Integrand size = 33, antiderivative size = 283 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b (A b+2 a B)}{d \sqrt {\cot (c+d x)}}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
2/3*b^2*B/d/cot(d*x+c)^(3/2)-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)-1/2*(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))*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^ (1/2)+2*b*(A*b+2*B*a)/d/cot(d*x+c)^(1/2)
Time = 0.61 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.80 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (6 \sqrt {2} \left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )+3 \sqrt {2} \left (a^2 (A-B)+b^2 (-A+B)-2 a b (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )-24 b (A b+2 a B) \sqrt {\tan (c+d x)}-8 b^2 B \tan ^{\frac {3}{2}}(c+d x)\right )}{12 d} \]
-1/12*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[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]]) - 24*b*(A*b + 2*a*B)*Sqrt [Tan[c + d*x]] - 8*b^2*B*Tan[c + d*x]^(3/2)))/d
Time = 0.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.83, 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, 4087, 25, 3042, 4111, 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.
| \(\displaystyle \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+b)^2 (A \cot (c+d x)+B)}{\cot ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4087 |
| \(\displaystyle \frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\int -\frac {a^2 A \cot ^2(c+d x)+\left (B a^2+2 A b a-b^2 B\right ) \cot (c+d x)+b (A b+2 a B)}{\cot ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {a^2 A \cot ^2(c+d x)+\left (B a^2+2 A b a-b^2 B\right ) \cot (c+d x)+b (A b+2 a B)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2 A \tan \left (c+d x+\frac {\pi }{2}\right )^2-\left (B a^2+2 A b a-b^2 B\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b (A b+2 a B)}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \int \frac {B a^2+2 A b a-b^2 B+\left (a^2 A-b (A b+2 a B)\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B a^2+2 A b a-b^2 B-\left (a^2 A-b (A b+2 a B)\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int -\frac {B a^2+2 A b a-b^2 B+\left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \int \frac {B a^2+2 A b a-b^2 B+\left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b (2 a B+A b)}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 B}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
(2*b^2*B)/(3*d*Cot[c + d*x]^(3/2)) + (2*b*(A*b + 2*a*B))/(d*Sqrt[Cot[c + d *x]]) + (2*(-1/2*((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])) + ((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]*Sq rt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/d
3.6.81.3.1 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_)])^2*((A_.) + (B_.)*tan[(e_.) + (f _.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[ (-(B*c - A*d))*(b*c - a*d)^2*((c + d*Tan[e + f*x])^(n + 1)/(f*d^2*(n + 1)*( c^2 + d^2))), x] + Simp[1/(d*(c^2 + d^2)) Int[(c + d*Tan[e + f*x])^(n + 1 )*Simp[B*(b*c - a*d)^2 + A*d*(a^2*c - b^2*c + 2*a*b*d) + d*(B*(a^2*c - b^2* c + 2*a*b*d) + A*(2*a*b*c - a^2*d + b^2*d))*Tan[e + f*x] + b^2*B*(c^2 + d^2 )*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b *c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((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[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
Time = 0.40 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(-\frac {-\frac {2 B \,b^{2}}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (A b +2 B a \right )}{\sqrt {\cot \left (d x +c \right )}}+\frac {\left (2 a b A +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(244\) |
| default | \(-\frac {-\frac {2 B \,b^{2}}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (A b +2 B a \right )}{\sqrt {\cot \left (d x +c \right )}}+\frac {\left (2 a b A +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(244\) |
-1/d*(-2/3*B*b^2/cot(d*x+c)^(3/2)-2*b*(A*b+2*B*a)/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. 4281 vs. \(2 (249) = 498\).
Time = 2.33 (sec) , antiderivative size = 4281, normalized size of antiderivative = 15.13 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
-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...
\[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{2} \sqrt {\cot {\left (c + d x \right )}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.90 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {6 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 8 \, {\left (B b^{2} + \frac {3 \, {\left (2 \, B a b + A b^{2}\right )}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{12 \, d} \]
-1/12*(6*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*arctan(1/2*sq rt(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*(B*b^2 + 3*(2*B*a*b + A*b^2)/tan(d*x + c))*tan(d*x + c)^(3/2))/d
\[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \sqrt {\cot \left (d x + c\right )} \,d x } \]
Timed out. \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int \sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \]