\(\int \frac {1}{x^2 (c+a^2 c x^2)^{5/2} \arctan (a x)^{5/2}} \, dx\) [1108]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20 a \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {20 a \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {8 \text {Int}\left (\frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right )}{a^2}+\frac {68}{3} \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right ) \]

[Out]

-2/3/a/c/x^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2)+20/9*a*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)/c^2/(a^2*c*x^2+c)^(1/2)+20*a*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)/c^2/(a^2*c*x^2+c)^(1/2)+8/3/a^2/c/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)+20/3/c/
x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)+8*Unintegrable(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)/a^2+68/3
*Unintegrable(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \]

[In]

Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c*x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]
]) + 20/(3*c*x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (20*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/P
i]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (20*a*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*
Sqrt[ArcTan[a*x]]])/(3*c^2*Sqrt[c + a^2*c*x^2]) + (8*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]
), x])/a^2 + (68*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}-\frac {4 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (10 a) \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {1}{3} \left (80 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {\left (80 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{3 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {\left (80 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {\left (80 a \sqrt {1+a^2 x^2}\right ) \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 )}{3 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {\left (20 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (20 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\frac {\left (40 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (40 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{3 a c x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {8}{3 a^2 c x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20}{3 c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {20 a \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {20 a \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {20}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+16 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \]

[In]

Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]

[In]

int(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x)

[Out]

int(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/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(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 254.93 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(1/(x^2*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(x^2*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(5/2)), x)