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

Optimal. Leaf size=263 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{2 a c^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {5 \sqrt {a^2 x^2+1} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 c^2 \sqrt {a^2 c x^2+c}}+\frac {9 \sqrt {a^2 x^2+1} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 c^2 \sqrt {a^2 c x^2+c}}+\frac {a x}{2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac {1}{2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}+\frac {a x}{2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]

[Out]

1/2*a*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2+3/2/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)+1/2*a*x/c^2/arctan(a*x)^2/(a
^2*c*x^2+c)^(1/2)-1/2/c^2/arctan(a*x)/(a^2*c*x^2+c)^(1/2)+5/8*Si(arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2
+c)^(1/2)+9/8*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-1/2*(a^2*c*x^2+c)^(1/2)/a/c^3/x/arct
an(a*x)^2-1/2*Unintegrable(1/x^2/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/a/c^2

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Rubi [A]  time = 1.72, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx \]

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*}

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Mathematica [A]  time = 3.80, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx \]

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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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )^{3}}, x\right ) \]

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

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maple [A]  time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{3}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

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]

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

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