Integrand size = 33, antiderivative size = 380 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 a \left (5 a^2 A-14 A b^2-15 a b B\right ) \sqrt {\cot (c+d x)}}{5 d}-\frac {2 a^2 (9 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x)}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}{5 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
-2/15*a^2*(9*A*b+5*B*a)*cot(d*x+c)^(3/2)/d-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3* a^2*b*(A+B)+b^3*(A+B))*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/2*( a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+2^(1/2)*cot(d*x+ c)^(1/2))/d*2^(1/2)+1/4*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))* ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/4*(3*a^2*b*(A-B)-b^3 *(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/ d*2^(1/2)+2/5*a*(5*A*a^2-14*A*b^2-15*B*a*b)*cot(d*x+c)^(1/2)/d-2/5*a*A*(b+ a*cot(d*x+c))^2*cot(d*x+c)^(1/2)/d
Time = 2.47 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.75 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 \sqrt {\cot (c+d x)} \left (-\frac {\left (a^3 (A-B)+3 a b^2 (-A+B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{2 \sqrt {2}}+\frac {\left (3 a^2 b (A-B)+b^3 (-A+B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{4 \sqrt {2}}-\frac {a^3 A}{5 \tan ^{\frac {5}{2}}(c+d x)}-\frac {a^2 (3 A b+a B)}{3 \tan ^{\frac {3}{2}}(c+d x)}+\frac {a \left (a^2 A-3 A b^2-3 a b B\right )}{\sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{d} \]
(2*Sqrt[Cot[c + d*x]]*(-1/2*((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + S qrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] + ((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2] ) - (a^3*A)/(5*Tan[c + d*x]^(5/2)) - (a^2*(3*A*b + a*B))/(3*Tan[c + d*x]^( 3/2)) + (a*(a^2*A - 3*A*b^2 - 3*a*b*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d *x]])/d
Time = 1.32 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.82, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4064, 3042, 4090, 27, 3042, 4120, 27, 3042, 4113, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} (a+b \tan (c+d x))^3 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+b)^3 (A \cot (c+d x)+B)}{\sqrt {\cot (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4090 |
| \(\displaystyle -\frac {2}{5} \int \frac {(b+a \cot (c+d x)) \left (-a (9 A b+5 a B) \cot ^2(c+d x)+5 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (a A-5 b B)\right )}{2 \sqrt {\cot (c+d x)}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \int \frac {(b+a \cot (c+d x)) \left (-a (9 A b+5 a B) \cot ^2(c+d x)+5 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (a A-5 b B)\right )}{\sqrt {\cot (c+d x)}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-a (9 A b+5 a B) \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 \left (A a^2-2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b (a A-5 b B)\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {1}{5} \left (-\frac {2}{3} \int \frac {3 \left ((a A-5 b B) b^2+a \left (5 A a^2-15 b B a-14 A b^2\right ) \cot ^2(c+d x)+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{2 \sqrt {\cot (c+d x)}}dx-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {(a A-5 b B) b^2+a \left (5 A a^2-15 b B a-14 A b^2\right ) \cot ^2(c+d x)+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {(a A-5 b B) b^2+a \left (5 A a^2-15 b B a-14 A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)-5 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )}{\sqrt {\cot (c+d x)}}dx+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {-5 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )-5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{5} \left (-\frac {2 \int \frac {5 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B-\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \int \frac {A a^3-3 b B a^2-3 A b^2 a+b^3 B-\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 \left (\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\) |
(-2*a*A*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])^2)/(5*d) + ((2*a*(5*a^2*A - 14*A*b^2 - 15*a*b*B)*Sqrt[Cot[c + d*x]])/d - (2*a^2*(9*A*b + 5*a*B)*Cot[ c + d*x]^(3/2))/(3*d) - (10*(((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + A rcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 + ((3*a^2*b*(A - B) - b^ 3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[ c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + C ot[c + d*x]]/(2*Sqrt[2])))/2))/d)/5
3.6.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp [g^(m + n) Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c *Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !Integer Q[p] && IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b *(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2 , 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && !LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1133\) vs. \(2(340)=680\).
Time = 1.22 (sec) , antiderivative size = 1134, normalized size of antiderivative = 2.98
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1134\) |
| default | \(\text {Expression too large to display}\) | \(1134\) |
-1/60/d*(1/tan(d*x+c))^(7/2)*tan(d*x+c)*(40*B*tan(d*x+c)*a^3-30*A*arctan(- 1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(5/2)*b^3-45*B*ln(-(1+2^(1/ 2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2 ^(1/2)*tan(d*x+c)^(5/2)*a*b^2+90*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1 /2)*tan(d*x+c)^(5/2)*a^2*b+90*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2) *tan(d*x+c)^(5/2)*a*b^2+90*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*t an(d*x+c)^(5/2)*a^2*b+90*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan (d*x+c)^(5/2)*a*b^2+45*A*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^ (1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(5/2)*a*b^2+90*B*ar ctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(5/2)*a^2*b-90*B*arct an(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(5/2)*a*b^2+45*B*ln(-(2 ^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+ c)))*2^(1/2)*tan(d*x+c)^(5/2)*a^2*b+90*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2) )*2^(1/2)*tan(d*x+c)^(5/2)*a^2*b-90*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2 ^(1/2)*tan(d*x+c)^(5/2)*a*b^2+45*A*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x +c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(5/2)*a^2 *b+24*A*a^3+360*A*tan(d*x+c)^2*a*b^2+360*B*tan(d*x+c)^2*a^2*b+120*A*tan(d* x+c)*a^2*b-30*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(5/ 2)*a^3-120*A*tan(d*x+c)^2*a^3-15*A*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x +c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(5/2)*...
Leaf count of result is larger than twice the leaf count of optimal. 6162 vs. \(2 (340) = 680\).
Time = 6.46 (sec) , antiderivative size = 6162, normalized size of antiderivative = 16.22 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 0.41 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.87 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {30 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {24 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {120 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}} + \frac {40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{60 \, d} \]
-1/60*(30*sqrt(2)*((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 + (A + B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 30*sqrt(2)* ((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 + (A + B)*b^3)*arctan(-1/ 2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 15*sqrt(2)*((A + B)*a^3 + 3* (A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 15*sqrt(2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3* (A + B)*a*b^2 - (A - B)*b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 24*A*a^3/tan(d*x + c)^(5/2) - 120*(A*a^3 - 3*B*a^2*b - 3*A*a*b^ 2)/sqrt(tan(d*x + c)) + 40*(B*a^3 + 3*A*a^2*b)/tan(d*x + c)^(3/2))/d
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]