Integrand size = 33, antiderivative size = 219 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {(i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {b \left (a^2 A+3 A b^2-2 a b B\right )}{a^2 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}} \]
(3*A*b-2*B*a)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d+(I*A+B)*ar ctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d-(I*A-B)*arctan h((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d-b*(A*a^2+3*A*b^2-2 *B*a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^(1/2)-A*cot(d*x+c)/a/d/(a+b*tan(d *x+c))^(1/2)
Time = 4.17 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+a^2 \left (\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2}}+\frac {(-i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2}}\right )-\frac {b \left (a^2 A+3 A b^2-2 a b B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {a A \cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}}{a^2 d} \]
(((3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + a^2 *(((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(a - I*b)^(3 /2) + (((-I)*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(a + I*b)^(3/2)) - (b*(a^2*A + 3*A*b^2 - 2*a*b*B))/((a^2 + b^2)*Sqrt[a + b*Tan[ c + d*x]]) - (a*A*Cot[c + d*x])/Sqrt[a + b*Tan[c + d*x]])/(a^2*d)
Time = 1.82 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.16, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4092, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^2 (a+b \tan (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle -\frac {\int \frac {\cot (c+d x) \left (3 A b \tan ^2(c+d x)+2 a A \tan (c+d x)+3 A b-2 a B\right )}{2 (a+b \tan (c+d x))^{3/2}}dx}{a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cot (c+d x) \left (3 A b \tan ^2(c+d x)+2 a A \tan (c+d x)+3 A b-2 a B\right )}{(a+b \tan (c+d x))^{3/2}}dx}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {3 A b \tan (c+d x)^2+2 a A \tan (c+d x)+3 A b-2 a B}{\tan (c+d x) (a+b \tan (c+d x))^{3/2}}dx}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {2 \int \frac {\cot (c+d x) \left (2 (a A+b B) \tan (c+d x) a^2+b \left (A a^2-2 b B a+3 A b^2\right ) \tan ^2(c+d x)+\left (a^2+b^2\right ) (3 A b-2 a B)\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}+\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\cot (c+d x) \left (2 (a A+b B) \tan (c+d x) a^2+b \left (A a^2-2 b B a+3 A b^2\right ) \tan ^2(c+d x)+\left (a^2+b^2\right ) (3 A b-2 a B)\right )}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}+\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {2 (a A+b B) \tan (c+d x) a^2+b \left (A a^2-2 b B a+3 A b^2\right ) \tan (c+d x)^2+\left (a^2+b^2\right ) (3 A b-2 a B)}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}+\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle -\frac {\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+\int \frac {2 \left (a^2 (a A+b B)-a^2 (A b-a B) \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}+\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+2 \int \frac {a^2 (a A+b B)-a^2 (A b-a B) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}+\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \int \frac {a^2 (a A+b B)-a^2 (A b-a B) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}+\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {1}{2} a^2 (a-i b) (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^2 (a+i b) (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {1}{2} a^2 (a-i b) (A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^2 (a+i b) (A-i B) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {i a^2 (a+i b) (A-i B) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^2 (a-i b) (A+i B) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {i a^2 (a-i b) (A+i B) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i a^2 (a+i b) (A-i B) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {a^2 (a-i b) (A+i B) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {a^2 (a+i b) (A-i B) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+2 \left (\frac {a^2 (a+i b) (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^2 (a-i b) (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {\left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+2 \left (\frac {a^2 (a+i b) (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^2 (a-i b) (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (a^2+b^2\right ) (3 A b-2 a B) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+2 \left (\frac {a^2 (a+i b) (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^2 (a-i b) (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {A \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2 A-2 a b B+3 A b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 \left (a^2+b^2\right ) (3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+2 \left (\frac {a^2 (a+i b) (A-i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^2 (a-i b) (A+i B) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}\) |
-((A*Cot[c + d*x])/(a*d*Sqrt[a + b*Tan[c + d*x]])) - ((2*((a^2*(a + I*b)*( A - I*B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + (a^2*(a - I*b)*(A + I*B)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - ( 2*(a^2 + b^2)*(3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/( Sqrt[a]*d))/(a*(a^2 + b^2)) + (2*b*(a^2*A + 3*A*b^2 - 2*a*b*B))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/(2*a)
3.4.55.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1) /(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^ 2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(8042\) vs. \(2(191)=382\).
Time = 0.24 (sec) , antiderivative size = 8043, normalized size of antiderivative = 36.73
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(8043\) |
| default | \(\text {Expression too large to display}\) | \(8043\) |
Leaf count of result is larger than twice the leaf count of optimal. 4530 vs. \(2 (186) = 372\).
Time = 17.37 (sec) , antiderivative size = 9075, normalized size of antiderivative = 41.44 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{2}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Time = 12.65 (sec) , antiderivative size = 38368, normalized size of antiderivative = 175.20 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
((2*(A*b^3 - B*a*b^2))/(a*b^2 + a^3) - ((a + b*tan(c + d*x))*(3*A*b^3 + A* a^2*b - 2*B*a*b^2))/(a*(a*b^2 + a^3)))/(d*(a + b*tan(c + d*x))^(3/2) - a*d *(a + b*tan(c + d*x))^(1/2)) + atan(-((((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 - 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 + 48*A*B*a^2*b*d^2)^ 2/4 - (A^4 + 2*A^2*B^2 + B^4)*(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 4*A^2*a^3*d^2 + 4*B^2*a^3*d^2 + 8*A*B*b^3*d^2 + 1 2*A^2*a*b^2*d^2 - 12*B^2*a*b^2*d^2 - 24*A*B*a^2*b*d^2)/(16*(a^6*d^4 + b^6* d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(((a + b*tan(c + d*x))^(1/2)* (576*A^2*a^15*b^28*d^7 + 5184*A^2*a^17*b^26*d^7 + 21568*A^2*a^19*b^24*d^7 + 53888*A^2*a^21*b^22*d^7 + 87808*A^2*a^23*b^20*d^7 + 94976*A^2*a^25*b^18* d^7 + 66304*A^2*a^27*b^16*d^7 + 27008*A^2*a^29*b^14*d^7 + 4288*A^2*a^31*b^ 12*d^7 - 832*A^2*a^33*b^10*d^7 - 320*A^2*a^35*b^8*d^7 + 256*B^2*a^17*b^26* d^7 + 1472*B^2*a^19*b^24*d^7 + 3712*B^2*a^21*b^22*d^7 + 6272*B^2*a^23*b^20 *d^7 + 9856*B^2*a^25*b^18*d^7 + 14336*B^2*a^27*b^16*d^7 + 15232*B^2*a^29*b ^14*d^7 + 10112*B^2*a^31*b^12*d^7 + 3712*B^2*a^33*b^10*d^7 + 576*B^2*a^35* b^8*d^7 - 768*A*B*a^16*b^27*d^7 - 6400*A*B*a^18*b^25*d^7 - 25856*A*B*a^20* b^23*d^7 - 66304*A*B*a^22*b^21*d^7 - 116480*A*B*a^24*b^19*d^7 - 141568*A*B *a^26*b^17*d^7 - 116480*A*B*a^28*b^15*d^7 - 61696*A*B*a^30*b^13*d^7 - 1894 4*A*B*a^32*b^11*d^7 - 2560*A*B*a^34*b^9*d^7) - ((((8*A^2*a^3*d^2 - 8*B^2*a ^3*d^2 - 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 + 48*A*B*...