Integrand size = 33, antiderivative size = 534 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt {\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
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)*co t(d*x+c)^(1/2))/(a^2+b^2)^3/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))/(a^2+b^2)^3/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))/(a^2+b^2)^3/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))/(a^2+b^ 2)^3/d*2^(1/2)-1/4*(A*a^4*b+18*A*a^2*b^3-15*A*b^5+3*B*a^5+6*B*a^3*b^2+35*B *a*b^4)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))*a^(1/2)/b^(5/2)/(a^2+b^2) ^3/d+1/2*a*(A*b-B*a)*cot(d*x+c)^(1/2)/b/(a^2+b^2)/d/(b+a*cot(d*x+c))^2-1/4 *a*(A*a^2*b-7*A*b^3+3*B*a^3+11*B*a*b^2)*cot(d*x+c)^(1/2)/b^2/(a^2+b^2)^2/d /(b+a*cot(d*x+c))
Time = 6.53 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \left (\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{4 \left (a^2+b^2\right )^3}-\frac {3 \sqrt {a} (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 b^{5/2} \left (a^2+b^2\right )}+\frac {\sqrt {a} \left (a^2 A b+3 A b^3-2 a^3 B-4 a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 b^{5/2} \left (a^2+b^2\right )^2}+\frac {\sqrt {a} \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \left (\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{8 \left (a^2+b^2\right )^3}-\frac {a^2 (A b-a B) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {3 a (A b-a B) \sqrt {\tan (c+d x)}}{8 b^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {a \left (a^2 A b+3 A b^3-2 a^3 B-4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{2 b^2 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{d} \]
(2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(4*(a^2 + b^2)^3 ) - (3*Sqrt[a]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/( 8*b^(5/2)*(a^2 + b^2)) + (Sqrt[a]*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B )*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*b^(5/2)*(a^2 + b^2)^2) + (Sqrt[a]*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B)*ArcTan[ (Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^3) + ((3*a^2*b *(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(Sqrt[2]*Log[1 - S qrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[T an[c + d*x]] + Tan[c + d*x]]))/(8*(a^2 + b^2)^3) - (a^2*(A*b - a*B)*Sqrt[T an[c + d*x]])/(4*b^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (3*a*(A*b - a*B )*Sqrt[Tan[c + d*x]])/(8*b^2*(a^2 + b^2)*(a + b*Tan[c + d*x])) + (a*(a^2*A *b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Sqrt[Tan[c + d*x]])/(2*b^2*(a^2 + b^2) ^2*(a + b*Tan[c + d*x]))))/d
Time = 2.40 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.87, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.788, Rules used = {3042, 4064, 3042, 4092, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
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.
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\cot (c+d x)^{5/2} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {A \cot (c+d x)+B}{\sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B-A \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\int -\frac {3 B a^2-3 (A b-a B) \cot ^2(c+d x) a+A b a+4 b^2 B+4 b (A b-a B) \cot (c+d x)}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 B a^2-3 (A b-a B) \cot ^2(c+d x) a+A b a+4 b^2 B+4 b (A b-a B) \cot (c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 B a^2-3 (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )^2 a+A b a+4 b^2 B-4 b (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 B a^4+A b a^3+3 b^2 B a^2+9 A b^3 a+\left (3 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \cot ^2(c+d x) a+8 b^4 B-8 b^2 \left (A a^2+2 b B a-A b^2\right ) \cot (c+d x)}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 B a^4+A b a^3+3 b^2 B a^2+9 A b^3 a+\left (3 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \cot ^2(c+d x) a+8 b^4 B-8 b^2 \left (A a^2+2 b B a-A b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 B a^4+A b a^3+3 b^2 B a^2+9 A b^3 a+\left (3 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 a+8 b^4 B+8 b^2 \left (A a^2+2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {8 \left (\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) b^2+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x) b^2\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) b^2+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x) b^2}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {b^2 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-b^2 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {b^2 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 \int \frac {b^2 \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \int \frac {A a^3+3 b B a^2-3 A b^2 a-b^3 B+\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 a \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}+\frac {\frac {\frac {16 b^2 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 \sqrt {a} \left (3 a^5 B+a^4 A b+6 a^3 b^2 B+18 a^2 A b^3+35 a b^4 B-15 A b^5\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {a \left (3 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 b \left (a^2+b^2\right )}\) |
