Integrand size = 33, antiderivative size = 371 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {(i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 a \left (a^2 A b+3 A b^3-2 a^3 B-4 a b^2 B\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (8 a^4 A b+17 a^2 A b^3+3 A b^5-16 a^5 B-30 a^3 b^2 B-8 a b^4 B\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}-\frac {2 \left (4 a^3 A b+10 a A b^3-8 a^4 B-15 a^2 b^2 B-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d} \]
-(I*A+B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d+(I* A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d+2/3*(8* A*a^4*b+17*A*a^2*b^3+3*A*b^5-16*B*a^5-30*B*a^3*b^2-8*B*a*b^4)*(a+b*tan(d*x +c))^(1/2)/b^4/(a^2+b^2)^2/d-2/3*(4*A*a^3*b+10*A*a*b^3-8*B*a^4-15*B*a^2*b^ 2-B*b^4)*(a+b*tan(d*x+c))^(1/2)*tan(d*x+c)/b^3/(a^2+b^2)^2/d+2*a*(A*a^2*b+ 3*A*b^3-2*B*a^3-4*B*a*b^2)*tan(d*x+c)^2/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c)) ^(1/2)+2/3*a*(A*b-B*a)*tan(d*x+c)^3/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.39 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {2 B \tan ^3(c+d x)}{3 b d (a+b \tan (c+d x))^{3/2}}+\frac {2 \left (\frac {3 (A b-2 a B) \tan ^2(c+d x)}{b d (a+b \tan (c+d x))^{3/2}}+\frac {2 \left (\frac {3 \left (4 a A b-8 a^2 B+b^2 B\right ) \tan (c+d x)}{2 b d (a+b \tan (c+d x))^{3/2}}-\frac {3 \left (-\frac {2 \left (8 a^2 A b+A b^3-16 a^3 B+2 a b^2 B\right )}{3 b (a+b \tan (c+d x))^{3/2}}+\frac {2 \left (\frac {\left (-\frac {3 A b^5}{2}+\frac {3}{2} a b^4 B\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )}{3 (i a+b) (a+b \tan (c+d x))^{3/2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )}{3 (i a-b) (a+b \tan (c+d x))^{3/2}}\right )}{b}-\frac {3}{2} b^3 B \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )}{(i a+b) \sqrt {a+b \tan (c+d x)}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )}{(i a-b) \sqrt {a+b \tan (c+d x)}}\right )\right )}{3 b}\right )}{4 b d}\right )}{b}\right )}{3 b} \]
(2*B*Tan[c + d*x]^3)/(3*b*d*(a + b*Tan[c + d*x])^(3/2)) + (2*((3*(A*b - 2* a*B)*Tan[c + d*x]^2)/(b*d*(a + b*Tan[c + d*x])^(3/2)) + (2*((3*(4*a*A*b - 8*a^2*B + b^2*B)*Tan[c + d*x])/(2*b*d*(a + b*Tan[c + d*x])^(3/2)) - (3*((- 2*(8*a^2*A*b + A*b^3 - 16*a^3*B + 2*a*b^2*B))/(3*b*(a + b*Tan[c + d*x])^(3 /2)) + (2*((((-3*A*b^5)/2 + (3*a*b^4*B)/2)*(-1/3*Hypergeometric2F1[-3/2, 1 , -1/2, (a + b*Tan[c + d*x])/(a - I*b)]/((I*a + b)*(a + b*Tan[c + d*x])^(3 /2)) + Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)]/(3 *(I*a - b)*(a + b*Tan[c + d*x])^(3/2))))/b - (3*b^3*B*(-(Hypergeometric2F1 [-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)]/((I*a + b)*Sqrt[a + b*Tan[c + d*x]])) + Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b )]/((I*a - b)*Sqrt[a + b*Tan[c + d*x]])))/2))/(3*b)))/(4*b*d)))/b))/(3*b)
Time = 2.17 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.06, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4088, 27, 3042, 4128, 27, 3042, 4130, 27, 3042, 4113, 3042, 4022, 3042, 4020, 25, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^4 (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {2 \int -\frac {3 \tan ^2(c+d x) \left (\left (-2 B a^2+A b a-b^2 B\right ) \tan ^2(c+d x)-b (A b-a B) \tan (c+d x)+2 a (A b-a B)\right )}{2 (a+b \tan (c+d x))^{3/2}}dx}{3 b \left (a^2+b^2\right )}+\frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\int \frac {\tan ^2(c+d x) \left (\left (-2 B a^2+A b a-b^2 B\right ) \tan ^2(c+d x)-b (A b-a B) \tan (c+d x)+2 a (A b-a B)\right )}{(a+b \tan (c+d x))^{3/2}}dx}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\int \frac {\tan (c+d x)^2 \left (\left (-2 B a^2+A b a-b^2 B\right ) \tan (c+d x)^2-b (A b-a B) \tan (c+d x)+2 a (A b-a B)\right )}{(a+b \tan (c+d x))^{3/2}}dx}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 \int \frac {\tan (c+d x) \left (\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) b^2+\left (-8 B a^4+4 A b a^3-15 b^2 B a^2+10 A b^3 a-b^4 B\right ) \tan ^2(c+d x)+4 a \left (-2 B a^3+A b a^2-4 b^2 B a+3 A b^3\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\int \frac {\tan (c+d x) \left (\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) b^2+\left (-8 B a^4+4 A b a^3-15 b^2 B a^2+10 A b^3 a-b^4 B\right ) \tan ^2(c+d x)+4 a \left (-2 B a^3+A b a^2-4 b^2 B a+3 A b^3\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\int \frac {\tan (c+d x) \left (\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) b^2+\left (-8 B a^4+4 A b a^3-15 b^2 B a^2+10 A b^3 a-b^4 B\right ) \tan (c+d x)^2+4 a \left (-2 B a^3+A b a^2-4 b^2 B a+3 A b^3\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\frac {2 \int -\frac {-3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) b^3+\left (-16 B a^5+8 A b a^4-30 b^2 B a^3+17 A b^3 a^2-8 b^4 B a+3 A b^5\right ) \tan ^2(c+d x)+2 a \left (-8 B a^4+4 A b a^3-15 b^2 B a^2+10 A b^3 a-b^4 B\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{3 b}+\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {-3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) b^3+\left (-16 B a^5+8 A b a^4-30 b^2 B a^3+17 A b^3 a^2-8 b^4 B a+3 A b^5\right ) \tan ^2(c+d x)+2 a \left (-8 B a^4+4 A b a^3-15 b^2 B a^2+10 A b^3 a-b^4 B\right )}{\sqrt {a+b \tan (c+d x)}}dx}{3 b}}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {-3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) b^3+\left (-16 B a^5+8 A b a^4-30 b^2 B a^3+17 A b^3 a^2-8 b^4 B a+3 A b^5\right ) \tan (c+d x)^2+2 a \left (-8 B a^4+4 A b a^3-15 b^2 B a^2+10 A b^3 a-b^4 B\right )}{\sqrt {a+b \tan (c+d x)}}dx}{3 b}}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {3 b^3 \left (A a^2+2 b B a-A b^2\right )-3 b^3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {3 b^3 \left (A a^2+2 b B a-A b^2\right )-3 b^3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {3}{2} b^3 (a-i b)^2 (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {3}{2} b^3 (a+i b)^2 (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {3}{2} b^3 (a-i b)^2 (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {3}{2} b^3 (a+i b)^2 (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {3 i b^3 (a+i b)^2 (A-i B) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {3 i b^3 (a-i b)^2 (A+i B) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {-\frac {3 i b^3 (a+i b)^2 (A-i B) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {3 i b^3 (a-i b)^2 (A+i B) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {3 b^2 (a-i b)^2 (A+i B) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{d}+\frac {3 b^2 (a+i b)^2 (A-i B) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{d}+\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 a \left (-2 a^3 B+a^2 A b-4 a b^2 B+3 A b^3\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (-8 a^4 B+4 a^3 A b-15 a^2 b^2 B+10 a A b^3-b^4 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {2 \left (-16 a^5 B+8 a^4 A b-30 a^3 b^2 B+17 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{b d}+\frac {3 b^3 (a+i b)^2 (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {3 b^3 (a-i b)^2 (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
(2*a*(A*b - a*B)*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^( 3/2)) - ((-2*a*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Tan[c + d*x]^2)/( b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + ((2*(4*a^3*A*b + 10*a*A*b^3 - 8*a^4*B - 15*a^2*b^2*B - b^4*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3* b*d) - ((3*(a + I*b)^2*b^3*(A - I*B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/( Sqrt[a - I*b]*d) + (3*(a - I*b)^2*b^3*(A + I*B)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) + (2*(8*a^4*A*b + 17*a^2*A*b^3 + 3*A*b^5 - 16* a^5*B - 30*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Tan[c + d*x]])/(b*d))/(3*b))/ (b*(a^2 + b^2)))/(b*(a^2 + b^2))
3.4.57.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x ])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d* (b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[ e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] & & LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(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])))
Leaf count of result is larger than twice the leaf count of optimal. \(4591\) vs. \(2(339)=678\).
Time = 0.20 (sec) , antiderivative size = 4592, normalized size of antiderivative = 12.38
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(4592\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(12953\) |
| default | \(\text {Expression too large to display}\) | \(12953\) |
A*(-3/4/d*b^3/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2) +2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)* a+1/d/b/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2) ^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)) *a^4-1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b ^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/ 2))*a^6-4/d*b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^ 2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^ (1/2))*a^4-1/d*b^3/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((( 2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)- 2*a)^(1/2))*a^2-1/4/d/b/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c)) ^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2 *a)^(1/2)*a^5+1/2/d*b/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^( 1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a )^(1/2)*a^3+3/4/d*b^3/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^( 1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a )^(1/2)*a-1/d/b/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a +b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a )^(1/2))*a^4+1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2 *(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/...
Leaf count of result is larger than twice the leaf count of optimal. 7358 vs. \(2 (334) = 668\).
Time = 2.20 (sec) , antiderivative size = 7358, normalized size of antiderivative = 19.83 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{4}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 47.63 (sec) , antiderivative size = 9547, normalized size of antiderivative = 25.73 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
(log((((a + b*tan(c + d*x))^(1/2)*(320*A^2*a^4*b^14*d^3 - 16*A^2*b^18*d^3 + 1024*A^2*a^6*b^12*d^3 + 1440*A^2*a^8*b^10*d^3 + 1024*A^2*a^10*b^8*d^3 + 320*A^2*a^12*b^6*d^3 - 16*A^2*a^16*b^2*d^3) - ((((320*A^4*a^2*b^8*d^4 - 16 *A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8* b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 40*A^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(a ^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a ^8*b^2*d^4))^(1/2)*(((((320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a ^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5 *d^2 + 40*A^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2 *b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(a + b* tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^1 3*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64 *a^21*b^2*d^5))/4 - 32*A*b^21*d^4 - 160*A*a^2*b^19*d^4 - 128*A*a^4*b^17*d^ 4 + 896*A*a^6*b^15*d^4 + 3136*A*a^8*b^13*d^4 + 4928*A*a^10*b^11*d^4 + 4480 *A*a^12*b^9*d^4 + 2432*A*a^14*b^7*d^4 + 736*A*a^16*b^5*d^4 + 96*A*a^18*b^3 *d^4))/4)*(((320*A^4*a^2*b^8*d^4 - 16*A^4*b^10*d^4 - 1760*A^4*a^4*b^6*d^4 + 1600*A^4*a^6*b^4*d^4 - 400*A^4*a^8*b^2*d^4)^(1/2) - 4*A^2*a^5*d^2 + 40*A ^2*a^3*b^2*d^2 - 20*A^2*a*b^4*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/4 + 96*A^3*a^3...