Integrand size = 30, antiderivative size = 166 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=-\frac {4 i a^2 (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a^2 (i c+d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f} \] Output:
-4*I*a^2*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+4*I *a^2*(c-I*d)^2*(c+d*tan(f*x+e))^(1/2)/f+4/3*a^2*(I*c+d)*(c+d*tan(f*x+e))^( 3/2)/f+4/5*I*a^2*(c+d*tan(f*x+e))^(5/2)/f-2/7*a^2*(c+d*tan(f*x+e))^(7/2)/d /f
Time = 1.71 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.96 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=-\frac {2 a^2 \left (210 i (c-i d)^{5/2} d \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} \left (15 c^3-322 i c^2 d-490 c d^2+210 i d^3+d \left (45 c^2-154 i c d-70 d^2\right ) \tan (e+f x)+3 (15 c-14 i d) d^2 \tan ^2(e+f x)+15 d^3 \tan ^3(e+f x)\right )\right )}{105 d f} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]
Output:
(-2*a^2*((210*I)*(c - I*d)^(5/2)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(15*c^3 - (322*I)*c^2*d - 490*c*d^2 + (210*I)*d^3 + d*(45*c^2 - (154*I)*c*d - 70*d^2)*Tan[e + f*x] + 3*(15*c - ( 14*I)*d)*d^2*Tan[e + f*x]^2 + 15*d^3*Tan[e + f*x]^3)))/(105*d*f)
Time = 1.02 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {3042, 4026, 3042, 4011, 3042, 4011, 3042, 4011, 3042, 4020, 27, 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.
| \(\displaystyle \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle -\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\int \left (2 i \tan (e+f x) a^2+2 a^2\right ) (c+d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\int \left (2 i \tan (e+f x) a^2+2 a^2\right ) (c+d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (c+d \tan (e+f x))^{3/2} \left (2 (c-i d) a^2+2 (i c+d) \tan (e+f x) a^2\right )dx-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d \tan (e+f x))^{3/2} \left (2 (c-i d) a^2+2 (i c+d) \tan (e+f x) a^2\right )dx-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \left (2 a^2 (c-i d)^2+2 i a^2 \tan (e+f x) (c-i d)^2\right ) \sqrt {c+d \tan (e+f x)}dx-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (2 a^2 (c-i d)^2+2 i a^2 \tan (e+f x) (c-i d)^2\right ) \sqrt {c+d \tan (e+f x)}dx-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {2 a^2 (c-i d)^3-2 a^2 (i c+d)^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a^2 (c-i d)^3-2 a^2 (i c+d)^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {4 i a^4 (c-i d)^6 \int \frac {1}{\sqrt {2} a^2 \sqrt {2 c+2 d \tan (e+f x)} \left (2 a^2 (i c+d)^6-2 a^2 (c-i d)^3 (i c+d)^3 \tan (e+f x)\right )}d\left (-2 a^2 (i c+d)^3 \tan (e+f x)\right )}{f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 i \sqrt {2} a^2 (c-i d)^6 \int \frac {1}{\sqrt {2 c+2 d \tan (e+f x)} \left (2 a^2 (i c+d)^6-2 a^2 (c-i d)^3 (i c+d)^3 \tan (e+f x)\right )}d\left (-2 a^2 (i c+d)^3 \tan (e+f x)\right )}{f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {4 i \sqrt {2} a^4 (d+i c)^3 (c-i d)^6 \int \frac {1}{\frac {2 a^2 (i c+d)^7}{d}+\frac {4 i a^6 (c-i d)^6 \tan ^2(e+f x) (i c+d)^6}{d}}d\sqrt {2 c+2 d \tan (e+f x)}}{d f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {4 a^2 (d+i c)^3 \text {arctanh}\left (\frac {\sqrt {2} a^2 (d+i c)^3 \tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 i a^2 (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]
Output:
(-4*a^2*(I*c + d)^3*ArcTanh[(Sqrt[2]*a^2*(I*c + d)^3*Tan[e + f*x])/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + ((4*I)*a^2*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x ]])/f + (4*a^2*(I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (((4*I)/5)*a^ 2*(c + d*Tan[e + f*x])^(5/2))/f - (2*a^2*(c + d*Tan[e + f*x])^(7/2))/(7*d* f)
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[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1002 vs. \(2 (139 ) = 278\).
