\(\int \frac {x^4}{(c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [667]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {1}{2 a^5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {3 x}{2 a^4 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {x}{a^4 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {5 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^3},x\right )}{a^4 c^2} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.97 (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 = {} \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

[In]

Int[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

-1/2*1/(a^5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + 1/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + (3*x)/(2*
a^4*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - x/(a^4*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*
CosIntegral[ArcTan[a*x]])/(8*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(
8*a^5*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^4*c^2)

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

Mathematica [N/A]

Not integrable

Time = 7.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

[In]

Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]

Maple [N/A] (verified)

Not integrable

Time = 18.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {x^{4}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}d x\]

[In]

int(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

[Out]

int(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 12.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^{4}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(x**4/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 227.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^4}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(x^4/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(x^4/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)