Integrand size = 33, antiderivative size = 294 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a (5 A b+3 a B) \sqrt {\cot (c+d x)}}{3 d}-\frac {2 a A \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}{3 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/ d*2^(1/2)+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1+2^(1/2)*cot(d*x+c )^(1/2))/d*2^(1/2)+1/4*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*ln(1+cot(d*x+c)-2 ^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/4*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*l n(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-2/3*a*(5*A*b+3*B*a)*cot (d*x+c)^(1/2)/d-2/3*a*A*(b+a*cot(d*x+c))*cot(d*x+c)^(1/2)/d
Time = 1.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.77 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\sqrt {\cot (c+d x)} \left (6 \sqrt {2} \left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )+3 \sqrt {2} \left (a^2 (A-B)+b^2 (-A+B)-2 a b (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )-\frac {8 a^2 A}{\tan ^{\frac {3}{2}}(c+d x)}-\frac {24 a (2 A b+a B)}{\sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{12 d} \]
(Sqrt[Cot[c + d*x]]*(6*Sqrt[2]*(2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B)) *(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 3*Sqrt[2]*(a^2*(A - B) + b^2*(-A + B) - 2*a*b*(A + B))*(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]]) - (8*a^2*A)/Tan[c + d*x]^(3/2) - (24*a*(2*A*b + a* B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(12*d)
Time = 0.96 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.85, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4064, 3042, 4090, 27, 3042, 4113, 3042, 4017, 27, 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.
| \(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{5/2} (a+b \tan (c+d x))^2 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+b)^2 (A \cot (c+d x)+B)}{\sqrt {\cot (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4090 |
| \(\displaystyle -\frac {2}{3} \int \frac {-a (5 A b+3 a B) \cot ^2(c+d x)+3 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (a A-3 b B)}{2 \sqrt {\cot (c+d x)}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} \int \frac {-a (5 A b+3 a B) \cot ^2(c+d x)+3 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (a A-3 b B)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{3} \int \frac {-a (5 A b+3 a B) \tan \left (c+d x+\frac {\pi }{2}\right )^2-3 \left (A a^2-2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b (a A-3 b B)}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{3} \left (-\int \frac {3 \left (B a^2+2 A b a-b^2 B\right )+3 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (-\int \frac {3 \left (B a^2+2 A b a-b^2 B\right )-3 \left (A a^2-2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{3} \left (-\frac {2 \int -\frac {3 \left (B a^2+2 A b a-b^2 B+\left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \int \frac {B a^2+2 A b a-b^2 B+\left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 a (3 a B+5 A b) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}\) |
(-2*a*A*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x]))/(3*d) + ((-2*a*(5*A*b + 3 *a*B)*Sqrt[Cot[c + d*x]])/d + (6*(((2*a*b*(A - B) + a^2*(A + B) - b^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^2*(A - B) - b^2*(A - B) - 2*a*b*( 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)/3
3.6.79.3.1 Defintions of rubi rules used
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[((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_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp [g^(m + n) Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c *Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !Integer Q[p] && IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b *(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2 , 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Time = 0.70 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(-\frac {\frac {2 A \,a^{2} \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+4 a b A \sqrt {\cot \left (d x +c \right )}+2 B \,a^{2} \sqrt {\cot \left (d x +c \right )}+\frac {\left (-2 a b A -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}+2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(252\) |
| default | \(-\frac {\frac {2 A \,a^{2} \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+4 a b A \sqrt {\cot \left (d x +c \right )}+2 B \,a^{2} \sqrt {\cot \left (d x +c \right )}+\frac {\left (-2 a b A -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}+2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(252\) |
-1/d*(2/3*A*a^2*cot(d*x+c)^(3/2)+4*a*b*A*cot(d*x+c)^(1/2)+2*B*a^2*cot(d*x+ c)^(1/2)+1/4*(-2*A*a*b-B*a^2+B*b^2)*2^(1/2)*(ln((1+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)))+2*arctan(1+2^(1/2)* cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/4*(-A*a^2+A*b^2 +2*B*a*b)*2^(1/2)*(ln((1+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)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan (-1+2^(1/2)*cot(d*x+c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 4305 vs. \(2 (258) = 516\).
Time = 2.31 (sec) , antiderivative size = 4305, normalized size of antiderivative = 14.64 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
1/6*(3*d*sqrt(-(2*A*B*a^4 - 12*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3 *b - 4*(A^2 - B^2)*a*b^3 + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^ 3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B ^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2)*log(((B*a^2 + 2*A*a*b - B*b^2)*d^3*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4) + ((A^3 - A*B^2)*a^6 - 2*(5*A^2*B - B^3)*a^5*b - (7*A^3 - 23*A*B^2)*a^4*b^2 + 4*(7*A^2*B - 3*B^3)*a^3*b^3 + (7*A^3 - 23* A*B^2)*a^2*b^4 - 2*(5*A^2*B - B^3)*a*b^5 - (A^3 - A*B^2)*b^6)*d)*sqrt(-(2* A*B*a^4 - 12*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2) *a*b^3 + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2) - ((A^4 - B^4)*a^8 - 8*(A^3*B + A*B^3)*a ^7*b - 4*(A^4 - B^4)*a^6*b^2 - 8*(A^3*B + A*B^3)*a^5*b^3 - 10*(A^4 - B^...
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.86 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {6 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, A a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {24 \, {\left (B a^{2} + 2 \, A a b\right )}}{\sqrt {\tan \left (d x + c\right )}}}{12 \, d} \]
1/12*(6*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*arctan(1/2*sqr t(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 6*sqrt(2)*((A + B)*a^2 + 2*(A - B )*a*b - (A + B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 3*sqrt(2)*((A - B)*a^2 - 2*(A + B)*a*b - (A - B)*b^2)*log(sqrt(2)/sqrt( tan(d*x + c)) + 1/tan(d*x + c) + 1) + 3*sqrt(2)*((A - B)*a^2 - 2*(A + B)*a *b - (A - B)*b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 8*A*a^2/tan(d*x + c)^(3/2) - 24*(B*a^2 + 2*A*a*b)/sqrt(tan(d*x + c)))/d
\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \]