Integrand size = 36, antiderivative size = 261 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {3 a^{5/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \] Output:
-3/64*a^(5/2)*(121*A-120*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d+ 4*2^(1/2)*a^(5/2)*(A-I*B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^( 1/2))/d+1/64*a^2*(149*I*A+152*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d+1/9 6*a^2*(107*A-104*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d-1/24*a^2*(11 *I*A+8*B)*cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/2)/d-1/4*a*A*cot(d*x+c)^4*(a+ I*a*tan(d*x+c))^(3/2)/d
Time = 3.01 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.68 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {-9 a^{5/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+768 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+a^2 \cot (c+d x) \left (447 i A+456 B+(214 A-208 i B) \cot (c+d x)+(-136 i A-64 B) \cot ^2(c+d x)-48 A \cot ^3(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}}{192 d} \] Input:
Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]) ,x]
Output:
(-9*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a] ] + 768*Sqrt[2]*a^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt [2]*Sqrt[a])] + a^2*Cot[c + d*x]*((447*I)*A + 456*B + (214*A - (208*I)*B)* Cot[c + d*x] + ((-136*I)*A - 64*B)*Cot[c + d*x]^2 - 48*A*Cot[c + d*x]^3)*S qrt[a + I*a*Tan[c + d*x]])/(192*d)
Time = 1.91 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.09, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4076, 27, 3042, 4076, 27, 3042, 4081, 27, 3042, 4081, 27, 3042, 4083, 3042, 3961, 219, 4082, 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.
| \(\displaystyle \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \cot ^4(c+d x) (i \tan (c+d x) a+a)^{3/2} (a (11 i A+8 B)-a (5 A-8 i B) \tan (c+d x))dx-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \cot ^4(c+d x) (i \tan (c+d x) a+a)^{3/2} (a (11 i A+8 B)-a (5 A-8 i B) \tan (c+d x))dx-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {(i \tan (c+d x) a+a)^{3/2} (a (11 i A+8 B)-a (5 A-8 i B) \tan (c+d x))}{\tan (c+d x)^4}dx-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int -\frac {1}{2} \cot ^3(c+d x) \sqrt {i \tan (c+d x) a+a} \left ((107 A-104 i B) a^2+(85 i A+88 B) \tan (c+d x) a^2\right )dx-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (-\frac {1}{6} \int \cot ^3(c+d x) \sqrt {i \tan (c+d x) a+a} \left ((107 A-104 i B) a^2+(85 i A+88 B) \tan (c+d x) a^2\right )dx-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (-\frac {1}{6} \int \frac {\sqrt {i \tan (c+d x) a+a} \left ((107 A-104 i B) a^2+(85 i A+88 B) \tan (c+d x) a^2\right )}{\tan (c+d x)^3}dx-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {\int \frac {3}{2} \cot ^2(c+d x) \sqrt {i \tan (c+d x) a+a} \left (a^3 (149 i A+152 B)-a^3 (107 A-104 i B) \tan (c+d x)\right )dx}{2 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \int \cot ^2(c+d x) \sqrt {i \tan (c+d x) a+a} \left (a^3 (149 i A+152 B)-a^3 (107 A-104 i B) \tan (c+d x)\right )dx}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (a^3 (149 i A+152 B)-a^3 (107 A-104 i B) \tan (c+d x)\right )}{\tan (c+d x)^2}dx}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (\frac {\int -\frac {1}{2} \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (3 (121 A-120 i B) a^4+(149 i A+152 B) \tan (c+d x) a^4\right )dx}{a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\int \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (3 (121 A-120 i B) a^4+(149 i A+152 B) \tan (c+d x) a^4\right )dx}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 (121 A-120 i B) a^4+(149 i A+152 B) \tan (c+d x) a^4\right )}{\tan (c+d x)}dx}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4083 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {512 a^4 (B+i A) \int \sqrt {i \tan (c+d x) a+a}dx+3 a^3 (121 A-120 i B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {512 a^4 (B+i A) \int \sqrt {i \tan (c+d x) a+a}dx+3 a^3 (121 A-120 i B) \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3961 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {3 a^3 (121 A-120 i B) \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx-\frac {1024 i a^5 (B+i A) \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {3 a^3 (121 A-120 i B) \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx-\frac {512 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\frac {3 a^5 (121 A-120 i B) \int \frac {\cot (c+d x)}{\sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}-\frac {512 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {-\frac {6 i a^4 (121 A-120 i B) \int \frac {1}{i-\frac {i (i \tan (c+d x) a+a)}{a}}d\sqrt {i \tan (c+d x) a+a}}{d}-\frac {512 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {-\frac {6 a^{9/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {512 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {a^3 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}\) |
