Integrand size = 21, antiderivative size = 141 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {(a+b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d} \] 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*b*cot(d*x+c)^(1/2)/d-2/3*a*cot(d*x+c)^(3 /2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.46 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {2 \sqrt {\cot (c+d x)} \left (a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )+3 b \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right )\right )}{3 d} \] Input:
Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x]),x]
Output:
(-2*Sqrt[Cot[c + d*x]]*(a*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Ta n[c + d*x]^2] + 3*b*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]))/(3* d)
Time = 0.62 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.28, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 4156, 3042, 4011, 3042, 4011, 3042, 4017, 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.
| \(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{5/2} (a+b \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} (b \cot (c+d x)-a)dx-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {-b-a \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \tan \left (c+d x+\frac {\pi }{2}\right )-b}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int \frac {b+a \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a-b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 b \sqrt {\cot (c+d x)}}{d}\) |
Input:
Int[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x]),x]
Output:
(-2*b*Sqrt[Cot[c + d*x]])/d - (2*a*Cot[c + d*x]^(3/2))/(3*d) + (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]))/2 - ((a - b)*(-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]] + C ot[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[(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. \(272\) vs. \(2(117)=234\).
Time = 0.25 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.94
| method | result | size |
| derivativedivides | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 a \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+6 a \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+6 b \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+6 a \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+6 b \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+3 b \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+24 b \tan \left (d x +c \right )+8 a \right )}{12 d}\) | \(273\) |
| default | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 a \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+6 a \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+6 b \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+6 a \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+6 b \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+3 b \sqrt {2}\, \tan \left (d x +c \right )^{\frac {3}{2}} \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+24 b \tan \left (d x +c \right )+8 a \right )}{12 d}\) | \(273\) |
Input:
int(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/12/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(3*a*2^(1/2)*tan(d*x+c)^(3/2)*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))+6*a*2^(1/2)*tan(d*x+c)^(3/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+6*b *2^(1/2)*tan(d*x+c)^(3/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+6*a*2^(1/2)*t an(d*x+c)^(3/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))+6*b*2^(1/2)*tan(d*x+c) ^(3/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))+3*b*2^(1/2)*tan(d*x+c)^(3/2)*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))+24*b*tan(d*x+c)+8*a)
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (117) = 234\).
Time = 0.12 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.04 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {6 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (a - b\right )} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) \tan \left (d x + c\right ) + 6 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (a - b\right )} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} - d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) \tan \left (d x + c\right ) - 3 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \log \left (2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} - {\left (a - b\right )} \tan \left (d x + c\right ) - a + b\right ) \tan \left (d x + c\right ) + 3 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \log \left (-2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} - {\left (a - b\right )} \tan \left (d x + c\right ) - a + b\right ) \tan \left (d x + c\right ) + \frac {4 \, {\left (3 \, b \tan \left (d x + c\right ) + a\right )}}{\sqrt {\tan \left (d x + c\right )}}}{6 \, d \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
-1/6*(6*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))*tan(d*x + c) + 6*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))*tan(d*x + c) - 3* 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)*tan(d*x + c) + 3*sqrt(1/2)*d*sqrt((a^2 - 2*a*b + b^2)/d^2)*log(-2*sqrt(1/2)*d*sqr t((a^2 - 2*a*b + b^2)/d^2)*sqrt(tan(d*x + c)) - (a - b)*tan(d*x + c) - a + b)*tan(d*x + c) + 4*(3*b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(d*tan(d*x + c))
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \cot ^{\frac {5}{2}}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**(5/2)*(a+b*tan(d*x+c)),x)
Output:
Integral((a + b*tan(c + d*x))*cot(c + d*x)**(5/2), x)
Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {6 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left (a - b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left (a - b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {24 \, b}{\sqrt {\tan \left (d x + c\right )}} - \frac {8 \, a}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
1/12*(6*sqrt(2)*(a + b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)) )) + 6*sqrt(2)*(a + b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)) )) - 3*sqrt(2)*(a - b)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1 ) + 3*sqrt(2)*(a - b)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1 ) - 24*b/sqrt(tan(d*x + c)) - 8*a/tan(d*x + c)^(3/2))/d
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)*cot(d*x + c)^(5/2), x)
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right ) \,d x \] Input:
int(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x)),x)
Output:
int(cot(c + d*x)^(5/2)*(a + b*tan(c + d*x)), x)
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) a \] Input:
int(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c)),x)
Output:
int(sqrt(cot(c + d*x))*cot(c + d*x)**2*tan(c + d*x),x)*b + int(sqrt(cot(c + d*x))*cot(c + d*x)**2,x)*a