Chapter 3
Listing of integrals

 3.1 \(\int x^3 \tan ^{-1}(a+b x^4) \, dx\)
 3.2 \(\int x^{-1+n} \tan ^{-1}(a+b x^n) \, dx\)
 3.3 \(\int x^5 \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.4 \(\int x^3 \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.5 \(\int x \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.6 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x} \, dx\)
 3.7 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^3} \, dx\)
 3.8 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^5} \, dx\)
 3.9 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^7} \, dx\)
 3.10 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^9} \, dx\)
 3.11 \(\int x^6 \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.12 \(\int x^4 \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.13 \(\int x^2 \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.14 \(\int \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.15 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^2} \, dx\)
 3.16 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^4} \, dx\)
 3.17 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^6} \, dx\)
 3.18 \(\int x^{9/2} \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.19 \(\int x^{5/2} \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.20 \(\int \sqrt{x} \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.21 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^{3/2}} \, dx\)
 3.22 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^{7/2}} \, dx\)
 3.23 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^{11/2}} \, dx\)
 3.24 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^{15/2}} \, dx\)
 3.25 \(\int x^{7/2} \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.26 \(\int x^{3/2} \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)
 3.27 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{\sqrt{x}} \, dx\)
 3.28 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^{5/2}} \, dx\)
 3.29 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^{9/2}} \, dx\)
 3.30 \(\int \frac{\tan ^{-1}(1+x+x^2)}{x^2} \, dx\)
 3.31 \(\int \frac{(a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))^n}{1-c^2 x^2} \, dx\)
 3.32 \(\int \frac{(a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))^3}{1-c^2 x^2} \, dx\)
 3.33 \(\int \frac{(a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))^2}{1-c^2 x^2} \, dx\)
 3.34 \(\int \frac{a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}})}{1-c^2 x^2} \, dx\)
 3.35 \(\int \frac{1}{(1-c^2 x^2) (a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))} \, dx\)
 3.36 \(\int \frac{1}{(1-c^2 x^2) (a+b \tan ^{-1}(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}))^2} \, dx\)
 3.37 \(\int x^m \tan ^{-1}(\tan (a+b x)) \, dx\)
 3.38 \(\int x^2 \tan ^{-1}(\tan (a+b x)) \, dx\)
 3.39 \(\int x \tan ^{-1}(\tan (a+b x)) \, dx\)
 3.40 \(\int \tan ^{-1}(\tan (a+b x)) \, dx\)
 3.41 \(\int \frac{\tan ^{-1}(\tan (a+b x))}{x} \, dx\)
 3.42 \(\int x^m \tan ^{-1}(\cot (a+b x)) \, dx\)
 3.43 \(\int x^2 \tan ^{-1}(\cot (a+b x)) \, dx\)
 3.44 \(\int x \tan ^{-1}(\cot (a+b x)) \, dx\)
 3.45 \(\int \tan ^{-1}(\cot (a+b x)) \, dx\)
 3.46 \(\int \frac{\tan ^{-1}(\cot (a+b x))}{x} \, dx\)
 3.47 \(\int \tan ^{-1}(\tan (a+b x)) \, dx\)
 3.48 \(\int x^2 \tan ^{-1}(c+d \tan (a+b x)) \, dx\)