Input:
Int[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
Output:
-1/4*(a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2))/d + (-1/3*(a^2*((11 *I)*A + 8*B)*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/d + ((a^2*(107*A - (104*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d) - (3*(-1/2*(( -6*a^(9/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] )/d - ((512*I)*Sqrt[2]*a^(9/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x] ]/(Sqrt[2]*Sqrt[a])])/d)/a - (a^3*((149*I)*A + 152*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d))/(4*a))/6)/8
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[(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_.)*(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[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a , b, c, d}, x] && 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] :> Sim p[(-a^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Simp[a/(d*(b*c + a*d)*(n + 1)) Int[ (a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b *d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, 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] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[( A*b + a*B)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[(B*c - A *d)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*T an[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Time = 0.31 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {2 a^{5} \left (-\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {5}{2}}}-\frac {\frac {\left (-\frac {19 i B}{16}+\frac {149 A}{128}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}+\left (\frac {145}{48} i a B -\frac {1127}{384} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-\frac {127}{48} i B \,a^{2}+\frac {1049}{384} A \,a^{2}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {13}{16} i B \,a^{3}-\frac {107}{128} A \,a^{3}\right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{4} \tan \left (d x +c \right )^{4}}+\frac {3 \left (-120 i B +121 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{2}}\right )}{d}\) | \(206\) |
| default | \(\frac {2 a^{5} \left (-\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {5}{2}}}-\frac {\frac {\left (-\frac {19 i B}{16}+\frac {149 A}{128}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}+\left (\frac {145}{48} i a B -\frac {1127}{384} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-\frac {127}{48} i B \,a^{2}+\frac {1049}{384} A \,a^{2}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {13}{16} i B \,a^{3}-\frac {107}{128} A \,a^{3}\right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{4} \tan \left (d x +c \right )^{4}}+\frac {3 \left (-120 i B +121 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{2}}\right )}{d}\) | \(206\) |
Input:
int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETUR NVERBOSE)
Output:
2/d*a^5*(-1/2*(4*I*B-4*A)/a^(5/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^( 1/2)*2^(1/2)/a^(1/2))-1/a^2*(((-19/16*I*B+149/128*A)*(a+I*a*tan(d*x+c))^(7 /2)+(145/48*I*a*B-1127/384*a*A)*(a+I*a*tan(d*x+c))^(5/2)+(-127/48*I*B*a^2+ 1049/384*A*a^2)*(a+I*a*tan(d*x+c))^(3/2)+(13/16*I*B*a^3-107/128*A*a^3)*(a+ I*a*tan(d*x+c))^(1/2))/a^4/tan(d*x+c)^4+3/128*(121*A-120*I*B)/a^(1/2)*arct anh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (206) = 412\).
Time = 0.12 (sec) , antiderivative size = 944, normalized size of antiderivative = 3.62 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algori thm="fricas")
Output:
1/768*(1536*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8* I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(4*((-I*A - B)*a^3*e^(I*d*x + I*c) - sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c ) + 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a^2)) - 1536*sqrt(2)*sqrt((A^2 - 2*I *A*B - B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6* d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(4*((-I*A - B)*a^3 *e^(I*d*x + I*c) - sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a^2)) + 9*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(d*e^(8* I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e ^(2*I*d*x + 2*I*c) + d)*log(16*(3*(-121*I*A - 120*B)*a^3*e^(2*I*d*x + 2*I* c) + (-121*I*A - 120*B)*a^3 + 2*sqrt(2)*sqrt((14641*A^2 - 29040*I*A*B - 14 400*B^2)*a^5/d^2)*(I*d*e^(3*I*d*x + 3*I*c) + I*d*e^(I*d*x + I*c))*sqrt(a/( e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((-121*I*A - 120*B)*a)) - 9*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I* c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(16*(3*(-121*I*A - 120*B)*a^3*e^(2*I*d*x + 2*I*c) + (-121*I *A - 120*B)*a^3 + 2*sqrt(2)*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5 /d^2)*(-I*d*e^(3*I*d*x + 3*I*c) - I*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d...
