Integrand size = 23, antiderivative size = 456 \[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d} \]
1/2*b*arctanh(((a+(a^2+b^2)^(1/2))^(1/2)-2^(1/2)*(a+b*tan(d*x+c))^(1/2))/( a-(a^2+b^2)^(1/2))^(1/2))/d*2^(1/2)/(a-(a^2+b^2)^(1/2))^(1/2)-1/2*b*arctan h(((a+(a^2+b^2)^(1/2))^(1/2)+2^(1/2)*(a+b*tan(d*x+c))^(1/2))/(a-(a^2+b^2)^ (1/2))^(1/2))/d*2^(1/2)/(a-(a^2+b^2)^(1/2))^(1/2)+1/4*b*ln(a+(a^2+b^2)^(1/ 2)-2^(1/2)*(a+(a^2+b^2)^(1/2))^(1/2)*(a+b*tan(d*x+c))^(1/2)+b*tan(d*x+c))/ d*2^(1/2)/(a+(a^2+b^2)^(1/2))^(1/2)-1/4*b*ln(a+(a^2+b^2)^(1/2)+2^(1/2)*(a+ (a^2+b^2)^(1/2))^(1/2)*(a+b*tan(d*x+c))^(1/2)+b*tan(d*x+c))/d*2^(1/2)/(a+( a^2+b^2)^(1/2))^(1/2)+2/105*(8*a^2-35*b^2)*(a+b*tan(d*x+c))^(3/2)/b^3/d-8/ 35*a*tan(d*x+c)*(a+b*tan(d*x+c))^(3/2)/b^2/d+2/7*tan(d*x+c)^2*(a+b*tan(d*x +c))^(3/2)/b/d
Result contains complex when optimal does not.
Time = 21.54 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.37 \[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {-105 i \sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+105 i \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\frac {2 \sqrt {a+b \tan (c+d x)} \left (8 a^3-38 a b^2-2 b \left (2 a^2+25 b^2\right ) \tan (c+d x)+3 b^2 \sec ^2(c+d x) (a+5 b \tan (c+d x))\right )}{b^3}}{105 d} \]
((-105*I)*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (105*I)*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2 *Sqrt[a + b*Tan[c + d*x]]*(8*a^3 - 38*a*b^2 - 2*b*(2*a^2 + 25*b^2)*Tan[c + d*x] + 3*b^2*Sec[c + d*x]^2*(a + 5*b*Tan[c + d*x])))/b^3)/(105*d)
Time = 1.23 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.14, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4049, 27, 3042, 4130, 27, 3042, 4114, 3042, 3966, 483, 1449, 1142, 25, 27, 1083, 219, 1103}
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.
| \(\displaystyle \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^4 \sqrt {a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {2 \int -\frac {1}{2} \tan (c+d x) \sqrt {a+b \tan (c+d x)} \left (4 a \tan ^2(c+d x)+7 b \tan (c+d x)+4 a\right )dx}{7 b}+\frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\int \tan (c+d x) \sqrt {a+b \tan (c+d x)} \left (4 a \tan ^2(c+d x)+7 b \tan (c+d x)+4 a\right )dx}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\int \tan (c+d x) \sqrt {a+b \tan (c+d x)} \left (4 a \tan (c+d x)^2+7 b \tan (c+d x)+4 a\right )dx}{7 b}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {2 \int -\frac {1}{2} \sqrt {a+b \tan (c+d x)} \left (8 a^2+\left (8 a^2-35 b^2\right ) \tan ^2(c+d x)\right )dx}{5 b}+\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\int \sqrt {a+b \tan (c+d x)} \left (8 a^2+\left (8 a^2-35 b^2\right ) \tan ^2(c+d x)\right )dx}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\int \sqrt {a+b \tan (c+d x)} \left (8 a^2+\left (8 a^2-35 b^2\right ) \tan (c+d x)^2\right )dx}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 4114 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {35 b^2 \int \sqrt {a+b \tan (c+d x)}dx+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {35 b^2 \int \sqrt {a+b \tan (c+d x)}dx+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {35 b^3 \int \frac {\sqrt {a+b \tan (c+d x)}}{\tan ^2(c+d x) b^2+b^2}d(b \tan (c+d x))}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 483 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \int \frac {b^2 \tan ^2(c+d x)}{b^4 \tan ^4(c+d x)-2 a b^2 \tan ^2(c+d x)+a^2+b^2}d\sqrt {a+b \tan (c+d x)}}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 1449 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {\int \frac {\sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {2} b \sqrt {a+\sqrt {a^2+b^2}} \tan (c+d x)+\sqrt {a^2+b^2}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\int \frac {\sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {2} b \sqrt {a+\sqrt {a^2+b^2}} \tan (c+d x)+\sqrt {a^2+b^2}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {1}{2} \int -\frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}\right )}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-b^2 \tan ^2(c+d x)}d\left (2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}\right )-\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-b^2 \tan ^2(c+d x)}d\left (\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )+\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {-\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}+\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}+2 \sqrt {a+b \tan (c+d x)}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac {\frac {8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac {\frac {70 b^3 \left (\frac {\frac {1}{2} \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+b^2 \tan ^2(c+d x)\right )-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}-\frac {\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}+2 \sqrt {a+b \tan (c+d x)}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}+\frac {1}{2} \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+b^2 \tan ^2(c+d x)\right )}{2 \sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}+\frac {2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{3 b d}}{5 b}}{7 b}\) |
(2*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2))/(7*b*d) - ((8*a*Tan[c + d*x] *(a + b*Tan[c + d*x])^(3/2))/(5*b*d) - ((70*b^3*((-((Sqrt[a + Sqrt[a^2 + b ^2]]*ArcTanh[(-(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]) + 2*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/Sqrt[a - Sqrt[a^2 + b^2]]) + Log[Sqrt[a^2 + b^2] + b^2*Tan[c + d*x]^2 - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2 ]]*Sqrt[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]) - (( Sqrt[a + Sqrt[a^2 + b^2]]*ArcTanh[(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]] + 2*S qrt[a + b*Tan[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/Sqrt[a - Sq rt[a^2 + b^2]] + Log[Sqrt[a^2 + b^2] + b^2*Tan[c + d*x]^2 + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2]*Sqrt[a + Sqrt[ a^2 + b^2]])))/d + (2*(8*a^2 - 35*b^2)*(a + b*Tan[c + d*x])^(3/2))/(3*b*d) )/(5*b))/(7*b)
3.6.3.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d Subst[Int[x^2/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x ] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r) Int[x^(m - 1)/(q - r*x + x^2), x], x] - Simp[1/(2*c*r) Int[x^(m - 1)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((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 - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Simp[(A - C) Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a , b, e, f, A, C, m}, x] && NeQ[A*b^2 + 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[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 = 1.75 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d \,b^{3}}-\frac {4 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d \,b^{3}}+\frac {2 a^{2} \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d \,b^{3}}-\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \ln \left (b \tan \left (d x +c \right )+a -\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \ln \left (b \tan \left (d x +c \right )+a -\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}\) | \(556\) |
| default | \(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d \,b^{3}}-\frac {4 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d \,b^{3}}+\frac {2 a^{2} \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d \,b^{3}}-\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \ln \left (b \tan \left (d x +c \right )+a -\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \ln \left (b \tan \left (d x +c \right )+a -\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4 d b}\) | \(556\) |
2/7/d/b^3*(a+b*tan(d*x+c))^(7/2)-4/5/d/b^3*a*(a+b*tan(d*x+c))^(5/2)+2/3/d/ b^3*a^2*(a+b*tan(d*x+c))^(3/2)-2/3*(a+b*tan(d*x+c))^(3/2)/b/d+1/4/d/b*(2*( a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*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))+1/d*b/(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))-1/4/d/b*(2*(a^2+b^2)^(1/2)+2*a)^(1/ 2)*a*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))-1/4/d/b*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*l n(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))+1/d*b/(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))+1/4/ d/b*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*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))
