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Chapter 3
Listing of integrals
3.1
\(\int \frac {1}{\sqrt {1-a x}} \, dx\)
3.2
\(\int \frac {-2 \log (-\sqrt {-1+a x})+\log (-1+a x)}{2 \pi \sqrt {-1+a x}} \, dx\)
3.3
\(\int \frac {1}{(2 x+\sqrt {1+x^2})^2} \, dx\)
3.4
\(\int \frac {1}{\sqrt {-1+x^2} (-4+3 x^2)^2} \, dx\)
3.5
\(\int \frac {1}{(2 \sqrt {x}+\sqrt {1+x})^2} \, dx\)
3.6
\(\int \frac {\sqrt {-1+x^2}}{(-i+x)^2} \, dx\)
3.7
\(\int \frac {1}{\sqrt {-1+x^2} (1+x^2)^2} \, dx\)
3.8
\(\int \frac {1}{(\sqrt {-1+x}+\sqrt {x})^2 \sqrt {-1+x}} \, dx\)
3.9
\(\int \frac {1}{\sqrt {-1+x^2} (\sqrt {x}+\sqrt {-1+x^2})^2} \, dx\)
3.10
\(\int \frac {(\sqrt {x}-\sqrt {-1+x^2})^2}{(1+x-x^2)^2 \sqrt {-1+x^2}} \, dx\)
3.11
\(\int (\frac {1}{\sqrt {2} (1+x)^2 \sqrt {-i+x^2}}+\frac {1}{\sqrt {2} (1+x)^2 \sqrt {i+x^2}}) \, dx\)
3.12
\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx\)
3.13
\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx\)
3.14
\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\)
3.15
\(\int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\)
3.16
\(\int \frac {(-1+x)^{3/2}+(1+x)^{3/2}}{(-1+x)^{3/2} (1+x)^{3/2}} \, dx\)
3.17
\(\int (x+\sqrt {a+x^2})^b \, dx\)
3.18
\(\int (x-\sqrt {a+x^2})^b \, dx\)
3.19
\(\int \frac {(x+\sqrt {a+x^2})^b}{\sqrt {a+x^2}} \, dx\)
3.20
\(\int \frac {(x-\sqrt {a+x^2})^b}{\sqrt {a+x^2}} \, dx\)
3.21
\(\int \frac {1}{(a+b e^{p x})^2} \, dx\)
3.22
\(\int \frac {1}{(b e^{-p x}+a e^{p x})^2} \, dx\)
3.23
\(\int \frac {x}{(b e^{-p x}+a e^{p x})^2} \, dx\)
3.24
\(\int \frac {1-x+3 x^2}{\sqrt {1-x+x^2} (1+x+x^2)^2} \, dx\)
3.25
\(\int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx\)
3.26
\(\int \frac {\sqrt {b x+\sqrt {a+b^2 x^2}}}{\sqrt {a+b^2 x^2}} \, dx\)
3.27
\(\int \frac {1}{x \sqrt {a^2+x^2} \sqrt {x+\sqrt {a^2+x^2}}} \, dx\)
3.28
\(\int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx\)
3.29
\(\int x^3 \log ^3(2+x) \log (3+x) \, dx\)
3.30
\(\int \frac {(x+\sqrt {b+x^2})^a}{\sqrt {b+x^2}} \, dx\)
3.31
\(\int (x+\sqrt {b+x^2})^a \, dx\)
3.32
\(\int (6+3 x^a+2 x^{2 a})^{\frac {1}{a}} (x^a+x^{2 a}+x^{3 a}) \, dx\)
3.33
\(\int \frac {1}{x \sqrt [3]{1-x^2}} \, dx\)
3.34
\(\int \frac {1}{x (1-x^2)^{2/3}} \, dx\)
3.35
\(\int \frac {1}{\sqrt [3]{1-x^3}} \, dx\)
3.36
\(\int \frac {1}{x \sqrt [3]{1-x^3}} \, dx\)
3.37
\(\int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx\)
3.38
\(\int \frac {x}{(1+x) \sqrt [3]{1-x^3}} \, dx\)
3.39
\(\int \frac {1}{x \sqrt [3]{2-3 x+x^2}} \, dx\)
3.40
