3.1.65 \(\int \frac {\text {ArcTan}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx\) [65]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\text {ArcTan}(a+b x)}{\sqrt [3]{1+(a+b x)^2}},x\right ) \]

[Out]

Unintegrable(arctan(b*x+a)/(1+(b*x+a)^2)^(1/3),x)

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Rubi [A]
time = 0.03, 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 {\text {ArcTan}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Subst][Defer[Int][ArcTan[x]/(1 + x^2)^(1/3), x], x, a + b*x]/b

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\tan ^{-1}(x)}{\sqrt [3]{1+x^2}} \, dx,x,a+b x\right )}{b}\\ \end {align*}

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(23)=46\).
time = 0.31, size = 163, normalized size = 7.09 \begin {gather*} \frac {6 \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right ) \left (15+10 (a+b x) \text {ArcTan}(a+b x)+\frac {4 (a+b x) \text {ArcTan}(a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}\right )+\frac {5 \sqrt [3]{2} \sqrt {\pi } \text {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}}{20 b \sqrt [3]{1+a^2+2 a b x+b^2 x^2} \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[
1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{
1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/
3)*Gamma[11/6]*Gamma[7/3])

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Maple [A]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (b x +a \right )}{\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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(b*x+a)/(b^2*x^2+2*a*b*x+a^2+1)^(1/3),x, algorithm="fricas")

[Out]

integral(arctan(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 + 1)^(1/3), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atan}{\left (a + b x \right )}}{\sqrt [3]{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(a + b*x)/(a**2 + 2*a*b*x + b**2*x**2 + 1)**(1/3), 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(b*x+a)/(b^2*x^2+2*a*b*x+a^2+1)^(1/3),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {atan}\left (a+b\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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