4.7 Test file Number [154] 5-Inverse-trig-functions/5.4-Inverse-cotangent/5.4.1-Inverse-cotangent-functions

4.7.1 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 = 0.37931 (sec), size = 177 ,normalized size = 8.05 \[ \frac{6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (4 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)+5 \left (a^2+2 a b x+b^2 x^2+1\right ) \left (2 (a+b x) \cot ^{-1}(a+b x)-3\right )\right )-5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{a^2+2 a b x+b^2 x^2+1}\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (a^2+2 a b x+b^2 x^2+1\right )^{4/3}} \]

[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 = 0.0850221 (sec), size = 180 ,normalized size = 7.5 \[ \frac{c \left (6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (4 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)+5 \left (a^2+2 a b x+b^2 x^2+1\right ) \left (2 (a+b x) \cot ^{-1}(a+b x)-3\right )\right )-5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{a^2+2 a b x+b^2 x^2+1}\right )\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (c \left (a^2+2 a b x+b^2 x^2+1\right )\right )^{4/3}} \]

[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 = 0.920116 (sec), size = 198 ,normalized size = 6.83 \[ \frac{3 \left (5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{a^2+2 a b x+b^2 x^2+1}\right )+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1} \left ((a+b x)^2+1\right )} \]

[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 = 0.204658 (sec), size = 200 ,normalized size = 6.45 \[ \frac{3 \left (5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{a^2+2 a b x+b^2 x^2+1}\right )+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left ((a+b x)^2+1\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]

[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])