Timed out. \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
Output:
Timed out
Time = 0.19 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.12 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {a^{4} {\left (\frac {768 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {9 \, {\left (121 \, A - 120 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} {\left (149 \, A - 152 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (1127 \, A - 1160 i \, B\right )} a + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (1049 \, A - 1016 i \, B\right )} a^{2} - 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (107 \, A - 104 i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} + 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + a^{5}}\right )}}{384 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algori thm="maxima")
Output:
-1/384*a^4*(768*sqrt(2)*(A - I*B)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a)))/a^(3/2) - 9*(1 21*A - 120*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d *x + c) + a) + sqrt(a)))/a^(3/2) + 2*(3*(I*a*tan(d*x + c) + a)^(7/2)*(149* A - 152*I*B) - (I*a*tan(d*x + c) + a)^(5/2)*(1127*A - 1160*I*B)*a + (I*a*t an(d*x + c) + a)^(3/2)*(1049*A - 1016*I*B)*a^2 - 3*sqrt(I*a*tan(d*x + c) + a)*(107*A - 104*I*B)*a^3)/((I*a*tan(d*x + c) + a)^4*a - 4*(I*a*tan(d*x + c) + a)^3*a^2 + 6*(I*a*tan(d*x + c) + a)^2*a^3 - 4*(I*a*tan(d*x + c) + a)* a^4 + a^5))/d
Exception generated. \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algori thm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Ar gument Ty
Time = 5.36 (sec) , antiderivative size = 3094, normalized size of antiderivative = 11.85 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^5*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)
Output:
(((107*A*a^6 - B*a^6*104i)*(a + a*tan(c + d*x)*1i)^(1/2))/(64*d) - ((149*A *a^3 - B*a^3*152i)*(a + a*tan(c + d*x)*1i)^(7/2))/(64*d) - ((1049*A*a^5 - B*a^5*1016i)*(a + a*tan(c + d*x)*1i)^(3/2))/(192*d) + ((1127*A*a^4 - B*a^4 *1160i)*(a + a*tan(c + d*x)*1i)^(5/2))/(192*d))/((a + a*tan(c + d*x)*1i)^4 - 4*a^3*(a + a*tan(c + d*x)*1i) - 4*a*(a + a*tan(c + d*x)*1i)^3 + 6*a^2*( a + a*tan(c + d*x)*1i)^2 + a^4) - 2*atanh((384*d^4*(a + a*tan(c + d*x)*1i) ^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229 *A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*341 53i)/(8192*d^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2 *a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*((485809*A^4*a^22)/(2 62144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + ( A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2))/((431 443*A^3*a^14*d)/256 - B^3*a^14*d*3542i + (21783*A*B^2*a^14*d)/4 + (A^2*B*a ^14*d*6993i)/32 + 214*A*a^3*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4 *a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(12 8*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) - B*a^3*d^3*((485809*A^4*a^ 22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^ 4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)* 208i) + (697*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22) /(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^...
\[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )^{3}d x \right ) b -\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )^{2}d x \right ) a +2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )^{2}d x \right ) b i +2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )d x \right ) a i +\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cot \left (d x +c \right )^{5} \tan \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \cot \left (d x +c \right )^{5}d x \right ) a \right ) \] Input:
int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)
Output:
sqrt(a)*a**2*( - int(sqrt(tan(c + d*x)*i + 1)*cot(c + d*x)**5*tan(c + d*x) **3,x)*b - int(sqrt(tan(c + d*x)*i + 1)*cot(c + d*x)**5*tan(c + d*x)**2,x) *a + 2*int(sqrt(tan(c + d*x)*i + 1)*cot(c + d*x)**5*tan(c + d*x)**2,x)*b*i + 2*int(sqrt(tan(c + d*x)*i + 1)*cot(c + d*x)**5*tan(c + d*x),x)*a*i + in t(sqrt(tan(c + d*x)*i + 1)*cot(c + d*x)**5*tan(c + d*x),x)*b + int(sqrt(ta n(c + d*x)*i + 1)*cot(c + d*x)**5,x)*a)