\(\int \frac {1}{x^4 (c+a^2 c x^2)^{5/2} \arctan (a x)^2} \, dx\) [596]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {a^3}{c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {2 a^3}{c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {11 a^3 \sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 a^3 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{c^2}-\frac {2 a^2 \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{c^2} \]

[Out]

-a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-2*a^3/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)-11/4*a^3*Si(arctan(a*x))*(a^2
*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-3/4*a^3*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+Unin
tegrable(1/x^4/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/c^2-2*a^2*Unintegrable(1/x^2/arctan(a*x)^2/(a^2*c*x^2+c)^(
1/2),x)/c^2

Rubi [N/A]

Not integrable

Time = 1.07 (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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]

[In]

Int[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

-(a^3/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (11*a^3*Sqrt[1
+ a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*a^3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcT
an[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 - (2*a^2*
Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/c^2

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

Mathematica [N/A]

Not integrable

Time = 7.73 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]

[In]

Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 33.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}d x\]

[In]

int(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

[Out]

int(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{4} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)/((a^6*c^3*x^10 + 3*a^4*c^3*x^8 + 3*a^2*c^3*x^6 + c^3*x^4)*arctan(a*x)^2), x)

Sympy [N/A]

Not integrable

Time = 31.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^{4} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x**4/(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{4} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

integrate(1/((a^2*c*x^2 + c)^(5/2)*x^4*arctan(a*x)^2), x)

Giac [N/A]

Not integrable

Time = 126.88 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{4} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(1/(x^4*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(x^4*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)), x)