Integrand size = 23, antiderivative size = 270 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
-8/5*a^2*b*cot(d*x+c)^(3/2)/d-2/5*a^2*cot(d*x+c)^(3/2)*(b+a*cot(d*x+c))/d- 1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/ 2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+1/4*( a-b)*(a^2+4*a*b+b^2)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1 /4*(a-b)*(a^2+4*a*b+b^2)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/ 2)+2*a*(a^2-3*b^2)*cot(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.83 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {2 a^2 b \cot ^{\frac {3}{2}}(c+d x)+\frac {2}{5} a^3 \cot ^{\frac {5}{2}}(c+d x)+\frac {2}{3} b \left (-3 a^2+b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-\frac {1}{4} a \left (a^2-3 b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{d} \]
-((2*a^2*b*Cot[c + d*x]^(3/2) + (2*a^3*Cot[c + d*x]^(5/2))/5 + (2*b*(-3*a^ 2 + b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2 ])/3 - (a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log [1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4)/d)
Time = 0.96 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.91, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4156, 3042, 4049, 27, 3042, 4113, 3042, 4011, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} (a+b \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle -\frac {2}{5} \int \frac {1}{2} \sqrt {\cot (c+d x)} \left (-12 a^2 b \cot ^2(c+d x)+5 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (3 a^2-5 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \int \sqrt {\cot (c+d x)} \left (-12 a^2 b \cot ^2(c+d x)+5 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (3 a^2-5 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-12 a^2 b \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b \left (3 a^2-5 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{5} \left (-\int \sqrt {\cot (c+d x)} \left (5 b \left (3 a^2-b^2\right )+5 a \left (a^2-3 b^2\right ) \cot (c+d x)\right )dx-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (5 b \left (3 a^2-b^2\right )-5 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {5 b \left (3 a^2-b^2\right ) \cot (c+d x)-5 a \left (a^2-3 b^2\right )}{\sqrt {\cot (c+d x)}}dx+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {-5 a \left (a^2-3 b^2\right )-5 b \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{5} \left (-\frac {2 \int \frac {5 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\) |
(-2*a^2*Cot[c + d*x]^(3/2)*(b + a*Cot[c + d*x]))/(5*d) + ((10*a*(a^2 - 3*b ^2)*Sqrt[Cot[c + d*x]])/d - (8*a^2*b*Cot[c + d*x]^(3/2))/d - (10*(((a + b) *(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 + ((a - b)*(a^2 + 4*a*b + b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/d)/5
3.9.14.3.1 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[(-(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_) + (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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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_.)*((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[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(232)=464\).
Time = 1.85 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.12
| method | result | size |
| derivativedivides | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right ) \left (15 \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{2} b -5 \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) b^{3}-10 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{3}+30 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{2} b +30 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a \,b^{2}-10 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) b^{3}-10 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{3}+30 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{2} b +30 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a \,b^{2}-10 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) b^{3}-5 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{3}+15 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a \,b^{2}-40 \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}+120 \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{2}+40 \tan \left (d x +c \right ) a^{2} b +8 a^{3}\right )}{20 d}\) | \(573\) |
| default | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right ) \left (15 \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{2} b -5 \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) b^{3}-10 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{3}+30 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{2} b +30 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a \,b^{2}-10 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) b^{3}-10 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{3}+30 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{2} b +30 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a \,b^{2}-10 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) b^{3}-5 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a^{3}+15 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) a \,b^{2}-40 \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}+120 \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{2}+40 \tan \left (d x +c \right ) a^{2} b +8 a^{3}\right )}{20 d}\) | \(573\) |
-1/20/d*(1/tan(d*x+c))^(7/2)*tan(d*x+c)*(15*2^(1/2)*ln(-(1+2^(1/2)*tan(d*x +c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*tan(d*x+c)^ (5/2)*a^2*b-5*2^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2) *tan(d*x+c)^(1/2)-tan(d*x+c)-1))*tan(d*x+c)^(5/2)*b^3-10*2^(1/2)*arctan(1+ 2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(5/2)*a^3+30*2^(1/2)*arctan(1+2^(1/2) *tan(d*x+c)^(1/2))*tan(d*x+c)^(5/2)*a^2*b+30*2^(1/2)*arctan(1+2^(1/2)*tan( d*x+c)^(1/2))*tan(d*x+c)^(5/2)*a*b^2-10*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c )^(1/2))*tan(d*x+c)^(5/2)*b^3-10*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2 ))*tan(d*x+c)^(5/2)*a^3+30*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*tan (d*x+c)^(5/2)*a^2*b+30*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x +c)^(5/2)*a*b^2-10*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^ (5/2)*b^3-5*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2) *tan(d*x+c)^(1/2)+tan(d*x+c)))*tan(d*x+c)^(5/2)*a^3+15*2^(1/2)*ln(-(2^(1/2 )*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))* tan(d*x+c)^(5/2)*a*b^2-40*tan(d*x+c)^2*a^3+120*tan(d*x+c)^2*a*b^2+40*tan(d *x+c)*a^2*b+8*a^3)
Leaf count of result is larger than twice the leaf count of optimal. 1428 vs. \(2 (232) = 464\).
Time = 0.36 (sec) , antiderivative size = 1428, normalized size of antiderivative = 5.29 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \]
-1/10*(5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^1 0*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) )/d^2)*log(((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^8*b - 46*a^6*b ^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^ 5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^ 8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27 *a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 - 5*d*sq rt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a ^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(-( (a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (3*a^8*b - 46*a^6*b^3 + 60*a^4* b^5 - 18*a^2*b^7 + b^9)*d)*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt (-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b ^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 1 2*a^2*b^10 - b^12)*sqrt(tan(d*x + c)))*tan(d*x + c)^2 - 5*d*sqrt((6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452 *a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log(((a^3 - 3*a*b^ 2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a...
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.89 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {40 \, a^{2} b}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {8 \, a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {40 \, {\left (a^{3} - 3 \, a b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}}}{20 \, d} \]
-1/20*(10*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt (2) + 2/sqrt(tan(d*x + c)))) + 10*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)* arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 5*sqrt(2)*(a^3 + 3 *a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 5*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 40*a^2*b/tan(d*x + c)^(3/2) + 8*a^3/tan(d*x + c)^(5/2) - 40*(a^3 - 3*a*b^2)/sqrt(tan(d*x + c)))/d
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]