 3.49 \(\int x \tan ^{-1}(c+d \tan (a+b x)) \, dx\)
 3.50 \(\int \tan ^{-1}(c+d \tan (a+b x)) \, dx\)
 3.51 \(\int \frac{\tan ^{-1}(c+d \tan (a+b x))}{x} \, dx\)
 3.52 \(\int x^2 \tan ^{-1}(c+(1+i c) \tan (a+b x)) \, dx\)
 3.53 \(\int x \tan ^{-1}(c+(1+i c) \tan (a+b x)) \, dx\)
 3.54 \(\int \tan ^{-1}(c+(1+i c) \tan (a+b x)) \, dx\)
 3.55 \(\int \frac{\tan ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx\)
 3.56 \(\int x^2 \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx\)
 3.57 \(\int x \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx\)
 3.58 \(\int \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx\)
 3.59 \(\int \frac{\tan ^{-1}(c+(-1+i c) \tan (a+b x))}{x} \, dx\)
 3.60 \(\int \tan ^{-1}(\cot (a+b x)) \, dx\)
 3.61 \(\int x^2 \tan ^{-1}(c+d \cot (a+b x)) \, dx\)
 3.62 \(\int x \tan ^{-1}(c+d \cot (a+b x)) \, dx\)
 3.63 \(\int \tan ^{-1}(c+d \cot (a+b x)) \, dx\)
 3.64 \(\int \frac{\tan ^{-1}(c+d \cot (a+b x))}{x} \, dx\)
 3.65 \(\int x^2 \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\)
 3.66 \(\int x \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\)
 3.67 \(\int \tan ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\)
 3.68 \(\int \frac{\tan ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx\)
 3.69 \(\int x^2 \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx\)
 3.70 \(\int x \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx\)
 3.71 \(\int \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx\)
 3.72 \(\int \frac{\tan ^{-1}(c+(-1-i c) \cot (a+b x))}{x} \, dx\)
 3.73 \(\int \tan ^{-1}(\sinh (x)) \, dx\)
 3.74 \(\int x \tan ^{-1}(\sinh (x)) \, dx\)
 3.75 \(\int x^2 \tan ^{-1}(\sinh (x)) \, dx\)
 3.76 \(\int (e+f x)^3 \tan ^{-1}(\tanh (a+b x)) \, dx\)
 3.77 \(\int (e+f x)^2 \tan ^{-1}(\tanh (a+b x)) \, dx\)
 3.78 \(\int (e+f x) \tan ^{-1}(\tanh (a+b x)) \, dx\)
 3.79 \(\int \tan ^{-1}(\tanh (a+b x)) \, dx\)
 3.80 \(\int \frac{\tan ^{-1}(\tanh (a+b x))}{e+f x} \, dx\)
 3.81 \(\int x^2 \tan ^{-1}(c+d \tanh (a+b x)) \, dx\)
 3.82 \(\int x \tan ^{-1}(c+d \tanh (a+b x)) \, dx\)
 3.83 \(\int \tan ^{-1}(c+d \tanh (a+b x)) \, dx\)
 3.84 \(\int \frac{\tan ^{-1}(c+d \tanh (a+b x))}{x} \, dx\)
 3.85 \(\int x^2 \tan ^{-1}(c+(i+c) \tanh (a+b x)) \, dx\)
 3.86 \(\int x \tan ^{-1}(c+(i+c) \tanh (a+b x)) \, dx\)
 3.87 \(\int \tan ^{-1}(c+(i+c) \tanh (a+b x)) \, dx\)
 3.88 \(\int \frac{\tan ^{-1}(c+(i+c) \tanh (a+b x))}{x} \, dx\)
 3.89 \(\int x^2 \tan ^{-1}(c-(i-c) \tanh (a+b x)) \, dx\)
 3.90 \(\int x \tan ^{-1}(c-(i-c) \tanh (a+b x)) \, dx\)
 3.91 \(\int \tan ^{-1}(c-(i-c) \tanh (a+b x)) \, dx\)
 3.92 \(\int \frac{\tan ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx\)
 3.93 \(\int (e+f x)^3 \tan ^{-1}(\coth (a+b x)) \, dx\)
 3.94 \(\int (e+f x)^2 \tan ^{-1}(\coth (a+b x)) \, dx\)
 3.95 \(\int (e+f x) \tan ^{-1}(\coth (a+b x)) \, dx\)
 3.96 \(\int \tan ^{-1}(\coth (a+b x)) \, dx\)
 3.97 \(\int \frac{\tan ^{-1}(\coth (a+b x))}{e+f x} \, dx\)
 3.98 \(\int x^2 \tan ^{-1}(c+d \coth (a+b x)) \, dx\)
 3.99 \(\int x \tan ^{-1}(c+d \coth (a+b x)) \, dx\)
 3.100 \(\int \tan ^{-1}(c+d \coth (a+b x)) \, dx\)
 3.101 \(\int \frac{\tan ^{-1}(c+d \coth (a+b x))}{x} \, dx\)
 3.102 \(\int x^2 \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx\)
 3.103 \(\int x \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx\)
 3.104 \(\int \tan ^{-1}(c+(i+c) \coth (a+b x)) \, dx\)