Time = 0.25 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.91 \[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {105 \, b^{3} d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d^{3} \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) - 105 \, b^{3} d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d^{3} \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) - 105 \, b^{3} d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d^{3} \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) + 105 \, b^{3} d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d^{3} \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \sqrt {-\frac {b^{2}}{d^{4}}} + \sqrt {b \tan \left (d x + c\right ) + a} b\right ) - 4 \, {\left (15 \, b^{3} \tan \left (d x + c\right )^{3} + 3 \, a b^{2} \tan \left (d x + c\right )^{2} + 8 \, a^{3} - 35 \, a b^{2} - {\left (4 \, a^{2} b + 35 \, b^{3}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right ) + a}}{210 \, b^{3} d} \]
-1/210*(105*b^3*d*sqrt(-(d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d^3*sqrt(-(d^2*s qrt(-b^2/d^4) + a)/d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b) - 105 *b^3*d*sqrt(-(d^2*sqrt(-b^2/d^4) + a)/d^2)*log(-d^3*sqrt(-(d^2*sqrt(-b^2/d ^4) + a)/d^2)*sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b) - 105*b^3*d*sqr t((d^2*sqrt(-b^2/d^4) - a)/d^2)*log(d^3*sqrt((d^2*sqrt(-b^2/d^4) - a)/d^2) *sqrt(-b^2/d^4) + sqrt(b*tan(d*x + c) + a)*b) + 105*b^3*d*sqrt((d^2*sqrt(- b^2/d^4) - a)/d^2)*log(-d^3*sqrt((d^2*sqrt(-b^2/d^4) - a)/d^2)*sqrt(-b^2/d ^4) + sqrt(b*tan(d*x + c) + a)*b) - 4*(15*b^3*tan(d*x + c)^3 + 3*a*b^2*tan (d*x + c)^2 + 8*a^3 - 35*a*b^2 - (4*a^2*b + 35*b^3)*tan(d*x + c))*sqrt(b*t an(d*x + c) + a))/(b^3*d)
\[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{4}{\left (c + d x \right )}\, dx \]
\[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Timed out} \]
Time = 21.52 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.81 \[ \int \tan ^4(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (2\,a\,\left (\frac {4\,a^2}{b^3\,d}-\frac {2\,\left (a^2+b^2\right )}{b^3\,d}\right )-\frac {8\,a^3}{b^3\,d}+\frac {4\,a\,\left (a^2+b^2\right )}{b^3\,d}\right )+\left (\frac {4\,a^2}{3\,b^3\,d}-\frac {2\,\left (a^2+b^2\right )}{3\,b^3\,d}\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}+\frac {2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{7/2}}{7\,b^3\,d}-\frac {4\,a\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{5\,b^3\,d}+\mathrm {atan}\left (\frac {d^3\,\left (\frac {16\,\left (b^4-a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}+\frac {16\,a\,b^2\,\left (a-b\,1{}\mathrm {i}\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}\right )\,\sqrt {-\frac {a-b\,1{}\mathrm {i}}{4\,d^2}}\,1{}\mathrm {i}}{8\,\left (a^2\,b^3+b^5\right )}\right )\,\sqrt {-\frac {a-b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {d^3\,\left (\frac {16\,\left (b^4-a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}+\frac {16\,a\,b^2\,\left (a+b\,1{}\mathrm {i}\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}\right )\,\sqrt {-\frac {a+b\,1{}\mathrm {i}}{4\,d^2}}\,1{}\mathrm {i}}{8\,\left (a^2\,b^3+b^5\right )}\right )\,\sqrt {-\frac {a+b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i} \]
atan((d^3*((16*(b^4 - a^2*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 + (16*a*b^2 *(a - b*1i)*(a + b*tan(c + d*x))^(1/2))/d^2)*(-(a - b*1i)/(4*d^2))^(1/2)*1 i)/(8*(b^5 + a^2*b^3)))*(-(a - b*1i)/(4*d^2))^(1/2)*2i + atan((d^3*((16*(b ^4 - a^2*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 + (16*a*b^2*(a + b*1i)*(a + b*tan(c + d*x))^(1/2))/d^2)*(-(a + b*1i)/(4*d^2))^(1/2)*1i)/(8*(b^5 + a^2* b^3)))*(-(a + b*1i)/(4*d^2))^(1/2)*2i + (a + b*tan(c + d*x))^(1/2)*(2*a*(( 4*a^2)/(b^3*d) - (2*(a^2 + b^2))/(b^3*d)) - (8*a^3)/(b^3*d) + (4*a*(a^2 + b^2))/(b^3*d)) + ((4*a^2)/(3*b^3*d) - (2*(a^2 + b^2))/(3*b^3*d))*(a + b*ta n(c + d*x))^(3/2) + (2*(a + b*tan(c + d*x))^(7/2))/(7*b^3*d) - (4*a*(a + b *tan(c + d*x))^(5/2))/(5*b^3*d)