Integrand size = 23, antiderivative size = 271 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 a^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d} \]
2*a^(7/2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(5/2)/(a^2+b^2)/d+1/2 *(a-b)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)+1/2*(a-b)*a rctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/4*(a+b)*ln(1-2^(1/ 2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)+1/4*(a+b)*ln(1+2^(1/2) *tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-2*a*tan(d*x+c)^(1/2)/b^2 /d+2/3*tan(d*x+c)^(3/2)/b/d
Time = 0.96 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.82 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {-\frac {6 \sqrt {2} (a-b) b^2 \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {24 a^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right )}-\frac {3 \sqrt {2} b^2 (a+b) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{a^2+b^2}-24 a \sqrt {\tan (c+d x)}+8 b \tan ^{\frac {3}{2}}(c+d x)}{12 b^2 d} \]
((-6*Sqrt[2]*(a - b)*b^2*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[ 1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (24*a^(7/2)*ArcTan[(Sqrt[b ]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)) - (3*Sqrt[2]*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) - 24*a*Sqrt[Tan[c + d*x] ] + 8*b*Tan[c + d*x]^(3/2))/(12*b^2*d)
Time = 1.17 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.94, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4049, 27, 3042, 4130, 27, 3042, 4137, 3042, 4017, 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.
| \(\displaystyle \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^{7/2}}{a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {2 \int -\frac {3 \sqrt {\tan (c+d x)} \left (a \tan ^2(c+d x)+b \tan (c+d x)+a\right )}{2 (a+b \tan (c+d x))}dx}{3 b}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\int \frac {\sqrt {\tan (c+d x)} \left (a \tan ^2(c+d x)+b \tan (c+d x)+a\right )}{a+b \tan (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\int \frac {\sqrt {\tan (c+d x)} \left (a \tan (c+d x)^2+b \tan (c+d x)+a\right )}{a+b \tan (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 \int -\frac {a^2+\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}+\frac {2 a \sqrt {\tan (c+d x)}}{b d}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\int \frac {a^2+\left (a^2-b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\int \frac {a^2+\left (a^2-b^2\right ) \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{b}}{b}\) |
| \(\Big \downarrow \) 4137 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {\int \frac {a b^2-b^3 \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {a^4 \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {\int \frac {a b^2-b^3 \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 \int \frac {b^2 (a-b \tan (c+d x))}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \int \frac {a-b \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{b}}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {2 b^2 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {a^4 \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {2 b^2 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 a^4 \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {2 b^2 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{b}}{b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}-\frac {\frac {2 a \sqrt {\tan (c+d x)}}{b d}-\frac {\frac {2 b^2 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 a^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{b}}{b}\) |
-((-(((2*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a ^2 + b^2)*d) + (2*b^2*(((a - b)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/ Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]))/2 + ((a + b)*( -1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)* d))/b) + (2*a*Sqrt[Tan[c + d*x]])/(b*d))/b) + (2*Tan[c + d*x]^(3/2))/(3*b* d)
3.6.83.3.1 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[((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_.)*((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_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Sim p[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*C)*T an[e + f*x], x], x], x] + Simp[(A*b^2 + 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, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.08 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+a \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{b^{2}}+\frac {\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}+\frac {2 a^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}}{d}\) | \(255\) |
| default | \(\frac {-\frac {2 \left (-\frac {b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+a \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{b^{2}}+\frac {\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}+\frac {2 a^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}}{d}\) | \(255\) |
1/d*(-2/b^2*(-1/3*b*tan(d*x+c)^(3/2)+a*tan(d*x+c)^(1/2))+2/(a^2+b^2)*(1/8* a*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c )^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1 /2)*tan(d*x+c)^(1/2)))-1/8*b*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d *x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x +c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))+2/b^2*a^4/(a^2+b^2)/(a* b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1415 vs. \(2 (227) = 454\).
Time = 0.36 (sec) , antiderivative size = 2856, normalized size of antiderivative = 10.54 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
[1/6*(6*a^3*sqrt(-a/b)*log((2*b*sqrt(-a/b)*sqrt(tan(d*x + c)) + b*tan(d*x + c) - a)/(b*tan(d*x + c) + a)) - 3*(a^2*b^2 + b^4)*d*sqrt(((a^4 + 2*a^2*b ^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4 *b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2 *b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^ 4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^2)*sqrt(tan(d*x + c))) + 3*(a^2*b^2 + b^4)*d*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a ^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^4*b + 2*a^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6* a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (a^3 - a*b^2)*d)*sqrt(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (a^2 - b^ 2)*sqrt(tan(d*x + c))) + 3*(a^2*b^2 + b^4)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4 )*d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2* b^6 + b^8)*d^4)) - 2*a*b)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^4*b + 2*a ^2*b^3 + b^5)*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4* b^4 + 4*a^2*b^6 + b^8)*d^4)) - (a^3 - a*b^2)*d)*sqrt(-((a^4 + 2*a^2*b^2...
Timed out. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {24 \, a^{4} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{2} + b^{2}} + \frac {8 \, {\left (b \tan \left (d x + c\right )^{\frac {3}{2}} - 3 \, a \sqrt {\tan \left (d x + c\right )}\right )}}{b^{2}}}{12 \, d} \]
1/12*(24*a^4*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2*b^2 + b^4)*sqrt( a*b)) + 3*(2*sqrt(2)*(a - b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*(a + b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(a + b)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/ (a^2 + b^2) + 8*(b*tan(d*x + c)^(3/2) - 3*a*sqrt(tan(d*x + c)))/b^2)/d
Timed out. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Timed out} \]
Time = 7.46 (sec) , antiderivative size = 4927, normalized size of antiderivative = 18.18 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
(2*tan(c + d*x)^(3/2))/(3*b*d) - atan((((((1/(b^2*d^2*1i - a^2*d^2*1i + 2* a*b*d^2))^(1/2)*(((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((32*( 12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) - (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1/(b^2*d^ 2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + (32*tan(c + d*x)^(1/2)*(14*a*b^ 9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4)))/2 - (32 *(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/ (b^3*d^5)))/2 - (32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4))*(1/(b^2*d ^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i)/2 - ((((1/(b^2*d^2*1i - a^2*d^2 *1i + 2*a*b*d^2))^(1/2)*(((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)* ((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) + (16 *tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^ 12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1 /(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - (32*tan(c + d*x)^(1/2)* (14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4))) /2 - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^ 2*d^2))/(b^3*d^5)))/2 + (32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4))*( 1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i)/2)/(((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d...