Integrand size = 23, antiderivative size = 198 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d} \] Output:
-1/2*(a^2-2*a*b-b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1/2*(a^ 2-2*a*b-b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1/2*(a^2+2*a*b-b ^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/d+2*(a^2-b^2) *cot(d*x+c)^(1/2)/d-4/3*a*b*cot(d*x+c)^(3/2)/d-2/5*a^2*cot(d*x+c)^(5/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.01 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.02 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {\frac {2}{5} a^2 \cot ^{\frac {5}{2}}(c+d x)-\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )-\frac {1}{4} \left (a^2-b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{d} \] Input:
Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2,x]
Output:
-(((2*a^2*Cot[c + d*x]^(5/2))/5 - (4*a*b*Cot[c + d*x]^(3/2)*(-1 + Hypergeo metric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 - ((a^2 - b^2)*(2*Sqrt[2]*ArcT an[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 = 0.83 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4156, 3042, 4026, 3042, 4011, 3042, 4011, 3042, 4017, 25, 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.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} (a+b \tan (c+d x))^2dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) \left (-a^2+2 b \cot (c+d x) a+b^2\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-a^2-2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a+b^2\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} \left (-2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-2 a b-\left (b^2-a^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {a^2-2 b \cot (c+d x) a-b^2}{\sqrt {\cot (c+d x)}}dx+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int -\frac {a^2-2 b \cot (c+d x) a-b^2}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \int \frac {a^2-2 b \cot (c+d x) a-b^2}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
Input:
Int[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2,x]
Output:
(2*(a^2 - b^2)*Sqrt[Cot[c + d*x]])/d - (4*a*b*Cot[c + d*x]^(3/2))/(3*d) - (2*a^2*Cot[c + d*x]^(5/2))/(5*d) + (2*(-1/2*((a^2 - 2*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])) - ((a^2 + 2*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]] + Co t[c + d*x]]/(2*Sqrt[2])))/2))/d
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_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
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]
Time = 0.37 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(-\frac {\frac {2 a^{2} \cot \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {4 a b \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-2 a^{2} \sqrt {\cot \left (d x +c \right )}+2 b^{2} \sqrt {\cot \left (d x +c \right )}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}}{d}\) | \(239\) |
| default | \(-\frac {\frac {2 a^{2} \cot \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {4 a b \cot \left (d x +c \right )^{\frac {3}{2}}}{3}-2 a^{2} \sqrt {\cot \left (d x +c \right )}+2 b^{2} \sqrt {\cot \left (d x +c \right )}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}}{d}\) | \(239\) |
Input:
int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/d*(2/5*a^2*cot(d*x+c)^(5/2)+4/3*a*b*cot(d*x+c)^(3/2)-2*a^2*cot(d*x+c)^( 1/2)+2*b^2*cot(d*x+c)^(1/2)+1/4*(a^2-b^2)*2^(1/2)*(ln((cot(d*x+c)+2^(1/2)* cot(d*x+c)^(1/2)+1)/(cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1))+2*arctan(1+2^ (1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))-1/2*a*b*2^( 1/2)*(ln((cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)+2^(1/2)*cot(d *x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*c ot(d*x+c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (170) = 340\).
Time = 0.12 (sec) , antiderivative size = 714, normalized size of antiderivative = 3.61 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-1/30*(30*sqrt(1/2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2 )*arctan(-(2*sqrt(1/2)*(a^2 + 2*a*b - b^2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b ^2 + 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^4 + 4*a^3*b + 2* a^2*b^2 - 4*a*b^3 + b^4)/d^2)*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2))/(a^4 - 6*a^2*b^2 + b^4))*tan(d*x + c)^2 + 30*sqrt(1/2)*d*sqrt(( a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2)*arctan(-(2*sqrt(1/2)*(a^2 + 2*a*b - b^2)*d*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2)*sqr t(tan(d*x + c)) - d^2*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d^2 )*sqrt((a^4 - 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + b^4)/d^2))/(a^4 - 6*a^2*b^2 + b^4))*tan(d*x + c)^2 - 15*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 + b^4)/d^2)*log(2*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2*a^2*b^2 - 4* a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) - a^2 - 2*a*b + b^2 - (a^2 + 2*a*b - b^2)*tan(d*x + c))*tan(d*x + c)^2 + 15*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2 *a^2*b^2 - 4*a*b^3 + b^4)/d^2)*log(-2*sqrt(1/2)*d*sqrt((a^4 + 4*a^3*b + 2* a^2*b^2 - 4*a*b^3 + b^4)/d^2)*sqrt(tan(d*x + c)) - a^2 - 2*a*b + b^2 - (a^ 2 + 2*a*b - b^2)*tan(d*x + c))*tan(d*x + c)^2 + 4*(10*a*b*tan(d*x + c) - 1 5*(a^2 - b^2)*tan(d*x + c)^2 + 3*a^2)/sqrt(tan(d*x + c)))/(d*tan(d*x + c)^ 2)
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(7/2)*(a+b*tan(d*x+c))**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {30 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {80 \, a b}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {120 \, {\left (a^{2} - b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}} + \frac {24 \, a^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{60 \, d} \] Input:
integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-1/60*(30*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt (tan(d*x + c)))) + 30*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqr t(2) - 2/sqrt(tan(d*x + c)))) + 15*sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2) /sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 15*sqrt(2)*(a^2 + 2*a*b - b^2) *log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 80*a*b/tan(d*x + c)^(3/2) - 120*(a^2 - b^2)/sqrt(tan(d*x + c)) + 24*a^2/tan(d*x + c)^(5/2)) /d
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)^2*cot(d*x + c)^(7/2), x)
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \] Input:
int(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2,x)
Output:
int(cot(c + d*x)^(7/2)*(a + b*tan(c + d*x))^2, x)
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {-2 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} a^{2}+10 \sqrt {\cot \left (d x +c \right )}\, a^{2}+5 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a^{2} d +5 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}d x \right ) b^{2} d +10 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )d x \right ) a b d}{5 d} \] Input:
int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^2,x)
Output:
( - 2*sqrt(cot(c + d*x))*cot(c + d*x)**2*a**2 + 10*sqrt(cot(c + d*x))*a**2 + 5*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*a**2*d + 5*int(sqrt(cot(c + d* x))*cot(c + d*x)**3*tan(c + d*x)**2,x)*b**2*d + 10*int(sqrt(cot(c + d*x))* cot(c + d*x)**3*tan(c + d*x),x)*a*b*d)/(5*d)