Integrand size = 34, antiderivative size = 171 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {4 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {4 a^2 (A-i B) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 a (A-i B) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d} \] Output:
-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+4*a^2*(A-I*B)*(a+I*a*tan(d*x+c))^(1/2)/d+2/3*a*(A-I*B)*(a+I*a*tan (d*x+c))^(3/2)/d+2/5*A*(a+I*a*tan(d*x+c))^(5/2)/d-2/7*I*B*(a+I*a*tan(d*x+c ))^(7/2)/a/d
Time = 0.97 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.77 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {-420 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+2 a^2 \sqrt {a+i a \tan (c+d x)} \left (266 A-260 i B+(77 i A+80 B) \tan (c+d x)+(-21 A+45 i B) \tan ^2(c+d x)-15 B \tan ^3(c+d x)\right )}{105 d} \] Input:
Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x ]
Output:
(-420*Sqrt[2]*a^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2 ]*Sqrt[a])] + 2*a^2*Sqrt[a + I*a*Tan[c + d*x]]*(266*A - (260*I)*B + ((77*I )*A + 80*B)*Tan[c + d*x] + (-21*A + (45*I)*B)*Tan[c + d*x]^2 - 15*B*Tan[c + d*x]^3))/(105*d)
Time = 0.68 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {3042, 4075, 3042, 4010, 3042, 3959, 3042, 3959, 3042, 3961, 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.
| \(\displaystyle \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4075 |
| \(\displaystyle \int (i \tan (c+d x) a+a)^{5/2} (A \tan (c+d x)-B)dx-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (i \tan (c+d x) a+a)^{5/2} (A \tan (c+d x)-B)dx-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 4010 |
| \(\displaystyle -(B+i A) \int (i \tan (c+d x) a+a)^{5/2}dx+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -(B+i A) \int (i \tan (c+d x) a+a)^{5/2}dx+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle -(B+i A) \left (2 a \int (i \tan (c+d x) a+a)^{3/2}dx+\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}\right )+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -(B+i A) \left (2 a \int (i \tan (c+d x) a+a)^{3/2}dx+\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}\right )+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle -(B+i A) \left (2 a \left (2 a \int \sqrt {i \tan (c+d x) a+a}dx+\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}\right )+\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}\right )+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -(B+i A) \left (2 a \left (2 a \int \sqrt {i \tan (c+d x) a+a}dx+\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}\right )+\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}\right )+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 3961 |
| \(\displaystyle -(B+i A) \left (2 a \left (\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}-\frac {4 i a^2 \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}\right )+\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}\right )+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -(B+i A) \left (2 a \left (\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}\right )+\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}\right )+\frac {2 A (a+i a \tan (c+d x))^{5/2}}{5 d}-\frac {2 i B (a+i a \tan (c+d x))^{7/2}}{7 a d}\) |
Input:
Int[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
Output:
(2*A*(a + I*a*Tan[c + d*x])^(5/2))/(5*d) - (((2*I)/7)*B*(a + I*a*Tan[c + d *x])^(7/2))/(a*d) - (I*A + B)*((((2*I)/3)*a*(a + I*a*Tan[c + d*x])^(3/2))/ d + 2*a*(((-2*I)*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[ 2]*Sqrt[a])])/d + ((2*I)*a*Sqrt[a + I*a*Tan[c + d*x]])/d))
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[2*a Int[(a + b*Tan[c + d* x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n , 1]
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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Simp [(b*c + a*d)/b Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e , f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B *d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f* x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && !LeQ[m, -1]
Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {2 A a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {2 i B \,a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {2 A \,a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 i a^{3} B \sqrt {a +i a \tan \left (d x +c \right )}+4 A \,a^{3} \sqrt {a +i a \tan \left (d x +c \right )}-4 a^{\frac {7}{2}} \left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{a d}\) | \(165\) |
| default | \(\frac {-\frac {2 i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {2 A a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {2 i B \,a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {2 A \,a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 i a^{3} B \sqrt {a +i a \tan \left (d x +c \right )}+4 A \,a^{3} \sqrt {a +i a \tan \left (d x +c \right )}-4 a^{\frac {7}{2}} \left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{a d}\) | \(165\) |
| parts | \(\frac {A \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+4 a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-4 a^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d}+\frac {2 i B \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}} a^{2}}{3}-2 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}+2 a^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d a}\) | \(187\) |
Input:
int(tan(d*x+c)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNV ERBOSE)
Output:
2/d/a*(-1/7*I*B*(a+I*a*tan(d*x+c))^(7/2)+1/5*A*a*(a+I*a*tan(d*x+c))^(5/2)- 1/3*I*B*a^2*(a+I*a*tan(d*x+c))^(3/2)+1/3*A*a^2*(a+I*a*tan(d*x+c))^(3/2)-2* I*B*a^3*(a+I*a*tan(d*x+c))^(1/2)+2*A*a^3*(a+I*a*tan(d*x+c))^(1/2)-2*a^(7/2 )*(A-I*B)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (130) = 260\).
Time = 0.11 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.79 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (105 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 105 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 2 \, \sqrt {2} {\left (2 \, {\left (91 \, A - 100 i \, B\right )} a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 7 \, {\left (61 \, A - 55 i \, B\right )} a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 350 \, {\left (A - i \, B\right )} a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 105 \, {\left (A - i \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )}}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorith m="fricas")
Output:
-2/105*(105*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6* I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*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)) - 105*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^( 6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*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)) - 2*sqrt(2)*(2*(91*A - 100*I*B)*a^2*e^(7*I* d*x + 7*I*c) + 7*(61*A - 55*I*B)*a^2*e^(5*I*d*x + 5*I*c) + 350*(A - I*B)*a ^2*e^(3*I*d*x + 3*I*c) + 105*(A - I*B)*a^2*e^(I*d*x + I*c))*sqrt(a/(e^(2*I *d*x + 2*I*c) + 1)))/(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3* d*e^(2*I*d*x + 2*I*c) + d)
\[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \tan {\left (c + d x \right )}\, dx \] Input:
integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
Output:
Integral((I*a*(tan(c + d*x) - I))**(5/2)*(A + B*tan(c + d*x))*tan(c + d*x) , x)
Time = 0.13 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2 \, {\left (105 \, \sqrt {2} {\left (A - i \, B\right )} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) - 15 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} B a + 21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} A a^{2} + 35 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (A - i \, B\right )} a^{3} + 210 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (A - i \, B\right )} a^{4}\right )}}{105 \, a^{2} d} \] Input:
integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorith m="maxima")
Output:
2/105*(105*sqrt(2)*(A - I*B)*a^(9/2)*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))) - 15*I*(I*a *tan(d*x + c) + a)^(7/2)*B*a + 21*(I*a*tan(d*x + c) + a)^(5/2)*A*a^2 + 35* (I*a*tan(d*x + c) + a)^(3/2)*(A - I*B)*a^3 + 210*sqrt(I*a*tan(d*x + c) + a )*(A - I*B)*a^4)/(a^2*d)
Exception generated. \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorith m="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 = 1.51 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.24 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2\,A\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,d}+\frac {2\,A\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,d}-\frac {B\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{3\,d}+\frac {4\,A\,a^2\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{d}-\frac {B\,a^2\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,4{}\mathrm {i}}{d}-\frac {B\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}\,2{}\mathrm {i}}{7\,a\,d}+\frac {\sqrt {2}\,B\,{\left (-a\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,4{}\mathrm {i}}{d}+\frac {\sqrt {2}\,A\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,4{}\mathrm {i}}{d} \] Input:
int(tan(c + d*x)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)
Output:
(2*A*(a + a*tan(c + d*x)*1i)^(5/2))/(5*d) + (2*A*a*(a + a*tan(c + d*x)*1i) ^(3/2))/(3*d) - (B*a*(a + a*tan(c + d*x)*1i)^(3/2)*2i)/(3*d) + (4*A*a^2*(a + a*tan(c + d*x)*1i)^(1/2))/d - (B*a^2*(a + a*tan(c + d*x)*1i)^(1/2)*4i)/ d - (B*(a + a*tan(c + d*x)*1i)^(7/2)*2i)/(7*a*d) + (2^(1/2)*B*(-a)^(5/2)*a tan((2^(1/2)*(a + a*tan(c + d*x)*1i)^(1/2))/(2*(-a)^(1/2)))*4i)/d + (2^(1/ 2)*A*a^(5/2)*atan((2^(1/2)*(a + a*tan(c + d*x)*1i)^(1/2)*1i)/(2*a^(1/2)))* 4i)/d
\[ \int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{4}d x \right ) b -\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{3}d x \right ) a +2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{3}d x \right ) b i +2 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}d x \right ) a i +\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )d x \right ) a \right ) \] Input:
int(tan(d*x+c)*(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)*tan(c + d*x)**4,x)*b - int(s qrt(tan(c + d*x)*i + 1)*tan(c + d*x)**3,x)*a + 2*int(sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**3,x)*b*i + 2*int(sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)** 2,x)*a*i + int(sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2,x)*b + int(sqrt(ta n(c + d*x)*i + 1)*tan(c + d*x),x)*a)