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

Optimal. Leaf size=176 \[ \frac{\text{Unintegrable}\left (\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{a^4 c^2}-\frac{7 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{x^3}{a^3 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.911595, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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 = 12.3128, size = 0, normalized size = 0. \[ \int \frac{x^5}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]

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]

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Maple [A]  time = 1.791, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \left ( \arctan \left ( ax \right ) \right ) ^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}

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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}

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]

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