Integrand size = 40, antiderivative size = 331 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:
(B*a^3-3*B*a*b^2+3*C*a^2*b-C*b^3)*x/(a^2+b^2)^3+(3*B*a^2*b-B*b^3-C*a^3+3*C *a*b^2)*ln(cos(d*x+c))/(a^2+b^2)^3/d+a^2*(B*a^4*b+3*B*a^2*b^3+6*B*b^5-3*C* a^5-9*C*a^3*b^2-10*C*a*b^4)*ln(a+b*tan(d*x+c))/b^4/(a^2+b^2)^3/d-(B*a^3*b+ 3*B*a*b^3-3*C*a^4-6*C*a^2*b^2-C*b^4)*tan(d*x+c)/b^3/(a^2+b^2)^2/d+1/2*a*(B *b-C*a)*tan(d*x+c)^3/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+1/2*a*(B*a^2*b+5*B*b ^3-3*C*a^3-7*C*a*b^2)*tan(d*x+c)^2/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 3.21 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.83 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {(B+i C) \log (i-\tan (c+d x))}{(-i a+b)^3}+\frac {(B-i C) \log (i+\tan (c+d x))}{(i a+b)^3}+\frac {2 a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3}+\frac {a^3 \left (-a b B+3 a^2 C+2 b^2 C\right )}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 C \tan ^3(c+d x)}{b (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (-2 a^3 b B-4 a b^3 B+6 a^4 C+11 a^2 b^2 C+3 b^4 C\right )}{b^4 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 d} \] Input:
Integrate[(Tan[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[ c + d*x])^3,x]
Output:
(((B + I*C)*Log[I - Tan[c + d*x]])/((-I)*a + b)^3 + ((B - I*C)*Log[I + Tan [c + d*x]])/(I*a + b)^3 + (2*a^2*(a^4*b*B + 3*a^2*b^3*B + 6*b^5*B - 3*a^5* C - 9*a^3*b^2*C - 10*a*b^4*C)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^3) + (a^3*(-(a*b*B) + 3*a^2*C + 2*b^2*C))/(b^4*(a^2 + b^2)*(a + b*Tan[c + d* x])^2) + (2*C*Tan[c + d*x]^3)/(b*(a + b*Tan[c + d*x])^2) - (2*a^2*(-2*a^3* b*B - 4*a*b^3*B + 6*a^4*C + 11*a^2*b^2*C + 3*b^4*C))/(b^4*(a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*d)
Time = 3.41 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.425, Rules used = {3042, 4115, 3042, 4088, 25, 3042, 4128, 27, 3042, 4130, 25, 3042, 4109, 3042, 3956, 4100, 16}
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.
| \(\displaystyle \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^3 \left (B \tan (c+d x)+C \tan (c+d x)^2\right )}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4115 |
| \(\displaystyle \int \frac {\tan ^4(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^4 (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {\int -\frac {\tan ^2(c+d x) \left (\left (-3 C a^2+b B a-2 b^2 C\right ) \tan ^2(c+d x)-2 b (b B-a C) \tan (c+d x)+3 a (b B-a C)\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}+\frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan ^2(c+d x) \left (\left (-3 C a^2+b B a-2 b^2 C\right ) \tan ^2(c+d x)-2 b (b B-a C) \tan (c+d x)+3 a (b B-a C)\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\int \frac {\tan (c+d x)^2 \left (\left (-3 C a^2+b B a-2 b^2 C\right ) \tan (c+d x)^2-2 b (b B-a C) \tan (c+d x)+3 a (b B-a C)\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {2 \tan (c+d x) \left (\left (B a^2+2 b C a-b^2 B\right ) \tan (c+d x) b^2+\left (-3 C a^4+b B a^3-6 b^2 C a^2+3 b^3 B a-b^4 C\right ) \tan ^2(c+d x)+a \left (-3 C a^3+b B a^2-7 b^2 C a+5 b^3 B\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \int \frac {\tan (c+d x) \left (\left (B a^2+2 b C a-b^2 B\right ) \tan (c+d x) b^2+\left (-3 C a^4+b B a^3-6 b^2 C a^2+3 b^3 B a-b^4 C\right ) \tan ^2(c+d