Integrand size = 33, antiderivative size = 438 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^{3/2} \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2 A+3 A b^2-a b B\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
b^(3/2)*(7*A*a^2*b+3*A*b^3-5*B*a^3-B*a*b^2)*arctan(a^(1/2)*cot(d*x+c)^(1/2 )/b^(1/2))/a^(5/2)/(a^2+b^2)^2/d+b*(A*b-B*a)*cot(d*x+c)^(3/2)/a/(a^2+b^2)/ d/(b+a*cot(d*x+c))+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(-1+2^(1/2) *cot(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A +B))*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(2*a*b*( A-B)-a^2*(A+B)+b^2*(A+B))*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b ^2)^2/d*2^(1/2)-1/4*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*ln(1+cot(d*x+c)+2^(1 /2)*cot(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)-(2*A*a^2+3*A*b^2-B*a*b)*cot(d* x+c)^(1/2)/a^2/(a^2+b^2)/d
Time = 5.88 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (\frac {2 \sqrt {2} \left (a^2 (A-B)+b^2 (-A+B)+2 a b (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{\left (a^2+b^2\right )^2}+\frac {4 b^{3/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )}-\frac {8 b^{3/2} \left (3 a^2 A b+A b^3-2 a^3 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2}-\frac {\sqrt {2} \left (2 a b (-A+B)+a^2 (A+B)-b^2 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{\left (a^2+b^2\right )^2}-\frac {8 A}{a^2 \sqrt {\tan (c+d x)}}+\frac {4 b^2 (-A b+a B) \sqrt {\tan (c+d x)}}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{4 d} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((2*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) + 2*a*b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^2 + (4*b^(3/2)*(-(A*b) + a*B)*A rcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)) - (8*b^ (3/2)*(3*a^2*A*b + A*b^3 - 2*a^3*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sq rt[a]])/(a^(5/2)*(a^2 + b^2)^2) - (Sqrt[2]*(2*a*b*(-A + B) + a^2*(A + B) - 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]]))/(a^2 + b^2)^2 - (8*A)/(a^2* Sqrt[Tan[c + d*x]]) + (4*b^2*(-(A*b) + a*B)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x]))))/(4*d)
Time = 2.11 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.86, 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, 4088, 27, 3042, 4130, 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.
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{3/2} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A \cot (c+d x)+B)}{(a \cot (c+d x)+b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\int -\frac {\sqrt {\cot (c+d x)} \left (\left (2 A a^2-b B a+3 A b^2\right ) \cot ^2(c+d x)-2 a (A b-a B) \cot (c+d x)+3 b (A b-a B)\right )}{2 (b+a \cot (c+d x))}dx}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {\cot (c+d x)} \left (\left (2 A a^2-b B a+3 A b^2\right ) \cot ^2(c+d x)-2 a (A b-a B) \cot (c+d x)+3 b (A b-a B)\right )}{b+a \cot (c+d x)}dx}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (\left (2 A a^2-b B a+3 A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+2 a (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )+3 b (A b-a B)\right )}{b-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {-\frac {2 \int \frac {2 (a A+b B) \cot (c+d x) a^2+\left (-2 B a^3+4 A b a^2-b^2 B a+3 A b^3\right ) \cot ^2(c+d x)+b \left (2 A a^2-b B a+3 A b^2\right )}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {2 (a A+b B) \cot (c+d x) a^2+\left (-2 B a^3+4 A b a^2-b^2 B a+3 A b^3\right ) \cot ^2(c+d x)+b \left (2 A a^2-b B a+3 A b^2\right )}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-2 (a A+b B) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (-2 B a^3+4 A b a^2-b^2 B a+3 A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 A a^2-b B a+3 A b^2\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {2 \left (\left (A a^2+2 b B a-A b^2\right ) a^2+\left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x) a^2\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {2 \int \frac {\left (A a^2+2 b B a-A b^2\right ) a^2+\left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x) a^2}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {2 \int \frac {a^2 \left (A a^2+2 b B a-A b^2\right )-a^2 \left (-B a^2+2 A b a+b^2 B\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {\frac {4 \int -\frac {a^2 \left (A a^2+2 b B a-A b^2+\left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \int \frac {a^2 \left (A a^2+2 b B a-A b^2+\left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \int \frac {A a^2+2 b B a-A b^2+\left (-B a^2+2 A b a+b^2 B\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {-\frac {2 b^2 \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b (A b-a B) \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {-\frac {2 \left (2 a^2 A-a b B+3 A b^2\right ) \sqrt {\cot (c+d x)}}{a d}-\frac {\frac {2 b^{3/2} \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {4 a^2 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}}{2 a \left (a^2+b^2\right )}\) |
(b*(A*b - a*B)*Cot[c + d*x]^(3/2))/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((-2*(2*a^2*A + 3*A*b^2 - a*b*B)*Sqrt[Cot[c + d*x]])/(a*d) - ((2*b^(3/2) *(7*a^2*A*b + 3*A*b^3 - 5*a^3*B - a*b^2*B)*ArcTan[(Sqrt[a]*Cot[c + d*x])/S qrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - (4*a^2*(((a^2*(A - B) - b^2*(A - B) + 2 *a*b*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[ 1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 - ((2*a*b*(A - B) - a^2*(A + B ) + 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))/a)/(2*a*(a^2 + b^2))
3.6.95.3.1 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*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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*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[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !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. \(2216\) vs. \(2(396)=792\).
