3.587 \(\int \frac {x^5}{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=177 \[ \frac {\text {Int}\left (\frac {x}{\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{a^4 c^2}-\frac {7 \sqrt {a^2 x^2+1} \text {Ci}\left (\tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {a^2 x^2+1} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{a^5 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}+\frac {x^3}{a^3 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

x^3/a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)+x/a^5/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)-7/4*Ci(arctan(a*x))*(a^2*x
^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/2)+3/4*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/2)+Un
integrable(x/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/a^4/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.91, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

x^3/(a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + x/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (7*Sqrt[1 + a^2*
x^2]*CosIntegral[ArcTan[a*x]])/(4*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x
]])/(4*a^6*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^4*c^2)

Rubi steps

\begin {align*} \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\frac {\int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{a^2}+\frac {\int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{a^2 c}\\ &=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{a^3}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{a^4 c^2}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{a^4 c}\\ &=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{a^4 c^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{a^5 c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{a^4 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{a^4 c^2}-\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{a^4 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {7 \sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{a^4 c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 12.61, size = 0, normalized size = 0.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{5}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^5/((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)^2), x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 5.70, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________