\(\int \frac {1}{x^3 (c+a^2 c x^2)^{5/2} \arctan (a x)^2} \, dx\) [595]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {2 a^3 x}{c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {2 a \sqrt {c+a^2 c x^2}}{c^3 x \arctan (a x)}+\frac {9 a^2 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{c^2}+\frac {2 a \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)},x\right )}{c^2} \]

[Out]

-a^3*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-2*a^3*x/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)+9/4*a^2*Ci(arctan(a*x))*(
a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+3/4*a^2*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+2
*a*(a^2*c*x^2+c)^(1/2)/c^3/x/arctan(a*x)+Unintegrable(1/x^3/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/c^2+2*a*Unint
egrable(1/x^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2),x)/c^2

Rubi [N/A]

Not integrable

Time = 1.40 (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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]

[In]

Int[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

-((a^3*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (2*a*Sqrt
[c + a^2*c*x^2])/(c^3*x*ArcTan[a*x]) + (9*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*
c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^3
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 + (2*a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c^2

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

Mathematica [N/A]

Not integrable

Time = 6.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]

[In]

Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 21.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}d x\]

[In]

int(1/x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x)

[Out]

int(1/x^3/(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 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^3/(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^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3)*arctan(a*x)^2), x)

Sympy [N/A]

Not integrable

Time = 24.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x**3/(a**2*c*x**2+c)**(5/2)/atan(a*x)**2,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^3/(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^3*arctan(a*x)^2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(1/(x^3*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(1/(x^3*atan(a*x)^2*(c + a^2*c*x^2)^(5/2)), x)