Integrand size = 36, antiderivative size = 268 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \] Output:
1/4*(23*A+12*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-1/4* (A-I*B)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^(3 /2)/d+1/3*(A+I*B)*cot(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(3/2)+1/6*(17*A+11*I*B )*cot(d*x+c)^2/a/d/(a+I*a*tan(d*x+c))^(1/2)+7/4*(3*I*A-2*B)*cot(d*x+c)*(a+ I*a*tan(d*x+c))^(1/2)/a^2/d-1/6*(22*A+13*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+ c))^(1/2)/a^2/d
Time = 3.51 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 (23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )-3 \sqrt {2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-\frac {\sqrt {a} \left (63 i A-42 B+(82 A+58 i B) \cot (c+d x)+3 (-3 i A+4 B) \cot ^2(c+d x)+6 A \cot ^3(c+d x)\right )}{(i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}}{12 a^{3/2} d} \] Input:
Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/ 2),x]
Output:
(3*(23*A + (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] - 3*Sqrt[ 2]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] - (Sqrt [a]*((63*I)*A - 42*B + (82*A + (58*I)*B)*Cot[c + d*x] + 3*((-3*I)*A + 4*B) *Cot[c + d*x]^2 + 6*A*Cot[c + d*x]^3))/((I + Cot[c + d*x])*Sqrt[a + I*a*Ta n[c + d*x]]))/(12*a^(3/2)*d)
Time = 1.88 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.07, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4079, 27, 3042, 4079, 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.
| \(\displaystyle \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^3 (a+i a \tan (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\int \frac {\cot ^3(c+d x) (2 a (5 A+2 i B)-7 a (i A-B) \tan (c+d x))}{2 \sqrt {i \tan (c+d x) a+a}}dx}{3 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot ^3(c+d x) (2 a (5 A+2 i B)-7 a (i A-B) \tan (c+d x))}{\sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a (5 A+2 i B)-7 a (i A-B) \tan (c+d x)}{\tan (c+d x)^3 \sqrt {i \tan (c+d x) a+a}}dx}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {1}{2} \cot ^3(c+d x) \sqrt {i \tan (c+d x) a+a} \left (4 a^2 (22 A+13 i B)-5 a^2 (17 i A-11 B) \tan (c+d x)\right )dx}{a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \cot ^3(c+d x) \sqrt {i \tan (c+d x) a+a} \left (4 a^2 (22 A+13 i B)-5 a^2 (17 i A-11 B) \tan (c+d x)\right )dx}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (4 a^2 (22 A+13 i B)-5 a^2 (17 i A-11 B) \tan (c+d x)\right )}{\tan (c+d x)^3}dx}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {\frac {\int -6 \cot ^2(c+d x) \sqrt {i \tan (c+d x) a+a} \left (7 (3 i A-2 B) a^3+(22 A+13 i B) \tan (c+d x) a^3\right )dx}{2 a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \cot ^2(c+d x) \sqrt {i \tan (c+d x) a+a} \left (7 (3 i A-2 B) a^3+(22 A+13 i B) \tan (c+d x) a^3\right )dx}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (7 (3 i A-2 B) a^3+(22 A+13 i B) \tan (c+d x) a^3\right )}{\tan (c+d x)^2}dx}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\int \frac {1}{2} \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (a^4 (23 A+12 i B)-7 a^4 (3 i A-2 B) \tan (c+d x)\right )dx}{a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\int \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (a^4 (23 A+12 i B)-7 a^4 (3 i A-2 B) \tan (c+d x)\right )dx}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (a^4 (23 A+12 i B)-7 a^4 (3 i A-2 B) \tan (c+d x)\right )}{\tan (c+d x)}dx}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4083 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {2 a^4 (B+i A) \int \sqrt {i \tan (c+d x) a+a}dx+a^3 (23 A+12 i B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {2 a^4 (B+i A) \int \sqrt {i \tan (c+d x) a+a}dx+a^3 (23 A+12 i B) \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3961 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {a^3 (23 A+12 i B) \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx-\frac {4 i a^5 (B+i A) \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {a^3 (23 A+12 i B) \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx-\frac {2 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\frac {a^5 (23 A+12 i B) \int \frac {\cot (c+d x)}{\sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}-\frac {2 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {-\frac {2 i a^4 (23 A+12 i B) \int \frac {1}{i-\frac {i (i \tan (c+d x) a+a)}{a}}d\sqrt {i \tan (c+d x) a+a}}{d}-\frac {2 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {-\frac {2 a^2 (22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{d}-\frac {3 \left (\frac {-\frac {2 a^{9/2} (23 A+12 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {2 i \sqrt {2} a^{9/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {7 a^3 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}\right )}{a}}{2 a^2}+\frac {a (17 A+11 i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}}{6 a^2}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}\) |
Input:
Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
((A + I*B)*Cot[c + d*x]^2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((a*(17*A + (11*I)*B)*Cot[c + d*x]^2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + ((-2*a^2*(22* A + (13*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/d - (3*(((-2*a^(9 /2)*(23*A + (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d - ((2 *I)*Sqrt[2]*a^(9/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]* Sqrt[a])])/d)/(2*a) - (7*a^3*((3*I)*A - 2*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan [c + d*x]])/d))/a)/(2*a^2))/(6*a^2)
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*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 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*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 = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {9}{2}}}-\frac {5 i B +7 A}{4 a^{4} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {i B +A}{6 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {7 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {9}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (12 i B +23 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )}{d}\) | \(198\) |
| default | \(\frac {2 a^{3} \left (-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {9}{2}}}-\frac {5 i B +7 A}{4 a^{4} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {i B +A}{6 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {7 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {9}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (12 i B +23 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )}{d}\) | \(198\) |
Input:
int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x,method=_RETUR NVERBOSE)
Output:
2/d*a^3*(-1/8/a^(9/2)*(A-I*B)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2) *2^(1/2)/a^(1/2))-1/4/a^4*(7*A+5*I*B)/(a+I*a*tan(d*x+c))^(1/2)-1/6/a^3*(A+ I*B)/(a+I*a*tan(d*x+c))^(3/2)+1/a^4*(-((-1/2*I*B-7/8*A)*(a+I*a*tan(d*x+c)) ^(3/2)+(1/2*I*a*B+9/8*a*A)*(a+I*a*tan(d*x+c))^(1/2))/a^2/tan(d*x+c)^2+1/8* (23*A+12*I*B)/a^(1/2)*arctanh((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. 903 vs. \(2 (209) = 418\).