Time = 0.42 (sec) , antiderivative size = 1003, normalized size of antiderivative = 6.04
| method | result | size |
| derivativedivides | \(\frac {2 a^{2} \left (\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 i c d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i \sqrt {c +d \tan \left (f x +e \right )}\, c^{2} d -2 i \sqrt {c +d \tan \left (f x +e \right )}\, d^{3}+\frac {2 d^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 \sqrt {c +d \tan \left (f x +e \right )}\, c \,d^{2}-2 d \left (\frac {\frac {\left (i c^{3} \sqrt {c^{2}+d^{2}}-3 i c \,d^{2} \sqrt {c^{2}+d^{2}}+i c^{4}-i d^{4}+3 c^{2} d \sqrt {c^{2}+d^{2}}-d^{3} \sqrt {c^{2}+d^{2}}+2 c^{3} d +2 c \,d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{4}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{4}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3} d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{3}-\frac {\left (i c^{3} \sqrt {c^{2}+d^{2}}-3 i c \,d^{2} \sqrt {c^{2}+d^{2}}+i c^{4}-i d^{4}+3 c^{2} d \sqrt {c^{2}+d^{2}}-d^{3} \sqrt {c^{2}+d^{2}}+2 c^{3} d +2 c \,d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (-i c^{3} \sqrt {c^{2}+d^{2}}+3 i c \,d^{2} \sqrt {c^{2}+d^{2}}-i c^{4}+i d^{4}-3 c^{2} d \sqrt {c^{2}+d^{2}}+d^{3} \sqrt {c^{2}+d^{2}}-2 c^{3} d -2 c \,d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{4}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{4}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3} d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{3}+\frac {\left (-i c^{3} \sqrt {c^{2}+d^{2}}+3 i c \,d^{2} \sqrt {c^{2}+d^{2}}-i c^{4}+i d^{4}-3 c^{2} d \sqrt {c^{2}+d^{2}}+d^{3} \sqrt {c^{2}+d^{2}}-2 c^{3} d -2 c \,d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )\right )}{f d}\) | \(1003\) |
| default | \(\frac {2 a^{2} \left (\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 i c d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i \sqrt {c +d \tan \left (f x +e \right )}\, c^{2} d -2 i \sqrt {c +d \tan \left (f x +e \right )}\, d^{3}+\frac {2 d^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 \sqrt {c +d \tan \left (f x +e \right )}\, c \,d^{2}-2 d \left (\frac {\frac {\left (i c^{3} \sqrt {c^{2}+d^{2}}-3 i c \,d^{2} \sqrt {c^{2}+d^{2}}+i c^{4}-i d^{4}+3 c^{2} d \sqrt {c^{2}+d^{2}}-d^{3} \sqrt {c^{2}+d^{2}}+2 c^{3} d +2 c \,d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{4}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{4}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3} d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{3}-\frac {\left (i c^{3} \sqrt {c^{2}+d^{2}}-3 i c \,d^{2} \sqrt {c^{2}+d^{2}}+i c^{4}-i d^{4}+3 c^{2} d \sqrt {c^{2}+d^{2}}-d^{3} \sqrt {c^{2}+d^{2}}+2 c^{3} d +2 c \,d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (-i c^{3} \sqrt {c^{2}+d^{2}}+3 i c \,d^{2} \sqrt {c^{2}+d^{2}}-i c^{4}+i d^{4}-3 c^{2} d \sqrt {c^{2}+d^{2}}+d^{3} \sqrt {c^{2}+d^{2}}-2 c^{3} d -2 c \,d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{4}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{4}+2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3} d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{3}+\frac {\left (-i c^{3} \sqrt {c^{2}+d^{2}}+3 i c \,d^{2} \sqrt {c^{2}+d^{2}}-i c^{4}+i d^{4}-3 c^{2} d \sqrt {c^{2}+d^{2}}+d^{3} \sqrt {c^{2}+d^{2}}-2 c^{3} d -2 c \,d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )\right )}{f d}\) | \(1003\) |
| parts | \(\text {Expression too large to display}\) | \(3625\) |
Input:
int((a+I*a*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