x)+a \left (-3 C a^3+b B a^2-7 b^2 C a+5 b^3 B\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \int \frac {\tan (c+d x) \left (\left (B a^2+2 b C a-b^2 B\right ) \tan (c+d x) b^2+\left (-3 C a^4+b B a^3-6 b^2 C a^2+3 b^3 B a-b^4 C\right ) \tan (c+d x)^2+a \left (-3 C a^3+b B a^2-7 b^2 C a+5 b^3 B\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\int -\frac {-\left (\left (-C a^2+2 b B a+b^2 C\right ) \tan (c+d x) b^3\right )+\left (a^2+b^2\right )^2 (b B-3 a C) \tan ^2(c+d x)+a \left (-3 C a^4+b B a^3-6 b^2 C a^2+3 b^3 B a-b^4 C\right )}{a+b \tan (c+d x)}dx}{b}+\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {\int \frac {-\left (\left (-C a^2+2 b B a+b^2 C\right ) \tan (c+d x) b^3\right )+\left (a^2+b^2\right )^2 (b B-3 a C) \tan ^2(c+d x)+a \left (-3 C a^4+b B a^3-6 b^2 C a^2+3 b^3 B a-b^4 C\right )}{a+b \tan (c+d x)}dx}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {\int \frac {-\left (\left (-C a^2+2 b B a+b^2 C\right ) \tan (c+d x) b^3\right )+\left (a^2+b^2\right )^2 (b B-3 a C) \tan (c+d x)^2+a \left (-3 C a^4+b B a^3-6 b^2 C a^2+3 b^3 B a-b^4 C\right )}{a+b \tan (c+d x)}dx}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {-\frac {b^3 \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^3 x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{a^2+b^2}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {-\frac {b^3 \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^3 x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{a^2+b^2}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^3 \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^3 x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{a^2+b^2}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}+\frac {b^3 \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^3 x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{a^2+b^2}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {2 \left (\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b d}-\frac {\frac {b^3 \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^3 x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{a^2+b^2}+\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}\right )}{b \left (a^2+b^2\right )}-\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}\) |
Input:
Int[(Tan[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d* x])^3,x]
Output:
(a*(b*B - a*C)*Tan[c + d*x]^3)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (-((a*(a^2*b*B + 5*b^3*B - 3*a^3*C - 7*a*b^2*C)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) + (2*(-(((b^3*(a^3*B - 3*a*b^2*B + 3*a^2*b *C - b^3*C)*x)/(a^2 + b^2) + (b^3*(3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)* Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (a^2*(a^4*b*B + 3*a^2*b^3*B + 6*b^5*B - 3*a^5*C - 9*a^3*b^2*C - 10*a*b^4*C)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d))/b) + ((a^3*b*B + 3*a*b^3*B - 3*a^4*C - 6*a^2*b^2*C - b^4*C)*Tan[c + d*x])/(b*d)))/(b*(a^2 + b^2)))/(2*b*(a^2 + b^2))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f) Subst[Int[(a + x)^m, x], x, b* Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 )/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*A + b*B - a *C)*(x/(a^2 + b^2)), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[( 1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(A*b - a*B - b*C)/( a^2 + b^2) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] & & NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C , 0]