\(\int \frac {1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx\)
3.41
\(\int \frac {1}{\sqrt [3]{x (-q+x^2)}} \, dx\)
3.42
\(\int \frac {1}{\sqrt [3]{(-1+x) (q-2 x+x^2)}} \, dx\)
3.43
\(\int \frac {1}{x \sqrt [3]{(-1+x) (q-2 q x+x^2)}} \, dx\)
3.44
\(\int \frac {2-(1+k) x}{\sqrt [3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx\)
3.45
\(\int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx\)
3.46
\(\int \frac {a+b x+c x^2}{(1-x+x^2) \sqrt [3]{1-x^3}} \, dx\)
3.47
\(\int \frac {1}{(3-2 x)^{11/2} (1+x+2 x^2)^5} \, dx\)
3.48
\(\int \frac {1}{(3-2 x)^{21/2} (1+x+2 x^2)^{10}} \, dx\)
3.49
\(\int \frac {1}{(3-2 x)^{41/2} (1+x+2 x^2)^{20}} \, dx\)
3.50
\(\int \frac {1}{(3-2 x+x^2)^{11/2} (1+x+2 x^2)^5} \, dx\)
3.51
\(\int \frac {1}{(3-2 x+x^2)^{21/2} (1+x+2 x^2)^{10}} \, dx\)
3.52
\(\int \frac {-a-\sqrt {1+a^2}+x}{(-a+\sqrt {1+a^2}+x) \sqrt {(-a+x) (1+x^2)}} \, dx\)
3.53
\(\int \frac {a+b x}{\sqrt [3]{1-x^2} (3+x^2)} \, dx\)
3.54
\(\int \frac {a+b x}{(3-x^2) \sqrt [3]{1+x^2}} \, dx\)
3.55
\(\int \frac {1}{x \sqrt [3]{4-6 x+3 x^2}} \, dx\)
3.56
\(\int x \sqrt [3]{1-x^3} \, dx\)
3.57
\(\int \frac {\sqrt [3]{1-x^3}}{x} \, dx\)
3.58
\(\int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx\)
3.59
\(\int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx\)
3.60
\(\int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx\)
3.61
\(\int \frac {2+x}{(1+x+x^2) \sqrt [3]{2+x^3}} \, dx\)
3.62
\(\int \frac {3-3 x+30 x^2+160 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)
3.63
\(\int \frac {3+12 x+20 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)
3.64
\(\int \frac {-84-576 x-400 x^2+2560 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)
3.65
\(\int \frac {\sqrt {1-x^4}}{1+x^4} \, dx\)
3.66
\(\int \frac {\sqrt {1+x^4}}{1-x^4} \, dx\)
3.67
\(\int \frac {\sqrt {1+p x^2+x^4}}{1-x^4} \, dx\)
3.68
\(\int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx\)
3.69
\(\int \frac {a+b x}{(2-x^2) \sqrt [4]{-1+x^2}} \, dx\)
3.70
\(\int \frac {a+b x}{\sqrt [4]{-1-x^2} (2+x^2)} \, dx\)
3.71
\(\int \frac {a+b x}{\sqrt [4]{1-x^2} (2-x^2)} \, dx\)
3.72
\(\int \frac {a+b x}{\sqrt [4]{1+x^2} (2+x^2)} \, dx\)
3.73
\(\int \frac {x}{\sqrt {1-x^3} (4-x^3)} \, dx\)
3.74
\(\int \frac {x}{(4-d x^3) \sqrt {-1+d x^3}} \, dx\)
3.75
\(\int \frac {x}{\sqrt {-1+x^3} (8+x^3)} \, dx\)
3.76
\(\int \frac {x}{(8-d x^3) \sqrt {1+d x^3}} \, dx\)
3.77
\(\int \frac {1}{\sqrt [3]{1-3 x^2} (3-x^2)} \, dx\)
3.78
\(\int \frac {1}{(3+x^2) \sqrt [3]{1+3 x^2}} \, dx\)
