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

Optimal. Leaf size=262 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{2 a c^2}+\frac{5 \sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{8 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{8 c^2 \sqrt{a^2 c x^2+c}}+\frac{a x}{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{1}{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{\sqrt{a^2 c x^2+c}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 1.72045, 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 = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac{1}{2} a \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+a^3 \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c^2}-\frac{a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{1}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-a \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+\frac{1}{2} \left (3 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}-\frac{a \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{2 c}+\frac{a \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\left (3 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac{a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 c}-\frac{a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c}+\frac{\left (3 a^2 \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 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac{\left (3 \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 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (a^2 \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 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (a^2 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a^2 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a 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 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{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{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 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 ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^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 )}{8 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 (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 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{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{8 c^2 \sqrt{c+a^2 c x^2}}+\frac{3 \sqrt{1+a^2 x^2} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{8 c^2 \sqrt{c+a^2 c x^2}}-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^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 )}{4 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 (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac{a x}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{\sqrt{c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{5 \sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{8 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 c^2 \sqrt{c+a^2 c x^2}}-\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ \end{align*}

Mathematica [A]  time = 3.46554, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.653, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

[Out]

int(1/x/(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{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(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*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}}{{\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(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^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)*arctan(a*x)^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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