Integrand size = 21, antiderivative size = 124 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a \sqrt {\cot (c+d x)}}{d} \] 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*a*cot(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.23 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {\sqrt {\cot (c+d x)} \left (8 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right )+\sqrt {2} b \left (2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right ) \sqrt {\tan (c+d x)}\right )}{4 d} \] Input:
Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x]),x]
Output:
-1/4*(Sqrt[Cot[c + d*x]]*(8*a*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x ]^2] + Sqrt[2]*b*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 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]])*Sqrt[Tan[c + d*x]]))/d
Time = 0.52 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4156, 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.
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{3/2} (a+b \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {b \cot (c+d x)-a}{\sqrt {\cot (c+d x)}}dx-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {-a-b \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int \frac {a-b \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d}\) |
Input:
Int[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x]),x]
Output:
(-2*a*Sqrt[Cot[c + d*x]])/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]] + 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[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. \(262\) vs. \(2(105)=210\).
Time = 0.21 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.12
| method | result | size |
| derivativedivides | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (b \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )-2 a \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 b \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-2 a \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 b \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-a \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )-8 a \right )}{4 d}\) | \(263\) |
| default | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (b \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )-2 a \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 b \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-2 a \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 b \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-a \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )-8 a \right )}{4 d}\) | \(263\) |
Input:
int(cot(d*x+c)^(3/2)*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/4/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(b*2^(1/2)*tan(d*x+c)^(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 ))-2*a*2^(1/2)*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*b*2^( 1/2)*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))-2*a*2^(1/2)*tan(d *x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))+2*b*2^(1/2)*tan(d*x+c)^(1/ 2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-a*2^(1/2)*tan(d*x+c)^(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 ))-8*a)
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (105) = 210\).
Time = 0.09 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.01 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) + 2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} d \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} - d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) + \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \log \left (2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + {\left (a + b\right )} \tan \left (d x + c\right ) + a + b\right ) - \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \log \left (-2 \, \sqrt {\frac {1}{2}} d \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\tan \left (d x + c\right )} + {\left (a + b\right )} \tan \left (d x + c\right ) + a + b\right ) - \frac {4 \, a}{\sqrt {\tan \left (d x + c\right )}}}{2 \, d} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
1/2*(2*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)) + 2*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)) + 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) - 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) - 4*a/sqrt(tan(d*x + c)))/d
\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \cot ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**(3/2)*(a+b*tan(d*x+c)),x)
Output:
Integral((a + b*tan(c + d*x))*cot(c + d*x)**(3/2), x)
Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.12 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, a}{\sqrt {\tan \left (d x + c\right )}}}{4 \, d} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
1/4*(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)) + 1/tan(d*x + c) + 1) - sqrt(2)*(a + b)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 8 *a/sqrt(tan(d*x + c)))/d
\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(3/2)*(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)*cot(d*x + c)^(3/2), x)
Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right ) \,d x \] Input:
int(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x)),x)
Output:
int(cot(c + d*x)^(3/2)*(a + b*tan(c + d*x)), x)
\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {-2 \sqrt {\cot \left (d x +c \right )}\, a -\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a d +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right ) \tan \left (d x +c \right )d x \right ) b d}{d} \] Input:
int(cot(d*x+c)^(3/2)*(a+b*tan(d*x+c)),x)
Output:
( - 2*sqrt(cot(c + d*x))*a - int(sqrt(cot(c + d*x))/cot(c + d*x),x)*a*d + int(sqrt(cot(c + d*x))*cot(c + d*x)*tan(c + d*x),x)*b*d)/d