3.79
\(\int \frac {1}{\sqrt [3]{1-x^2} (3+x^2)} \, dx\)
3.80
\(\int \frac {1}{(3-x^2) \sqrt [3]{1+x^2}} \, dx\)
3.81
\(\int \frac {a+x}{(-a+x) \sqrt {a^2 x-(1+a^2) x^2+x^3}} \, dx\)
3.82
\(\int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+(-1-2 a+a^2) x^2+x^3}} \, dx\)
3.83
\(\int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-(-1+2 a+a^2) x^2+(-1+2 a) x^3}} \, dx\)
3.84
\(\int \frac {1-\sqrt [3]{2} x}{(2^{2/3}+x) \sqrt {1+x^3}} \, dx\)
3.85
\(\int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx\)
3.86
\(\int \frac {x}{\sqrt {1+x^3} (10+6 \sqrt {3}+x^3)} \, dx\)
3.87
\(\int \frac {x}{\sqrt {1+x^3} (10-6 \sqrt {3}+x^3)} \, dx\)
3.88
\(\int \frac {x}{\sqrt {-1+x^3} (-10-6 \sqrt {3}+x^3)} \, dx\)
3.89
\(\int \frac {x}{\sqrt {-1+x^3} (-10+6 \sqrt {3}+x^3)} \, dx\)
3.90
\(\int \frac {1-\sqrt {3}+x}{(1+\sqrt {3}+x) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx\)
3.91
\(\int \frac {1+\sqrt {3}+x}{(1-\sqrt {3}+x) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx\)
3.92
\(\int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx\)
3.93
\(\int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx\)
3.94
\(\int \frac {1}{(1-x^3) \sqrt [3]{a+b x^3}} \, dx\)
3.95
\(\int \frac {1+x}{(1+x+x^2) \sqrt [3]{a+b x^3}} \, dx\)
3.96
\(\int \frac {x^2}{(1-x^3) \sqrt [3]{a+b x^3}} \, dx\)
3.97
\(\int \frac {1}{\sqrt [3]{1-x^3} (1+x^3)} \, dx\)
3.98
\(\int \frac {x}{\sqrt [3]{1-x^3} (1+x^3)} \, dx\)
3.99
\(\int \frac {x^2}{\sqrt [3]{1-x^3} (1+x^3)} \, dx\)
3.100
\(\int \frac {1+x}{(1-x+x^2) \sqrt [3]{1-x^3}} \, dx\)
3.101
\(\int \frac {(1+x)^2}{\sqrt [3]{1-x^3} (1+x^3)} \, dx\)
3.102
\(\int \frac {1-x}{(1+x+x^2) \sqrt [3]{1+x^3}} \, dx\)
3.103
\(\int \frac {(1-x^3)^{2/3}}{(1+x+x^2)^2} \, dx\)
3.104
\(\int \frac {1-x}{(1+x+x^2) \sqrt [3]{1-x^3}} \, dx\)
3.105
\(\int \frac {(1-x)^2}{(1-x^3)^{4/3}} \, dx\)
3.106
\(\int (1-x^3)^{2/3} \, dx\)
3.107
\(\int \frac {(1-x^3)^{2/3}}{x} \, dx\)
3.108
\(\int \frac {(1-x^3)^{2/3}}{a+b x} \, dx\)
3.109
\(\int \frac {(1-x^3)^{2/3}}{(1-x+x^2)^2} \, dx\)
3.110
\(\int \frac {(1-2 x) (1-x^3)^{2/3}}{(1-x+x^2)^2} \, dx\)
3.111
\(\int \frac {(1-x^3)^{2/3}}{1+x} \, dx\)
3.112
\(\int \frac {(1-x+x^2) (1-x^3)^{2/3}}{1+x^3} \, dx\)
3.113
\(\int \frac {(1-x^3)^{2/3}}{1+x^3} \, dx\)
3.114
\(\int \frac {x (1-x^3)^{2/3}}{1+x^3} \, dx\)
3.115
\(\int \frac {(1-x) (1-x^3)^{2/3}}{1+x^3} \, dx\)
3.116
\(\int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx\)
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