 3.105 \(\int \frac{\tan ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx\)
 3.106 \(\int x^2 \tan ^{-1}(c-(i-c) \coth (a+b x)) \, dx\)
 3.107 \(\int x \tan ^{-1}(c-(i-c) \coth (a+b x)) \, dx\)
 3.108 \(\int \tan ^{-1}(c-(i-c) \coth (a+b x)) \, dx\)
 3.109 \(\int \frac{\tan ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx\)
 3.110 \(\int \tan ^{-1}(e^x) \, dx\)
 3.111 \(\int x \tan ^{-1}(e^x) \, dx\)
 3.112 \(\int x^2 \tan ^{-1}(e^x) \, dx\)
 3.113 \(\int \tan ^{-1}(e^{a+b x}) \, dx\)
 3.114 \(\int x \tan ^{-1}(e^{a+b x}) \, dx\)
 3.115 \(\int x^2 \tan ^{-1}(e^{a+b x}) \, dx\)
 3.116 \(\int \tan ^{-1}(a+b f^{c+d x}) \, dx\)
 3.117 \(\int x \tan ^{-1}(a+b f^{c+d x}) \, dx\)
 3.118 \(\int x^2 \tan ^{-1}(a+b f^{c+d x}) \, dx\)
 3.119 \(\int e^{-x} \tan ^{-1}(e^x) \, dx\)
 3.120 \(\int \frac{\tan ^{-1}(x)}{(-1+x)^3} \, dx\)
 3.121 \(\int \frac{\tan ^{-1}(1+2 x)}{(4+3 x)^3} \, dx\)
 3.122 \(\int \tan ^{-1}(\sqrt{1+x}) \, dx\)
 3.123 \(\int \frac{1}{(1+x^2) (2+\tan ^{-1}(x))} \, dx\)
 3.124 \(\int \frac{1}{(a+a x^2) (b-2 b \tan ^{-1}(x))} \, dx\)
 3.125 \(\int \frac{x+x^3+(1+x)^2 \tan ^{-1}(x)}{(1+x)^2 (1+x^2)} \, dx\)
 3.126 \(\int -x^3 \tan ^{-1}(\sqrt{x}-\sqrt{1+x}) \, dx\)
 3.127 \(\int -x^2 \tan ^{-1}(\sqrt{x}-\sqrt{1+x}) \, dx\)
 3.128 \(\int -x \tan ^{-1}(\sqrt{x}-\sqrt{1+x}) \, dx\)
 3.129 \(\int -\tan ^{-1}(\sqrt{x}-\sqrt{1+x}) \, dx\)
 3.130 \(\int -\frac{\tan ^{-1}(\sqrt{x}-\sqrt{1+x})}{x} \, dx\)
 3.131 \(\int -\frac{\tan ^{-1}(\sqrt{x}-\sqrt{1+x})}{x^2} \, dx\)
 3.132 \(\int -\frac{\tan ^{-1}(\sqrt{x}-\sqrt{1+x})}{x^3} \, dx\)
 3.133 \(\int -\frac{\tan ^{-1}(\sqrt{x}-\sqrt{1+x})}{x^4} \, dx\)
 3.134 \(\int \frac{\tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})^m}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx\)
 3.135 \(\int \frac{\tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})^2}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx\)
 3.136 \(\int \frac{\tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx\)
 3.137 \(\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})} \, dx\)
 3.138 \(\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})^2} \, dx\)
 3.139 \(\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})^3} \, dx\)
 3.140 \(\int \frac{\tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})^m}{\sqrt{a+b x^2}} \, dx\)
 3.141 \(\int \frac{\tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})^2}{\sqrt{a+b x^2}} \, dx\)
 3.142 \(\int \frac{\tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})}{\sqrt{a+b x^2}} \, dx\)
 3.143 \(\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})} \, dx\)
 3.144 \(\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})^2} \, dx\)
 3.145 \(\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})^3} \, dx\)
 3.146 \(\int \frac{\tan ^{-1}(c (a+b x)) \log (d (a+b x))}{a+b x} \, dx\)
 3.147 \(\int e^{c (a+b x)} \tan ^{-1}(\sinh (a c+b c x)) \, dx\)
 3.148 \(\int e^{c (a+b x)} \tan ^{-1}(\cosh (a c+b c x)) \, dx\)
 3.149 \(\int e^{c (a+b x)} \tan ^{-1}(\tanh (a c+b c x)) \, dx\)
 3.150 \(\int e^{c (a+b x)} \tan ^{-1}(\coth (a c+b c x)) \, dx\)
 3.151 \(\int e^{c (a+b x)} \tan ^{-1}(\text{sech}(a c+b c x)) \, dx\)
 3.152 \(\int e^{c (a+b x)} \tan ^{-1}(\text{csch}(a c+b c x)) \, dx\)
 3.153 \(\int \frac{(a+b \tan ^{-1}(c x^n)) (d+e \log (f x^m))}{x} \, dx\)