Time = 0.39 (sec) , antiderivative size = 2217, normalized size of antiderivative = 5.06
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2217\) |
| default | \(\text {Expression too large to display}\) | \(2217\) |
-1/4/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(12*A*(a*b)^(1/2)*tan(d*x+c)*b^5+12 *A*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*b^6+16*A*(a*b)^ (1/2)*a^3*b^2+8*A*(a*b)^(1/2)*a*b^4+8*A*(a*b)^(1/2)*a^5+12*A*arctan(b*tan( d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(1/2)*a*b^5-4*B*(a*b)^(1/2)*tan(d*x+c )*a^3*b^2-4*B*(a*b)^(1/2)*tan(d*x+c)*a*b^4-20*B*arctan(b*tan(d*x+c)^(1/2)/ (a*b)^(1/2))*tan(d*x+c)^(1/2)*a^4*b^2-4*B*arctan(b*tan(d*x+c)^(1/2)/(a*b)^ (1/2))*tan(d*x+c)^(1/2)*a^2*b^4+8*A*(a*b)^(1/2)*a^4*b*tan(d*x+c)+28*A*arct an(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*a^2*b^4-20*B*arctan(b* tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*a^3*b^3-4*B*arctan(b*tan(d* x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(3/2)*a*b^5+20*A*(a*b)^(1/2)*tan(d*x+c) *a^2*b^3+28*A*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*tan(d*x+c)^(1/2)*a^3* b^3+4*A*(a*b)^(1/2)*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^ (3/2)*a^3*b^2-2*A*(a*b)^(1/2)*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*t an(d*x+c)^(3/2)*a^2*b^3+2*A*(a*b)^(1/2)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+ c)^(1/2))*tan(d*x+c)^(3/2)*a^4*b+4*A*(a*b)^(1/2)*2^(1/2)*arctan(-1+2^(1/2) *tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*a^3*b^2+2*A*(a*b)^(1/2)*2^(1/2)*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))*tan(d*x+c)^(1/2)*a^4*b+4*A*(a*b)^(1/2)*2^(1/2)*arctan(1+2^(1/2)*tan(d *x+c)^(1/2))*tan(d*x+c)^(1/2)*a^4*b-2*A*(a*b)^(1/2)*2^(1/2)*arctan(1+2^(1/ 2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(1/2)*a^3*b^2+4*A*(a*b)^(1/2)*2^(1/2)*a...
Leaf count of result is larger than twice the leaf count of optimal. 5937 vs. \(2 (397) = 794\).
Time = 37.59 (sec) , antiderivative size = 11904, normalized size of antiderivative = 27.18 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (5 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3} + B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, {\left (B a b^{2} - A b^{3}\right )}}{{\left (a^{4} b + a^{2} b^{3} + \frac {a^{5} + a^{3} b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}} + \frac {8 \, A}{a^{2} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]
-1/4*(4*(5*B*a^3*b^2 - 7*A*a^2*b^3 + B*a*b^4 - 3*A*b^5)*arctan(a/(sqrt(a*b )*sqrt(tan(d*x + c))))/((a^6 + 2*a^4*b^2 + a^2*b^4)*sqrt(a*b)) - (2*sqrt(2 )*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B )*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)* log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4) - 4*(B*a*b^2 - A*b^3)/((a^4*b + a^2*b^3 + (a^5 + a^3*b^2)/tan(d*x + c ))*sqrt(tan(d*x + c))) + 8*A/(a^2*sqrt(tan(d*x + c))))/d
\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]