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[1/b^2 Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
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*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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])))
Time = 0.26 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (d x +c \right ) C}{b^{3}}+\frac {a^{2} \left (B \,a^{4} b +3 B \,a^{2} b^{3}+6 B \,b^{5}-3 C \,a^{5}-9 C \,a^{3} b^{2}-10 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (B b -C a \right )}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (2 B \,a^{2} b +4 B \,b^{3}-3 C \,a^{3}-5 C a \,b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(263\) |
| default | \(\frac {\frac {\tan \left (d x +c \right ) C}{b^{3}}+\frac {a^{2} \left (B \,a^{4} b +3 B \,a^{2} b^{3}+6 B \,b^{5}-3 C \,a^{5}-9 C \,a^{3} b^{2}-10 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} \left (B b -C a \right )}{2 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (2 B \,a^{2} b +4 B \,b^{3}-3 C \,a^{3}-5 C a \,b^{2}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(263\) |
| norman | \(\frac {\frac {C \tan \left (d x +c \right )^{3}}{d b}+\frac {\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) a^{2} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {a \left (2 B \,a^{4} b +4 B \,a^{2} b^{3}-6 C \,a^{5}-11 C \,a^{3} b^{2}-3 C a \,b^{4}\right ) \tan \left (d x +c \right )}{d \,b^{3} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {a^{2} \left (3 B \,a^{4} b +7 B \,a^{2} b^{3}-9 C \,a^{5}-17 C \,a^{3} b^{2}-4 C a \,b^{4}\right )}{2 d \,b^{4} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {2 b \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} \left (B \,a^{4} b +3 B \,a^{2} b^{3}+6 B \,b^{5}-3 C \,a^{5}-9 C \,a^{3} b^{2}-10 C a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) b^{4} d}-\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}\) | \(512\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1180\) |
| risch | \(\text {Expression too large to display}\) | \(1551\) |
Input:
int(tan(d*x+c)^3*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x,method =_RETURNVERBOSE)
Output:
1/d*(tan(d*x+c)*C/b^3+1/b^4*a^2*(B*a^4*b+3*B*a^2*b^3+6*B*b^5-3*C*a^5-9*C*a ^3*b^2-10*C*a*b^4)/(a^2+b^2)^3*ln(a+b*tan(d*x+c))-1/2/b^4*a^4*(B*b-C*a)/(a ^2+b^2)/(a+b*tan(d*x+c))^2+1/b^4*a^3*(2*B*a^2*b+4*B*b^3-3*C*a^3-5*C*a*b^2) /(a^2+b^2)^2/(a+b*tan(d*x+c))+1/(a^2+b^2)^3*(1/2*(-3*B*a^2*b+B*b^3+C*a^3-3 *C*a*b^2)*ln(1+tan(d*x+c)^2)+(B*a^3-3*B*a*b^2+3*C*a^2*b-C*b^3)*arctan(tan( d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (328) = 656\).
Time = 0.18 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.69 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(tan(d*x+c)^3*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-1/2*(3*C*a^7*b^2 - B*a^6*b^3 + 9*C*a^5*b^4 - 7*B*a^4*b^5 - 2*(C*a^6*b^3 + 3*C*a^4*b^5 + 3*C*a^2*b^7 + C*b^9)*tan(d*x + c)^3 - 2*(B*a^5*b^4 + 3*C*a^ 4*b^5 - 3*B*a^3*b^6 - C*a^2*b^7)*d*x - (9*C*a^7*b^2 - 3*B*a^6*b^3 + 23*C*a ^5*b^4 - 9*B*a^4*b^5 + 12*C*a^3*b^6 + 4*C*a*b^8 + 2*(B*a^3*b^6 + 3*C*a^2*b ^7 - 3*B*a*b^8 - C*b^9)*d*x)*tan(d*x + c)^2 + (3*C*a^9 - B*a^8*b + 9*C*a^7 *b^2 - 3*B*a^6*b^3 + 10*C*a^5*b^4 - 6*B*a^4*b^5 + (3*C*a^7*b^2 - B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 + 10*C*a^3*b^6 - 6*B*a^2*b^7)*tan(d*x + c)^2 + 2*(3*C*a^8*b - B*a^7*b^2 + 9*C*a^6*b^3 - 3*B*a^5*b^4 + 10*C*a^4*b^5 - 6* B*a^3*b^6)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^ 2)/(tan(d*x + c)^2 + 1)) - (3*C*a^9 - B*a^8*b + 9*C*a^7*b^2 - 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 + 3*C*a^3*b^6 - B*a^2*b^7 + (3*C*a^7*b^2 - B*a ^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 + 3*C*a*b^8 - B*b^9)*tan(d*x + c)^2 + 2*(3*C*a^8*b - B*a^7*b^2 + 9*C*a^6*b^3 - 3*B*a^ 5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 + 3*C*a^2*b^7 - B*a*b^8)*tan(d*x + c))*l og(1/(tan(d*x + c)^2 + 1)) - 2*(3*C*a^8*b - B*a^7*b^2 + 6*C*a^6*b^3 - 3*B* a^5*b^4 - 2*C*a^4*b^5 + 4*B*a^3*b^6 + C*a^2*b^7 + 2*(B*a^4*b^5 + 3*C*a^3*b ^6 - 3*B*a^2*b^7 - C*a*b^8)*d*x)*tan(d*x + c))/((a^6*b^6 + 3*a^4*b^8 + 3*a ^2*b^10 + b^12)*d*tan(d*x + c)^2 + 2*(a^7*b^5 + 3*a^5*b^7 + 3*a^3*b^9 + a* b^11)*d*tan(d*x + c) + (a^8*b^4 + 3*a^6*b^6 + 3*a^4*b^8 + a^2*b^10)*d)
Exception generated. \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(tan(d*x+c)**3*(B*tan(d*x+c)+C*tan(d*x+c)**2)/(a+b*tan(d*x+c))**3 ,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Time = 0.13 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, C a^{7} - 3 \, B a^{6} b + 9 \, C a^{5} b^{2} - 7 \, B a^{4} b^{3} + 2 \, {\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + 5 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, C \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \] Input:
integrate(tan(d*x+c)^3*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*(2*(B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(3*C*a^7 - B*a^6*b + 9*C*a^5*b^2 - 3*B*a^4*b^3 + 10 *C*a^3*b^4 - 6*B*a^2*b^5)*log(b*tan(d*x + c) + a)/(a^6*b^4 + 3*a^4*b^6 + 3 *a^2*b^8 + b^10) + (C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*log(tan(d*x + c )^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (5*C*a^7 - 3*B*a^6*b + 9*C* a^5*b^2 - 7*B*a^4*b^3 + 2*(3*C*a^6*b - 2*B*a^5*b^2 + 5*C*a^4*b^3 - 4*B*a^3 *b^4)*tan(d*x + c))/(a^6*b^4 + 2*a^4*b^6 + a^2*b^8 + (a^4*b^6 + 2*a^2*b^8 + b^10)*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*tan(d*x + c)) + 2 *C*tan(d*x + c)/b^3)/d
Time = 0.71 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.16 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} - \frac {{\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} d + 3 \, a^{4} b^{6} d + 3 \, a^{2} b^{8} d + b^{10} d} + \frac {C \tan \left (d x + c\right )}{b^{3} d} - \frac {5 \, C a^{9} - 3 \, B a^{8} b + 14 \, C a^{7} b^{2} - 10 \, B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 7 \, B a^{4} b^{5} + 2 \, {\left (3 \, C a^{8} b - 2 \, B a^{7} b^{2} + 8 \, C a^{6} b^{3} - 6 \, B a^{5} b^{4} + 5 \, C a^{4} b^{5} - 4 \, B a^{3} b^{6}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{4} d} \] Input:
integrate(tan(d*x+c)^3*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
(B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 3 *a^2*b^4*d + b^6*d) + 1/2*(C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*log(tan( d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) - (3*C*a^7 - B *a^6*b + 9*C*a^5*b^2 - 3*B*a^4*b^3 + 10*C*a^3*b^4 - 6*B*a^2*b^5)*log(abs(b *tan(d*x + c) + a))/(a^6*b^4*d + 3*a^4*b^6*d + 3*a^2*b^8*d + b^10*d) + C*t an(d*x + c)/(b^3*d) - 1/2*(5*C*a^9 - 3*B*a^8*b + 14*C*a^7*b^2 - 10*B*a^6*b ^3 + 9*C*a^5*b^4 - 7*B*a^4*b^5 + 2*(3*C*a^8*b - 2*B*a^7*b^2 + 8*C*a^6*b^3 - 6*B*a^5*b^4 + 5*C*a^4*b^5 - 4*B*a^3*b^6)*tan(d*x + c))/((a^2 + b^2)^3*(b *tan(d*x + c) + a)^2*b^4*d)
