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] $$\int \frac{{\cot }^{-1}(a+bx)}{\sqrt[3]{1+a^2+2abx+b^2{x}^2}}dx$$

[B]   time = 0.37931 (sec), size = 177 ,normalized size = 8.05 $$\frac{6\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right)(4(a+bx){}_2{F}_1(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2bxa+b^2{x}^2+1}){\cot }^{-1}(a+bx)+5(a^2+2abx+b^2{x}^2+1)(2(a+bx){\cot }^{-1}(a+bx)-3))-5\sqrt[3]{2}\sqrt{\pi }\text{Gamma}\left(\frac{5}{3}\right)\text{HypergeometricPFQ}(\{1,\frac{4}{3},\frac{4}{3}\},\{\frac{11}{6},\frac{7}{3}\},\frac{1}{a^2+2abx+b^2{x}^2+1})}{20b\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right){(a^2+2abx+b^2{x}^2+1)}^{4/3}}$$

[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] $$\int \frac{{\cot }^{-1}(a+bx)}{\sqrt[3]{(1+a^2)c+2abcx+b^{2}c{x}^2}}dx$$

[B]   time = 0.0850221 (sec), size = 180 ,normalized size = 7.5 $$\frac{c(6\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right)(4(a+bx){}_2{F}_1(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2bxa+b^2{x}^2+1}){\cot }^{-1}(a+bx)+5(a^2+2abx+b^2{x}^2+1)(2(a+bx){\cot }^{-1}(a+bx)-3))-5\sqrt[3]{2}\sqrt{\pi }\text{Gamma}\left(\frac{5}{3}\right)\text{HypergeometricPFQ}(\{1,\frac{4}{3},\frac{4}{3}\},\{\frac{11}{6},\frac{7}{3}\},\frac{1}{a^2+2abx+b^2{x}^2+1}))}{20b\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right){\left(c(a^2+2abx+b^2{x}^2+1)\right)}^{4/3}}$$

[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] $$\int \frac{(a+bx{)}^2{\cot }^{-1}(a+bx)}{\sqrt[3]{1+a^2+2abx+b^2{x}^2}}dx$$

[B]   time = 0.920116 (sec), size = 198 ,normalized size = 6.83 $$\frac{3(5\sqrt[3]{2}\sqrt{\pi }\text{Gamma}\left(\frac{5}{3}\right)\text{HypergeometricPFQ}(\{1,\frac{4}{3},\frac{4}{3}\},\{\frac{11}{6},\frac{7}{3}\},\frac{1}{a^2+2abx+b^2{x}^2+1})+\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right)(5((a+bx{)}^2+1)(3((a+bx{)}^2+7)+4(a+bx)((a+bx{)}^2-2){\cot }^{-1}(a+bx))-24(a+bx){}_2{F}_1(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2bxa+b^2{x}^2+1}){\cot }^{-1}(a+bx)))}{140b\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right)\sqrt[3]{a^2+2abx+b^2{x}^2+1}((a+bx{)}^2+1)}$$

[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] $$\int \frac{(a+bx{)}^2{\cot }^{-1}(a+bx)}{\sqrt[3]{(1+a^2)c+2abcx+b^{2}c{x}^2}}dx$$

[B]   time = 0.204658 (sec), size = 200 ,normalized size = 6.45 $$\frac{3(5\sqrt[3]{2}\sqrt{\pi }\text{Gamma}\left(\frac{5}{3}\right)\text{HypergeometricPFQ}(\{1,\frac{4}{3},\frac{4}{3}\},\{\frac{11}{6},\frac{7}{3}\},\frac{1}{a^2+2abx+b^2{x}^2+1})+\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right)(5((a+bx{)}^2+1)(3((a+bx{)}^2+7)+4(a+bx)((a+bx{)}^2-2){\cot }^{-1}(a+bx))-24(a+bx){}_2{F}_1(1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2bxa+b^2{x}^2+1}){\cot }^{-1}(a+bx)))}{140b\text{Gamma}\left(\frac{11}{6}\right)\text{Gamma}\left(\frac{7}{3}\right)((a+bx{)}^2+1)\sqrt[3]{c(a^2+2abx+b^2{x}^2+1)}}$$

[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])