Integrand size = 21, antiderivative size = 160 \[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}} \] Output:
1/2*(a-b)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d+1/2*(a-b)*arctan(1 +2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d+1/2*(a+b)*arctanh(2^(1/2)*cot(d*x+c)^ (1/2)/(1+cot(d*x+c)))*2^(1/2)/d+2/5*b/d/cot(d*x+c)^(5/2)+2/3*a/d/cot(d*x+c )^(3/2)-2*b/d/cot(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.58 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-40 a \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right )\right ) \tan ^{\frac {3}{2}}(c+d x)+3 b \left (-10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-40 \sqrt {\tan (c+d x)}+8 \tan ^{\frac {5}{2}}(c+d x)\right )\right )}{60 d} \] Input:
Integrate[(a + b*Tan[c + d*x])/Cot[c + d*x]^(5/2),x]
Output:
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-40*a*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Tan[c + d*x]^(3/2) + 3*b*(-10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 5 *Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 40*Sqrt[Tan[ c + d*x]] + 8*Tan[c + d*x]^(5/2))))/(60*d)
Time = 0.72 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.24, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.952, Rules used = {3042, 4156, 3042, 4012, 3042, 4012, 25, 3042, 4012, 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.
| \(\displaystyle \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \tan (c+d x)}{\cot (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {a \cot (c+d x)+b}{\cot ^{\frac {7}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {b-a \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int \frac {a-b \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)}dx+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int -\frac {b+a \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {b+a \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {b-a \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int \frac {a-b \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a+b \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {2 \int -\frac {a-b \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int \frac {a-b \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2 b}{d \sqrt {\cot (c+d x)}}\) |
Input:
Int[(a + b*Tan[c + d*x])/Cot[c + d*x]^(5/2),x]
Output:
(2*b)/(5*d*Cot[c + d*x]^(5/2)) + (2*a)/(3*d*Cot[c + d*x]^(3/2)) - (2*b)/(d *Sqrt[Cot[c + d*x]]) - (2*(-1/2*((a - b)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) - ((a + b)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + L og[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[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[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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[(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 = 0.19 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.55
| method | result | size |
| derivativedivides | \(\frac {24 b \tan \left (d x +c \right )^{\frac {5}{2}}+40 a \tan \left (d x +c \right )^{\frac {3}{2}}+15 b \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )-30 a \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 b \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-30 a \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 b \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-15 a \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )-120 b \sqrt {\tan \left (d x +c \right )}}{60 d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right )^{\frac {5}{2}}}\) | \(248\) |
| default | \(\frac {24 b \tan \left (d x +c \right )^{\frac {5}{2}}+40 a \tan \left (d x +c \right )^{\frac {3}{2}}+15 b \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )-30 a \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 b \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-30 a \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 b \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-15 a \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )-120 b \sqrt {\tan \left (d x +c \right )}}{60 d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right )^{\frac {5}{2}}}\) | \(248\) |
Input:
int((a+b*tan(d*x+c))/cot(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/60/d*(24*b*tan(d*x+c)^(5/2)+40*a*tan(d*x+c)^(3/2)+15*b*2^(1/2)*ln(-(tan( d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1) )-30*a*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+30*b*2^(1/2)*arctan(1+2^ (1/2)*tan(d*x+c)^(1/2))-30*a*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))+3 0*b*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-15*a*2^(1/2)*ln(-(2^(1/2)* tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))-12 0*b*tan(d*x+c)^(1/2))/(1/tan(d*x+c))^(5/2)/tan(d*x+c)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (133) = 266\).
Time = 0.09 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.53 \[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\frac {30 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) + 30 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} - d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) + 15 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \log \left (2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + {\left (a + b\right )} \tan \left (d x + c\right ) + a + b\right ) - 15 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \log \left (-2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + {\left (a + b\right )} \tan \left (d x + c\right ) + a + b\right ) + \frac {4 \, {\left (3 \, b \tan \left (d x + c\right )^{3} + 5 \, a \tan \left (d x + c\right )^{2} - 15 \, b \tan \left (d x + c\right )\right )}}{\sqrt {\tan \left (d x + c\right )}}}{30 \, d} \] Input:
integrate((a+b*tan(d*x+c))/cot(d*x+c)^(5/2),x, algorithm="fricas")
Output:
1/30*(30*sqrt(1/2)*d*sqrt((a^2 - 2*a*b + b^2)/d^2)*arctan(-(2*sqrt(1/2)*(a + b)*d*sqrt((a^2 - 2*a*b + b^2)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^2 + 2*a*b + b^2)/d^2)*sqrt((a^2 - 2*a*b + b^2)/d^2))/(a^2 - b^2)) + 30*sqrt(1 /2)*d*sqrt((a^2 - 2*a*b + b^2)/d^2)*arctan(-(2*sqrt(1/2)*(a + b)*d*sqrt((a ^2 - 2*a*b + b^2)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt((a^2 + 2*a*b + b^2)/d ^2)*sqrt((a^2 - 2*a*b + b^2)/d^2))/(a^2 - b^2)) + 15*sqrt(1/2)*d*sqrt((a^2 + 2*a*b + b^2)/d^2)*log(2*sqrt(1/2)*d*sqrt((a^2 + 2*a*b + b^2)/d^2)*sqrt( tan(d*x + c)) + (a + b)*tan(d*x + c) + a + b) - 15*sqrt(1/2)*d*sqrt((a^2 + 2*a*b + b^2)/d^2)*log(-2*sqrt(1/2)*d*sqrt((a^2 + 2*a*b + b^2)/d^2)*sqrt(t an(d*x + c)) + (a + b)*tan(d*x + c) + a + b) + 4*(3*b*tan(d*x + c)^3 + 5*a *tan(d*x + c)^2 - 15*b*tan(d*x + c))/sqrt(tan(d*x + c)))/d
\[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {a + b \tan {\left (c + d x \right )}}{\cot ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*tan(d*x+c))/cot(d*x+c)**(5/2),x)
Output:
Integral((a + b*tan(c + d*x))/cot(c + d*x)**(5/2), x)
Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\frac {8 \, {\left (3 \, b + \frac {5 \, a}{\tan \left (d x + c\right )} - \frac {15 \, b}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 30 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt {2} {\left (a + b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt {2} {\left (a + b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{60 \, d} \] Input:
integrate((a+b*tan(d*x+c))/cot(d*x+c)^(5/2),x, algorithm="maxima")
Output:
1/60*(8*(3*b + 5*a/tan(d*x + c) - 15*b/tan(d*x + c)^2)*tan(d*x + c)^(5/2) + 30*sqrt(2)*(a - b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 30*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 15*sqrt(2)*(a + b)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 15*sqrt(2)*(a + b)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1 ))/d
\[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {b \tan \left (d x + c\right ) + a}{\cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))/cot(d*x+c)^(5/2),x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)/cot(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {a+b\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*tan(c + d*x))/cot(c + d*x)^(5/2),x)
Output:
int((a + b*tan(c + d*x))/cot(c + d*x)^(5/2), x)
\[ \int \frac {a+b \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{3}}d x \right ) a +\left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )}{\cot \left (d x +c \right )^{3}}d x \right ) b \] Input:
int((a+b*tan(d*x+c))/cot(d*x+c)^(5/2),x)
Output:
int(sqrt(cot(c + d*x))/cot(c + d*x)**3,x)*a + int((sqrt(cot(c + d*x))*tan( c + d*x))/cot(c + d*x)**3,x)*b