2) + 3*a**2*x**3*atan(a*x)**(3/2) + x*atan(
a*x)**(3/2)), x)/c**3

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 204.94 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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