\(\int \frac {\arctan (a+b x)}{\sqrt [3]{(1+a^2) c+2 a b c x+b^2 c x^2}} \, dx\) [66]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\text {Int}\left (\frac {\arctan (a+b x)}{\sqrt [3]{c+c (a+b x)^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\arctan (x)}{\sqrt [3]{c+c x^2}} \, dx,x,a+b x\right )}{b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(25)=50\).

Time = 0.15 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.00 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\frac {6 \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (15+10 (a+b x) \arctan (a+b x)+\frac {4 (a+b x) \arctan (a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}\right )+\frac {5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{1+(a+b x)^2}}{20 b \sqrt [3]{c \left (1+a^2+2 a b x+b^2 x^2\right )} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]

[In]

Integrate[ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*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*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))
^(1/3)*Gamma[11/6]*Gamma[7/3])

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94

\[\int \frac {\arctan \left (b x +a \right )}{{\left (\left (a^{2}+1\right ) c +2 a b c x +b^{2} c \,x^{2}\right )}^{\frac {1}{3}}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\operatorname {atan}{\left (a + b x \right )}}{\sqrt [3]{c \left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 59.74 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.09 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{{\left (c\,b^2\,x^2+2\,a\,c\,b\,x+c\,\left (a^2+1\right )\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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