\(\int \frac {1}{x^2 (c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [673]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {7 a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3},x\right )}{c^2} \]

[Out]

1/2*a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2-3/2*a^2*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)+1/2*a/c^2/arctan(a*x)^2/
(a^2*c*x^2+c)^(1/2)-1/2*a^2*x/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)+7/8*a*Ci(arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/
(a^2*c*x^2+c)^(1/2)+9/8*a*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+Unintegrable(1/x^2/arcta
n(a*x)^3/(a^2*c*x^2+c)^(1/2),x)/c^2

Rubi [N/A]

Not integrable

Time = 0.80 (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 = {} \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

[In]

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

[Out]

a/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + a/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (3*a^2*x)/(2*c*(c
+ a^2*c*x^2)^(3/2)*ArcTan[a*x]) - (a^2*x)/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*a*Sqrt[1 + a^2*x^2]*Cos
Integral[ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + (9*a*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*c^2
*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c^2

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx\right )+\frac {\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx}{c} \\ & = \frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {1}{2} \left (3 a^3\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2}-\frac {a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx}{c} \\ & = \frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{2} \left (3 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx-\left (3 a^4\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2}+\frac {a^3 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{2 c} \\ & = \frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2}+\frac {a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx}{2 c}+\frac {\left (3 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a^4 \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}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2}+\frac {\left (3 a \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 \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}}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2}+\frac {\left (a \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 (3 a \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 )}{2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a \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}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2}+\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arctan (a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a \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 {\left (3 a \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 (9 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {7 a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3} \, dx}{c^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 13.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 29.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 176.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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