3.666 \(\int \frac{x^5}{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=240 \[ \frac{\text{Unintegrable}\left (\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right )}{a^4 c^2}+\frac{7 \sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{8 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{8 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{2 a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{2}{a^6 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{x^3}{2 a^3 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

x^3/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + x/(2*a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - 3/(2*a^6
*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + 2/(a^6*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*Sqrt[1 + a^2*x^2]*Sin
Integral[ArcTan[a*x]])/(8*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a
^6*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^4*c^2)

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Rubi [A]  time = 1.35826, 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 = {} \[ \int \frac{x^5}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

x^3/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + x/(2*a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - 3/(2*a^6
*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + 2/(a^6*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*Sqrt[1 + a^2*x^2]*Sin
Integral[ArcTan[a*x]])/(8*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a
^6*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), 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)^3} \, dx &=-\frac{\int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx}{a^2}+\frac{\int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{a^2 c}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac{3 \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{2 a^3}+\frac{\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{a^4 c^2}-\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{a^4 c}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}+\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{2 a^5}+\frac{\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{a^4 c^2}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{2 a^5 c}-\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{2 a^5 c}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{2}{a^6 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{9 \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 a^4}+\frac{\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{a^4 c^2}+\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^4 c}+\frac{3 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^4 c}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{2}{a^6 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)^3} \, dx}{a^4 c^2}+\frac{\sqrt{1+a^2 x^2} \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(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}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{2}{a^6 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)^3} \, dx}{a^4 c^2}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 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{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^6 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{2}{a^6 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{2 \sqrt{1+a^2 x^2} \text{Si}\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)^3} \, dx}{a^4 c^2}-\frac{\left (9 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^6 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{2}{a^6 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{2 \sqrt{1+a^2 x^2} \text{Si}\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)^3} \, dx}{a^4 c^2}-\frac{\left (9 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^6 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{x}{2 a^5 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{2}{a^6 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{7 \sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{8 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{9 \sqrt{1+a^2 x^2} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{8 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)^3} \, dx}{a^4 c^2}\\ \end{align*}

Mathematica [A]  time = 7.71025, size = 0, normalized size = 0. \[ \int \frac{x^5}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.859, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \left ( \arctan \left ( ax \right ) \right ) ^{3}} \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)^3,x)

[Out]

int(x^5/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,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 )^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

integrate(x^5/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3), 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 )^{3}}, 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)^3,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)^3), 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}^{3}{\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)**3,x)

[Out]

Integral(x**5/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), 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 )^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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