Integrand size = 23, antiderivative size = 216 \[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}} \] 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+2/7*b^2/d/co t(d*x+c)^(7/2)+4/5*a*b/d/cot(d*x+c)^(5/2)+2/3*(a^2-b^2)/d/cot(d*x+c)^(3/2) -4*a*b/d/cot(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.37 \[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 \left (-5 \left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},1,-\frac {3}{4},-\cot ^2(c+d x)\right )+a \left (5 a+14 b \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right )}{35 d \cot ^{\frac {7}{2}}(c+d x)} \] Input:
Integrate[(a + b*Tan[c + d*x])^2/Cot[c + d*x]^(5/2),x]
Output:
(2*(-5*(a^2 - b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2] + a*( 5*a + 14*b*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2]) ))/(35*d*Cot[c + d*x]^(7/2))
Time = 0.99 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.14, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4156, 3042, 4025, 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.
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\cot (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+b)^2}{\cot ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \int \frac {2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac {7}{2}}(c+d x)}dx+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int \frac {a^2-2 b \cot (c+d x) a-b^2}{\cot ^{\frac {5}{2}}(c+d x)}dx+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int -\frac {2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {2 a b-\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int \frac {a^2-2 b \cot (c+d x) a-b^2}{\sqrt {\cot (c+d x)}}dx+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a^2+2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {2 \int -\frac {a^2-2 b \cot (c+d x) a-b^2}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int \frac {a^2-2 b \cot (c+d x) a-b^2}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[(a + b*Tan[c + d*x])^2/Cot[c + d*x]^(5/2),x]
Output:
(2*b^2)/(7*d*Cot[c + d*x]^(7/2)) + (4*a*b)/(5*d*Cot[c + d*x]^(5/2)) + (2*( a^2 - b^2))/(3*d*Cot[c + d*x]^(3/2)) - (4*a*b)/(d*Sqrt[Cot[c + d*x]]) - (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] + Lo g[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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*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^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^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 = 0.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(-\frac {\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}-\frac {2 b^{2}}{7 \cot \left (d x +c \right )^{\frac {7}{2}}}-\frac {2 \left (a^{2}-b^{2}\right )}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 a b}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 a b}{\sqrt {\cot \left (d x +c \right )}}}{d}\) | \(244\) |
| default | \(-\frac {\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}-\frac {2 b^{2}}{7 \cot \left (d x +c \right )^{\frac {7}{2}}}-\frac {2 \left (a^{2}-b^{2}\right )}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 a b}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 a b}{\sqrt {\cot \left (d x +c \right )}}}{d}\) | \(244\) |
Input:
int((a+b*tan(d*x+c))^2/cot(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/d*(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*arc tan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))-2/7 *b^2/cot(d*x+c)^(7/2)-2/3*(a^2-b^2)/cot(d*x+c)^(3/2)-4/5*a*b/cot(d*x+c)^(5 /2)+4*a*b/cot(d*x+c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (185) = 370\).
Time = 0.11 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.21 \[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^2/cot(d*x+c)^(5/2),x, algorithm="fricas")
Output:
1/210*(210*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)) + 210*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)) - 105 *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*s qrt(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)) + 105*sq rt(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*sqr t(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)) + 4*(15*b^ 2*tan(d*x + c)^4 + 42*a*b*tan(d*x + c)^3 - 210*a*b*tan(d*x + c) + 35*(a^2 - b^2)*tan(d*x + c)^2)/sqrt(tan(d*x + c)))/d
\[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{2}}{\cot ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**2/cot(d*x+c)**(5/2),x)
Output:
Integral((a + b*tan(c + d*x))**2/cot(c + d*x)**(5/2), x)
Time = 0.12 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\frac {8 \, {\left (15 \, b^{2} + \frac {42 \, a b}{\tan \left (d x + c\right )} - \frac {210 \, a b}{\tan \left (d x + c\right )^{3}} + \frac {35 \, {\left (a^{2} - b^{2}\right )}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 210 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 105 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 105 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{420 \, d} \] Input:
integrate((a+b*tan(d*x+c))^2/cot(d*x+c)^(5/2),x, algorithm="maxima")
Output:
1/420*(8*(15*b^2 + 42*a*b/tan(d*x + c) - 210*a*b/tan(d*x + c)^3 + 35*(a^2 - b^2)/tan(d*x + c)^2)*tan(d*x + c)^(7/2) + 210*sqrt(2)*(a^2 - 2*a*b - b^2 )*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 210*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 105 *sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 105*sqrt(2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/d
\[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}{\cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^2/cot(d*x+c)^(5/2),x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)^2/cot(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*tan(c + d*x))^2/cot(c + d*x)^(5/2),x)
Output:
int((a + b*tan(c + d*x))^2/cot(c + d*x)^(5/2), x)
\[ \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{3}}d x \right ) a^{2}+\left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}}{\cot \left (d x +c \right )^{3}}d x \right ) b^{2}+2 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )}{\cot \left (d x +c \right )^{3}}d x \right ) a b \] Input:
int((a+b*tan(d*x+c))^2/cot(d*x+c)^(5/2),x)
Output:
int(sqrt(cot(c + d*x))/cot(c + d*x)**3,x)*a**2 + int((sqrt(cot(c + d*x))*t an(c + d*x)**2)/cot(c + d*x)**3,x)*b**2 + 2*int((sqrt(cot(c + d*x))*tan(c + d*x))/cot(c + d*x)**3,x)*a*b