3.7 Test file Number [154]

3.7.1 Mathematica

Integral number [116] \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 0.289474 (sec), size = 177 ,normalized size = 7.7 \[ \frac {6 \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+a^2+2 a b x+b^2 x^2\right ) \left (-3+2 (a+b x) \cot ^{-1}(a+b x)\right )+4 (a+b x) \cot ^{-1}(a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )-5 \sqrt [3]{2} \sqrt {\pi } \text {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )}{20 b \left (1+a^2+2 a b x+b^2 x^2\right )^{4/3} \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right )} \]

[In]

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

Integral number [117] \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.125618 (sec), size = 180 ,normalized size = 7.2 \[ \frac {c \left (6 \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+a^2+2 a b x+b^2 x^2\right ) \left (-3+2 (a+b x) \cot ^{-1}(a+b x)\right )+4 (a+b x) \cot ^{-1}(a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )-5 \sqrt [3]{2} \sqrt {\pi } \text {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{20 b \left (c \left (1+a^2+2 a b x+b^2 x^2\right )\right )^{4/3} \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right )} \]

[In]

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

[Out]

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

Integral number [120] \[ \int \frac {(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

[B]   time = 0.640545 (sec), size = 198 ,normalized size = 6.6 \[ \frac {3 \left (\text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+(a+b x)^2\right ) \left (3 \left (7+(a+b x)^2\right )+4 (a+b x) \left (-2+(a+b x)^2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \cot ^{-1}(a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )+5 \sqrt [3]{2} \sqrt {\pi } \text {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{140 b \sqrt [3]{1+a^2+2 a b x+b^2 x^2} \left (1+(a+b x)^2\right ) \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right )} \]

[In]

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

[Out]

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + b*x)*(-2 + (a + b*x)^2)*ArcCot[a
+ b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) +
5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1
)]))/(140*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b*x)^2)*Gamma[11/6]*Gamma[7/3])

Integral number [121] \[ \int \frac {(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

[B]   time = 0.21293 (sec), size = 200 ,normalized size = 6.25 \[ \frac {3 \left (\text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+(a+b x)^2\right ) \left (3 \left (7+(a+b x)^2\right )+4 (a+b x) \left (-2+(a+b x)^2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \cot ^{-1}(a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )+5 \sqrt [3]{2} \sqrt {\pi } \text {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{140 b \sqrt [3]{c \left (1+a^2+2 a b x+b^2 x^2\right )} \left (1+(a+b x)^2\right ) \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right )} \]

[In]

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

[Out]

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + b*x)*(-2 + (a + b*x)^2)*ArcCot[a
+ b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) +
5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1
)]))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*(1 + (a + b*x)^2)*Gamma[11/6]*Gamma[7/3])