(a*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(b + a*Cot[c + d*x]) ^2) + (-((a*(a^2*A*b - 7*A*b^3 + 3*a^3*B + 11*a*b^2*B)*Sqrt[Cot[c + d*x]]) /(b*(a^2 + b^2)*d*(b + a*Cot[c + d*x]))) + ((2*Sqrt[a]*(a^4*A*b + 18*a^2*A *b^3 - 15*A*b^5 + 3*a^5*B + 6*a^3*b^2*B + 35*a*b^4*B)*ArcTan[(Sqrt[a]*Cot[ c + d*x])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)*d) + (16*b^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[C ot[c + d*x]]]/Sqrt[2]) + ArcTan[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]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/ (2*b*(a^2 + b^2)))/(4*b*(a^2 + b^2))
3.7.4.3.1 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_))^(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, 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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*(A*b - a*B)*(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[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
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]
Time = 0.50 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {a \left (A \,a^{4} b -6 A \,a^{2} b^{3}-7 A \,b^{5}+3 B \,a^{5}+14 B \,a^{3} b^{2}+11 B a \,b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}}{8 b^{2}}-\frac {\left (A \,a^{4} b +10 A \,a^{2} b^{3}+9 A \,b^{5}-5 B \,a^{5}-18 B \,a^{3} b^{2}-13 B a \,b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 b}}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (A \,a^{4} b +18 A \,a^{2} b^{3}-15 A \,b^{5}+3 B \,a^{5}+6 B \,a^{3} b^{2}+35 B a \,b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(451\) |
| default | \(\frac {-\frac {2 a \left (\frac {\frac {a \left (A \,a^{4} b -6 A \,a^{2} b^{3}-7 A \,b^{5}+3 B \,a^{5}+14 B \,a^{3} b^{2}+11 B a \,b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}}{8 b^{2}}-\frac {\left (A \,a^{4} b +10 A \,a^{2} b^{3}+9 A \,b^{5}-5 B \,a^{5}-18 B \,a^{3} b^{2}-13 B a \,b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 b}}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (A \,a^{4} b +18 A \,a^{2} b^{3}-15 A \,b^{5}+3 B \,a^{5}+6 B \,a^{3} b^{2}+35 B a \,b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(451\) |
1/d*(-2*a/(a^2+b^2)^3*((1/8*a*(A*a^4*b-6*A*a^2*b^3-7*A*b^5+3*B*a^5+14*B*a^ 3*b^2+11*B*a*b^4)/b^2*cot(d*x+c)^(3/2)-1/8*(A*a^4*b+10*A*a^2*b^3+9*A*b^5-5 *B*a^5-18*B*a^3*b^2-13*B*a*b^4)/b*cot(d*x+c)^(1/2))/(b+a*cot(d*x+c))^2+1/8 *(A*a^4*b+18*A*a^2*b^3-15*A*b^5+3*B*a^5+6*B*a^3*b^2+35*B*a*b^4)/b^2/(a*b)^ (1/2)*arctan(a*cot(d*x+c)^(1/2)/(a*b)^(1/2)))-2/(a^2+b^2)^3*(1/8*(-A*a^3+3 *A*a*b^2-3*B*a^2*b+B*b^3)*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/ 2))/(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c) ^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/8*(-3*A*a^2*b+A*b^3+B*a^3 -3*B*a*b^2)*2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x +c)+2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arct an(-1+2^(1/2)*cot(d*x+c)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 8923 vs. \(2 (482) = 964\).
Time = 106.83 (sec) , antiderivative size = 17872, normalized size of antiderivative = 33.47 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.04 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, B a^{6} + A a^{5} b + 6 \, B a^{4} b^{2} + 18 \, A a^{3} b^{3} + 35 \, B a^{2} b^{4} - 15 \, A a b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a b}} - \frac {2 \, \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 ) + 2 \, \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 ) + \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 ) - \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 )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {\frac {5 \, B a^{4} b - A a^{3} b^{2} + 13 \, B a^{2} b^{3} - 9 \, A a b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {3 \, B a^{5} + A a^{4} b + 11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8} + \frac {2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )}}{\tan \left (d x + c\right )} + \frac {a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \]
-1/4*((3*B*a^6 + A*a^5*b + 6*B*a^4*b^2 + 18*A*a^3*b^3 + 35*B*a^2*b^4 - 15* A*a*b^5)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*sqrt(a*b)) - (2*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)))) + 2*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)))) + 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) - 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))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + ((5*B*a^4* b - A*a^3*b^2 + 13*B*a^2*b^3 - 9*A*a*b^4)/sqrt(tan(d*x + c)) + (3*B*a^5 + A*a^4*b + 11*B*a^3*b^2 - 7*A*a^2*b^3)/tan(d*x + c)^(3/2))/(a^4*b^4 + 2*a^2 *b^6 + b^8 + 2*(a^5*b^3 + 2*a^3*b^5 + a*b^7)/tan(d*x + c) + (a^6*b^2 + 2*a ^4*b^4 + a^2*b^6)/tan(d*x + c)^2))/d
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]