3.8.41 \(\int \frac {\sqrt {\text {ArcTan}(a x)}}{x^4 \sqrt {c+a^2 c x^2}} \, dx\) [741]

Optimal. Leaf size=138 \[ -\frac {\sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}{3 c x}+\frac {1}{6} a \text {Int}\left (\frac {1}{x^3 \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}},x\right )-\frac {1}{3} a^3 \text {Int}\left (\frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}},x\right ) \]

[Out]

-1/3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/c/x^3+2/3*a^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/c/x+1/6*a*Unint
egrable(1/x^3/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)-1/3*a^3*Unintegrable(1/x/(a^2*c*x^2+c)^(1/2)/arctan(a*x
)^(1/2),x)

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Rubi [A]
time = 0.28, 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 = {} \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/3*(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*c*x) +
(a*Defer[Int][1/(x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/6 - (a^3*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]
*Sqrt[ArcTan[a*x]]), x])/3

Rubi steps

\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^4 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{3 c x^3}+\frac {1}{6} a \int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2 \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{3 c x}+\frac {1}{6} a \int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {1}{3} a^3 \int \frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ \end {align*}

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Mathematica [A]
time = 14.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 4.86, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{x^{4} \sqrt {a^{2} c \,x^{2}+c}}\, dx\]

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)^(1/2),x)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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)^(1/2),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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \end {gather*}

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)**(1/2),x)

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}

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)^(1/2)),x)

[Out]

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

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