Integrand size = 25, antiderivative size = 264 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (5 a^2-b^2\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{5 a d}-\frac {4 b \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d}-\frac {2 a \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{5 d} \]
-I*(I*a-b)^(3/2)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1 /2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+I*(I*a+b)^(3/2)*arctanh((I*a+b)^( 1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^ (1/2)/d-4/5*b*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/d-2/5*a*cot(d*x+c)^( 5/2)*(a+b*tan(d*x+c))^(1/2)/d+2/5*(5*a^2-b^2)*cot(d*x+c)^(1/2)*(a+b*tan(d* x+c))^(1/2)/a/d
Time = 1.65 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.83 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\cot ^{\frac {5}{2}}(c+d x) \left (-5 \sqrt [4]{-1} a (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {5}{2}}(c+d x)+5 \sqrt [4]{-1} a (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {5}{2}}(c+d x)+2 \sqrt {a+b \tan (c+d x)} \left (-a^2-2 a b \tan (c+d x)+\left (5 a^2-b^2\right ) \tan ^2(c+d x)\right )\right )}{5 a d} \]
(Cot[c + d*x]^(5/2)*(-5*(-1)^(1/4)*a*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*S qrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^( 5/2) + 5*(-1)^(1/4)*a*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqr t[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-a^2 - 2*a*b*Tan[c + d*x] + (5*a^2 - b^2)*Tan[c + d*x]^2 )))/(5*a*d)
Time = 1.64 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.06, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 4729, 3042, 4050, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 216, 219}
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.
| \(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{7/2} (a+b \tan (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {7}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan (c+d x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4050 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{5} \int -\frac {-4 a b \tan ^2(c+d x)-5 \left (a^2-b^2\right ) \tan (c+d x)+6 a b}{2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \int \frac {-4 a b \tan ^2(c+d x)-5 \left (a^2-b^2\right ) \tan (c+d x)+6 a b}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \int \frac {-4 a b \tan (c+d x)^2-5 \left (a^2-b^2\right ) \tan (c+d x)+6 a b}{\tan (c+d x)^{5/2} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \left (-\frac {2 \int \frac {3 \left (10 b \tan (c+d x) a^2+4 b^2 \tan ^2(c+d x) a+\left (5 a^2-b^2\right ) a\right )}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \left (-\frac {\int \frac {10 b \tan (c+d x) a^2+4 b^2 \tan ^2(c+d x) a+\left (5 a^2-b^2\right ) a}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \left (-\frac {\int \frac {10 b \tan (c+d x) a^2+4 b^2 \tan (c+d x)^2 a+\left (5 a^2-b^2\right ) a}{\tan (c+d x)^{3/2} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \left (-\frac {-\frac {2 \int -\frac {5 \left (2 a^3 b-a^2 \left (a^2-b^2\right ) \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \left (-\frac {\frac {5 \int \frac {2 a^3 b-a^2 \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{5} \left (-\frac {\frac {5 \int \frac {2 a^3 b-a^2 \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{a}-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {5 \left (\frac {1}{2} i a^2 (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i a^2 (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {5 \left (\frac {1}{2} i a^2 (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i a^2 (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {5 \left (\frac {i a^2 (a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}-\frac {i a^2 (a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {5 \left (\frac {i a^2 (a-i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {i a^2 (a+i b)^2 \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {5 \left (\frac {i a^2 (a-i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {i a^2 (a+i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {4 b \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (5 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {5 \left (\frac {i a^2 (a-i b)^2 \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}-\frac {i a^2 (a+i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a}}{a}\right )\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*a*Sqrt[a + b*Tan[c + d*x]])/(5* d*Tan[c + d*x]^(5/2)) + ((-4*b*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^( 3/2)) - ((5*(((-I)*a^2*(a + I*b)^2*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x] ])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (I*a^2*(a - I*b)^2*ArcTa nh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)))/a - (2*(5*a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/a)/5)
3.9.47.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ (n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 *a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^ 2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 2*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[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
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[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
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[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(2794\) vs. \(2(216)=432\).
Time = 39.87 (sec) , antiderivative size = 2795, normalized size of antiderivative = 10.59
1/20/d*csc(d*x+c)*(-1/(1-cos(d*x+c))*(csc(d*x+c)*(1-cos(d*x+c))^2-sin(d*x+ c)))^(7/2)*(1-cos(d*x+c))*((csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c )-cot(d*x+c))-a)/(csc(d*x+c)^2*(1-cos(d*x+c))^2-1))^(1/2)*(csc(d*x+c)^4*a^ 2*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c) )-a)*(1-cos(d*x+c)))^(1/2)*(1-cos(d*x+c))^4*(-b+(a^2+b^2)^(1/2))^(1/2)-5*c sc(d*x+c)^3*ln(-1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+ c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c ))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1 -cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))*(a^2+b^2)^(1/2)*(b+(a^2+b^2 )^(1/2))^(1/2)*b*(1-cos(d*x+c))^3*(-b+(a^2+b^2)^(1/2))^(1/2)+5*csc(d*x+c)^ 3*ln(1/(1-cos(d*x+c))*(-csc(d*x+c)*a*(1-cos(d*x+c))^2+2*(a^2+b^2)^(1/2)*(1 -cos(d*x+c))+2*sin(d*x+c)*(-csc(d*x+c)*(csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2* b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x+c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/ 2)+2*b*(1-cos(d*x+c))+sin(d*x+c)*a))*(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2))^( 1/2)*b*(1-cos(d*x+c))^3*(-b+(a^2+b^2)^(1/2))^(1/2)-5*csc(d*x+c)^3*a^2*ln(- 1/(1-cos(d*x+c))*(csc(d*x+c)*a*(1-cos(d*x+c))^2+2*sin(d*x+c)*(-csc(d*x+c)* (csc(d*x+c)^2*a*(1-cos(d*x+c))^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*(1-cos(d*x +c)))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b *(1-cos(d*x+c))-sin(d*x+c)*a))*(b+(a^2+b^2)^(1/2))^(1/2)*(1-cos(d*x+c))^3* (-b+(a^2+b^2)^(1/2))^(1/2)+5*csc(d*x+c)^3*ln(-1/(1-cos(d*x+c))*(csc(d*x...
Leaf count of result is larger than twice the leaf count of optimal. 3666 vs. \(2 (210) = 420\).
Time = 0.60 (sec) , antiderivative size = 3666, normalized size of antiderivative = 13.89 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]
-1/40*(5*a*d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4) /d^4))/d^2)*log((((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5 *b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4) *d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2 *sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12 *a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2 *(a^3*b + 2*a*b^3)*d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan( d*x + c))*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 + 5*a*d*sqrt(( 3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(( (a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 1 2*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*ta n(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^ 4*b^2 + 9*a^2*b^4)/d^4))/d^2) + 2*((a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c)^2 + 2*(a^6*b - a^4*b^3 - 6*a^2*b^5)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3) *d^2*tan(d*x + c)^2 - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2*tan(d*x + c))*sqrt(-(a ^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1))*tan(d*x + c)^2 - 5*a*d*sqrt((3*a^2*b - b^3 +...
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2} \,d x \]