3.1006 \(\int \frac {1}{x^4 (c+a^2 c x^2)^3 \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=93 \[ -\frac {8 \text {Int}\left (\frac {1}{x^5 \left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right )}{a}-16 a \text {Int}\left (\frac {1}{x^3 \left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right )-\frac {2}{a c^3 x^4 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}} \]

[Out]

-2/a/c^3/x^4/(a^2*x^2+1)^2/arctan(a*x)^(1/2)-8*Unintegrable(1/x^5/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)/a-16*a*
Unintegrable(1/x^3/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.20, 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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]

[Out]

-2/(a*c^3*x^4*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (8*Defer[Int][1/(x^5*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),
x])/a - 16*a*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c^3 x^4 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {8 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(16 a) \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

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Mathematica [A]  time = 7.56, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]

[Out]

Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]

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fricas [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^4/(a^2*c*x^2+c)^3/arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 4.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(3/2),x)

[Out]

int(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^2*c*x^2+c)^3/arctan(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*atan(a*x)^(3/2)*(c + a^2*c*x^2)^3),x)

[Out]

int(1/(x^4*atan(a*x)^(3/2)*(c + a^2*c*x^2)^3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{6} x^{10} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{8} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a**2*c*x**2+c)**3/atan(a*x)**(3/2),x)

[Out]

Integral(1/(a**6*x**10*atan(a*x)**(3/2) + 3*a**4*x**8*atan(a*x)**(3/2) + 3*a**2*x**6*atan(a*x)**(3/2) + x**4*a
tan(a*x)**(3/2)), x)/c**3

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