3.706 \(\int \frac {\sqrt {\tan ^{-1}(a x)}}{x^4 (c+a^2 c x^2)} \, dx\)

Optimal. Leaf size=61 \[ -\frac {a^2 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x^2},x\right )}{c}+\frac {\text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x^4},x\right )}{c}+\frac {2 a^3 \tan ^{-1}(a x)^{3/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.19, 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 = {} \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.51, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^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(arctan(a*x)^(1/2)/x^4/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 3.79, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arctan \left (a x \right )}}{x^{4} \left (a^{2} c \,x^{2}+c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arctan(a*x)^(1/2)/x^4/(a^2*c*x^2+c),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(arctan(a*x)^(1/2)/x^4/(a^2*c*x^2+c),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 = -0.02 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(atan(a*x))/(a**2*x**6 + x**4), x)/c

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