Time = 0.15 (sec) , antiderivative size = 903, normalized size of antiderivative = 3.37 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x, algori thm="fricas")
Output:
1/48*(12*sqrt(1/2)*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c ) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log(- 4*(sqrt(2)*sqrt(1/2)*(I*a^2*d*e^(2*I*d*x + 2*I*c) + I*a^2*d)*sqrt(a/(e^(2* I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2)) + (-I*A - B)*a* e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(I*A + B)) - 12*sqrt(1/2)*(a^2*d*e^(7*I* d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sq rt((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log(-4*(sqrt(2)*sqrt(1/2)*(-I*a^2*d*e^ (2*I*d*x + 2*I*c) - I*a^2*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2)) + (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c )/(I*A + B)) + 3*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2 ))*log(-16*(3*(23*I*A - 12*B)*a^2*e^(2*I*d*x + 2*I*c) + (23*I*A - 12*B)*a^ 2 + 2*sqrt(2)*(I*a^3*d*e^(3*I*d*x + 3*I*c) + I*a^3*d*e^(I*d*x + I*c))*sqrt (a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^ 2)))*e^(-2*I*d*x - 2*I*c)/(-23*I*A + 12*B)) - 3*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2))*log(-16*(3*(23*I*A - 12*B)*a^2*e^(2*I*d* x + 2*I*c) + (23*I*A - 12*B)*a^2 + 2*sqrt(2)*(-I*a^3*d*e^(3*I*d*x + 3*I*c) - I*a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((529*A^ 2 + 552*I*A*B - 144*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-23*I*A + 12...
\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Integral((A + B*tan(c + d*x))*cot(c + d*x)**3/(I*a*(tan(c + d*x) - I))**(3 /2), x)
Time = 0.14 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left (21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} {\left (3 \, A + 2 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (107 \, A + 68 i \, B\right )} a + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (17 \, A + 11 i \, B\right )} a^{2} + 4 \, {\left (A + i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5}} - \frac {3 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {7}{2}}} + \frac {3 \, {\left (23 \, A + 12 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}}{24 \, d} \] Input:
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x, algori thm="maxima")
Output:
-1/24*a^2*(2*(21*(I*a*tan(d*x + c) + a)^3*(3*A + 2*I*B) - (I*a*tan(d*x + c ) + a)^2*(107*A + 68*I*B)*a + 2*(I*a*tan(d*x + c) + a)*(17*A + 11*I*B)*a^2 + 4*(A + I*B)*a^3)/((I*a*tan(d*x + c) + a)^(7/2)*a^3 - 2*(I*a*tan(d*x + c ) + a)^(5/2)*a^4 + (I*a*tan(d*x + c) + a)^(3/2)*a^5) - 3*sqrt(2)*(A - I*B) *log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sq rt(I*a*tan(d*x + c) + a)))/a^(7/2) + 3*(23*A + 12*I*B)*log((sqrt(I*a*tan(d *x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(7/2))/d
Exception generated. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),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 TypeDone
Time = 5.49 (sec) , antiderivative size = 3106, normalized size of antiderivative = 11.59 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^(3/2),x)
Output:
2*atanh((48*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ( (277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*B^ 2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2)*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/(B^3*a^2*d*3124i - 25296*A^3*a^2*d + 190 48*A*B^2*a^2*d - A^2*B*a^2*d*38282i + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5 041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^ 3*B*a^6*146506i)/d^4)^(1/2)*52i)/a) - (4216*A^2*d^2*(a + a*tan(c + d*x)*1i )^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a ^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*1 46506i)/d^4)^(1/2)/(64*a^6) - (73*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d ^2))^(1/2))/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)/a - (A^2* B*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d ^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - ( 114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4 )^(1/2)*52i)/a^4) + (1136*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A...
\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\left (\int \frac {\cot \left (d x +c \right )^{3}}{\sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right ) i +\sqrt {\tan \left (d x +c \right ) i +1}}d x \right ) a +\left (\int \frac {\cot \left (d x +c \right )^{3} \tan \left (d x +c \right )}{\sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right ) i +\sqrt {\tan \left (d x +c \right ) i +1}}d x \right ) b}{\sqrt {a}\, a} \] Input:
int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
(int(cot(c + d*x)**3/(sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)*i + sqrt(tan(c + d*x)*i + 1)),x)*a + int((cot(c + d*x)**3*tan(c + d*x))/(sqrt(tan(c + d* x)*i + 1)*tan(c + d*x)*i + sqrt(tan(c + d*x)*i + 1)),x)*b)/(sqrt(a)*a)