Integrand size = 33, antiderivative size = 278 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {(a-i b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \] Output:
1/8*(24*A*a^2*b+A*b^3+16*B*a^3-6*B*a*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a ^(1/2))/a^(3/2)/d-(a-I*b)^(3/2)*(I*A+B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a- I*b)^(1/2))/d+(a+I*b)^(3/2)*(I*A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b) ^(1/2))/d+1/8*(8*A*a^2-A*b^2-10*B*a*b)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a /d-1/12*(7*A*b+6*B*a)*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d-1/3*a*A*cot(d* x+c)^3*(a+b*tan(d*x+c))^(1/2)/d
Time = 3.85 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.87 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {3 \left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {a} \left (-24 i a (a-i b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+24 i a (a+i b)^{3/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-\cot (c+d x) \left (-24 a^2 A+3 A b^2+30 a b B+2 a (7 A b+6 a B) \cot (c+d x)+8 a^2 A \cot ^2(c+d x)\right ) \sqrt {a+b \tan (c+d x)}\right )}{24 a^{3/2} d} \] Input:
Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x ]
Output:
(3*(24*a^2*A*b + A*b^3 + 16*a^3*B - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + Sqrt[a]*((-24*I)*a*(a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt [a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (24*I)*a*(a + I*b)^(3/2)*(A + I*B)*A rcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - Cot[c + d*x]*(-24*a^2*A + 3*A*b^2 + 30*a*b*B + 2*a*(7*A*b + 6*a*B)*Cot[c + d*x] + 8*a^2*A*Cot[c + d *x]^2)*Sqrt[a + b*Tan[c + d*x]]))/(24*a^(3/2)*d)
Time = 2.37 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.01, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4088, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
| \(\displaystyle \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {1}{3} \int \frac {\cot ^3(c+d x) \left (-b (5 a A-6 b B) \tan ^2(c+d x)-6 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (7 A b+6 a B)\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\cot ^3(c+d x) \left (-b (5 a A-6 b B) \tan ^2(c+d x)-6 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (7 A b+6 a B)\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {-b (5 a A-6 b B) \tan (c+d x)^2-6 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (7 A b+6 a B)}{\tan (c+d x)^3 \sqrt {a+b \tan (c+d x)}}dx-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {3 \cot ^2(c+d x) \left (a b (7 A b+6 a B) \tan ^2(c+d x)+8 a \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (8 A a^2-10 b B a-A b^2\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \int \frac {\cot ^2(c+d x) \left (a b (7 A b+6 a B) \tan ^2(c+d x)+8 a \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (8 A a^2-10 b B a-A b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \int \frac {a b (7 A b+6 a B) \tan (c+d x)^2+8 a \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (8 A a^2-10 b B a-A b^2\right )}{\tan (c+d x)^2 \sqrt {a+b \tan (c+d x)}}dx}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \left (-\frac {\int -\frac {\cot (c+d x) \left (-16 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x) a^2-b \left (8 A a^2-10 b B a-A b^2\right ) \tan ^2(c+d x) a+\left (16 B a^3+24 A b a^2-6 b^2 B a+A b^3\right ) a\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {\int \frac {\cot (c+d x) \left (-16 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x) a^2-b \left (8 A a^2-10 b B a-A b^2\right ) \tan ^2(c+d x) a+\left (16 B a^3+24 A b a^2-6 b^2 B a+A b^3\right ) a\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {\int \frac {-16 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x) a^2-b \left (8 A a^2-10 b B a-A b^2\right ) \tan (c+d x)^2 a+\left (16 B a^3+24 A b a^2-6 b^2 B a+A b^3\right ) a}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {\int -\frac {16 \left (\left (A a^2-2 b B a-A b^2\right ) a^2+\left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x) a^2\right )}{\sqrt {a+b \tan (c+d x)}}dx+a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx-16 \int \frac {\left (A a^2-2 b B a-A b^2\right ) a^2+\left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x) a^2}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \int \frac {\left (A a^2-2 b B a-A b^2\right ) a^2+\left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x) a^2}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {1}{2} a^2 (a-i b)^2 (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^2 (a+i b)^2 (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {1}{2} a^2 (a-i b)^2 (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^2 (a+i b)^2 (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {i a^2 (a-i b)^2 (A-i B) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^2 (a+i b)^2 (A+i B) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {i a^2 (a+i b)^2 (A+i B) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i a^2 (a-i b)^2 (A-i B) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {a^2 (a-i b)^2 (A-i B) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {a^2 (a+i b)^2 (A+i B) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {a^2 (a-i b)^{3/2} (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{3/2} (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {\frac {a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}-16 \left (\frac {a^2 (a-i b)^{3/2} (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{3/2} (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {\frac {2 a \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}-16 \left (\frac {a^2 (a-i b)^{3/2} (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{3/2} (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {-\frac {2 \sqrt {a} \left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-16 \left (\frac {a^2 (a-i b)^{3/2} (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{3/2} (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\) |
Input:
Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]
Output:
-1/3*(a*A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/d + (-1/2*((7*A*b + 6*a *B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d - (3*((-16*((a^2*(a - I*b)^ (3/2)*(A - I*B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d + (a^2*(a + I*b)^(3/ 2)*(A + I*B)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d) - (2*Sqrt[a]*(24*a^2*A *b + A*b^3 + 16*a^3*B - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a ]])/d)/(2*a) - ((8*a^2*A - A*b^2 - 10*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d))/(4*a))/6
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x ])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d* (b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[ e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] & & LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1984\) vs. \(2(240)=480\).
Time = 0.15 (sec) , antiderivative size = 1985, normalized size of antiderivative = 7.14
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1985\) |
| default | \(\text {Expression too large to display}\) | \(1985\) |
Input:
int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x,method=_RETURNV ERBOSE)
Output:
1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+ c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-1/ 4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/ 2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-1/3/ d/tan(d*x+c)^3*A*(a+b*tan(d*x+c))^(3/2)+2/d*a^(3/2)*arctanh((a+b*tan(d*x+c ))^(1/2)/a^(1/2))*B-1/2/d*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2 +b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+ 1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2 +b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2-1/4/d*ln((a+b *tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^ (1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)+1/2/d*ln((a+b*tan(d *x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2)) *B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta n(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1 /2)-2*a)^(1/2))*B*a^2+1/d*b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^ 2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^ (1/2))*B+1/4/d*b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b *tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/4/d*b*ln( b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b ^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/4/d*ln(b*tan(d*x+c)+a+(a+b...
Leaf count of result is larger than twice the leaf count of optimal. 3215 vs. \(2 (234) = 468\).
Time = 14.72 (sec) , antiderivative size = 6449, normalized size of antiderivative = 23.20 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorith m="fricas")
Output:
Too large to include
\[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)
Output:
Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**(3/2)*cot(c + d*x)**4, x)
Timed out. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorith m="maxima")
Output:
Timed out
Exception generated. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorith m="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 6.43 (sec) , antiderivative size = 25789, normalized size of antiderivative = 92.77 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(3/2),x)
Output:
atan(((((((256*A*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1 536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(( (8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^ 2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1 /2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2* a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/(4*a^2*d^4))*(-(((8*A^2*a^3*d ^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2* A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^ 4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^ 3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^ 3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b ^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/ (4*a^2*d^4))*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2* a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + ...
\[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int \cot \left (d x +c \right )^{4} \left (a +\tan \left (d x +c \right ) b \right )^{\frac {3}{2}} \left (A +B \tan \left (d x +c \right )\right )d x \] Input:
int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x)
Output:
int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x)