Time = 6.98 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {C\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {5\,C\,a^7-3\,B\,a^6\,b+9\,C\,a^5\,b^2-7\,B\,a^4\,b^3}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,C\,a^6-2\,B\,a^5\,b+5\,C\,a^4\,b^2-4\,B\,a^3\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,C\,a^5+B\,a^4\,b-9\,C\,a^3\,b^2+3\,B\,a^2\,b^3-10\,C\,a\,b^4+6\,B\,b^5\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3} \] Input:
int((tan(c + d*x)^3*(B*tan(c + d*x) + C*tan(c + d*x)^2))/(a + b*tan(c + d* x))^3,x)
Output:
(log(tan(c + d*x) - 1i)*(B*1i - C))/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1 i)) - ((5*C*a^7 - 7*B*a^4*b^3 + 9*C*a^5*b^2 - 3*B*a^6*b)/(2*b*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(3*C*a^6 - 4*B*a^3*b^3 + 5*C*a^4*b^2 - 2*B*a^ 5*b))/(a^4 + b^4 + 2*a^2*b^2))/(d*(a^2*b^3 + b^5*tan(c + d*x)^2 + 2*a*b^4* tan(c + d*x))) + (log(tan(c + d*x) + 1i)*(B - C*1i))/(2*d*(a*b^2*3i - 3*a^ 2*b - a^3*1i + b^3)) + (C*tan(c + d*x))/(b^3*d) + (a^2*log(a + b*tan(c + d *x))*(6*B*b^5 - 3*C*a^5 + 3*B*a^2*b^3 - 9*C*a^3*b^2 + B*a^4*b - 10*C*a*b^4 ))/(b^4*d*(a^2 + b^2)^3)
Time = 0.16 (sec) , antiderivative size = 1268, normalized size of antiderivative = 3.83 \[ \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(tan(d*x+c)^3*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x)
Output:
(log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**3*b**6*c - 3*log(tan(c + d*x) **2 + 1)*tan(c + d*x)**2*a**2*b**8 - 3*log(tan(c + d*x)**2 + 1)*tan(c + d* x)**2*a*b**8*c + log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*b**10 + 2*log(ta n(c + d*x)**2 + 1)*tan(c + d*x)*a**4*b**5*c - 6*log(tan(c + d*x)**2 + 1)*t an(c + d*x)*a**3*b**7 - 6*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**2*b**7* c + 2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a*b**9 + log(tan(c + d*x)**2 + 1)*a**5*b**4*c - 3*log(tan(c + d*x)**2 + 1)*a**4*b**6 - 3*log(tan(c + d*x )**2 + 1)*a**3*b**6*c + log(tan(c + d*x)**2 + 1)*a**2*b**8 - 6*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**7*b**2*c + 2*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**6*b**4 - 18*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**5*b** 4*c + 6*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**4*b**6 - 20*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**3*b**6*c + 12*log(tan(c + d*x)*b + a)*tan( c + d*x)**2*a**2*b**8 - 12*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**8*b*c + 4*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**7*b**3 - 36*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**6*b**3*c + 12*log(tan(c + d*x)*b + a)*tan(c + d*x)*a* *5*b**5 - 40*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**4*b**5*c + 24*log(tan (c + d*x)*b + a)*tan(c + d*x)*a**3*b**7 - 6*log(tan(c + d*x)*b + a)*a**9*c + 2*log(tan(c + d*x)*b + a)*a**8*b**2 - 18*log(tan(c + d*x)*b + a)*a**7*b **2*c + 6*log(tan(c + d*x)*b + a)*a**6*b**4 - 20*log(tan(c + d*x)*b + a)*a **5*b**4*c + 12*log(tan(c + d*x)*b + a)*a**4*b**6 + 2*tan(c + d*x)**3*a...