2/f*a^2/d*(2/5*I*d*(c+d*tan(f*x+e))^(5/2)-1/7*(c+d*tan(f*x+e))^(7/2)+2/3*I *c*d*(c+d*tan(f*x+e))^(3/2)+2*I*(c+d*tan(f*x+e))^(1/2)*c^2*d-2*I*(c+d*tan( f*x+e))^(1/2)*d^3+2/3*d^2*(c+d*tan(f*x+e))^(3/2)+4*(c+d*tan(f*x+e))^(1/2)* c*d^2-2*d*(1/2/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(1/2*(I*c^3*( c^2+d^2)^(1/2)-3*I*c*d^2*(c^2+d^2)^(1/2)+I*c^4-I*d^4+3*c^2*d*(c^2+d^2)^(1/ 2)-d^3*(c^2+d^2)^(1/2)+2*c^3*d+2*c*d^3)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e)) ^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(I*(2*(c^2+d^2)^(1 /2)+2*c)^(1/2)*c^4-I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^4+2*(2*(c^2+d^2)^(1/2 )+2*c)^(1/2)*c^3*d+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^3-1/2*(I*c^3*(c^2+d ^2)^(1/2)-3*I*c*d^2*(c^2+d^2)^(1/2)+I*c^4-I*d^4+3*c^2*d*(c^2+d^2)^(1/2)-d^ 3*(c^2+d^2)^(1/2)+2*c^3*d+2*c*d^3)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+ d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+ 2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/2/(2*(c^2+d^2)^(1/2)+2*c)^(1 /2)/(c^2+d^2)^(1/2)*(1/2*(-I*c^3*(c^2+d^2)^(1/2)+3*I*c*d^2*(c^2+d^2)^(1/2) -I*c^4+I*d^4-3*c^2*d*(c^2+d^2)^(1/2)+d^3*(c^2+d^2)^(1/2)-2*c^3*d-2*c*d^3)* ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^ 2+d^2)^(1/2))+2*(I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4-I*(2*(c^2+d^2)^(1/2)+ 2*c)^(1/2)*d^4+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d+2*(2*(c^2+d^2)^(1/2)+ 2*c)^(1/2)*c*d^3+1/2*(-I*c^3*(c^2+d^2)^(1/2)+3*I*c*d^2*(c^2+d^2)^(1/2)-I*c ^4+I*d^4-3*c^2*d*(c^2+d^2)^(1/2)+d^3*(c^2+d^2)^(1/2)-2*c^3*d-2*c*d^3)*(...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (132) = 264\).
Time = 0.31 (sec) , antiderivative size = 917, normalized size of antiderivative = 5.52 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((a+I*a*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas ")
Output:
-1/105*(105*(d*f*e^(6*I*f*x + 6*I*e) + 3*d*f*e^(4*I*f*x + 4*I*e) + 3*d*f*e ^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^4*c^5 - 5*I*a^4*c^4*d - 10*a^4*c^3*d^2 + 10*I*a^4*c^2*d^3 + 5*a^4*c*d^4 - I*a^4*d^5)/f^2)*log(2*(a^2*c^3 - 2*I*a^ 2*c^2*d - a^2*c*d^2 - (I*f*e^(2*I*f*x + 2*I*e) + I*f)*sqrt(((c - I*d)*e^(2 *I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(a^4*c^5 - 5*I *a^4*c^4*d - 10*a^4*c^3*d^2 + 10*I*a^4*c^2*d^3 + 5*a^4*c*d^4 - I*a^4*d^5)/ f^2) + (a^2*c^3 - 3*I*a^2*c^2*d - 3*a^2*c*d^2 + I*a^2*d^3)*e^(2*I*f*x + 2* I*e))*e^(-2*I*f*x - 2*I*e)/(a^2*c^2 - 2*I*a^2*c*d - a^2*d^2)) - 105*(d*f*e ^(6*I*f*x + 6*I*e) + 3*d*f*e^(4*I*f*x + 4*I*e) + 3*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^4*c^5 - 5*I*a^4*c^4*d - 10*a^4*c^3*d^2 + 10*I*a^4*c^2*d^3 + 5*a^4*c*d^4 - I*a^4*d^5)/f^2)*log(2*(a^2*c^3 - 2*I*a^2*c^2*d - a^2*c*d^ 2 - (-I*f*e^(2*I*f*x + 2*I*e) - I*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(a^4*c^5 - 5*I*a^4*c^4*d - 10*a ^4*c^3*d^2 + 10*I*a^4*c^2*d^3 + 5*a^4*c*d^4 - I*a^4*d^5)/f^2) + (a^2*c^3 - 3*I*a^2*c^2*d - 3*a^2*c*d^2 + I*a^2*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a^2*c^2 - 2*I*a^2*c*d - a^2*d^2)) + 2*(15*a^2*c^3 - 277*I*a^2*c ^2*d - 381*a^2*c*d^2 + 167*I*a^2*d^3 + (15*a^2*c^3 - 367*I*a^2*c^2*d - 689 *a^2*c*d^2 + 337*I*a^2*d^3)*e^(6*I*f*x + 6*I*e) + (45*a^2*c^3 - 1011*I*a^2 *c^2*d - 1579*a^2*c*d^2 + 613*I*a^2*d^3)*e^(4*I*f*x + 4*I*e) + (45*a^2*c^3 - 921*I*a^2*c^2*d - 1271*a^2*c*d^2 + 563*I*a^2*d^3)*e^(2*I*f*x + 2*I*e...
\[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=- a^{2} \left (\int \left (- c^{2} \sqrt {c + d \tan {\left (e + f x \right )}}\right )\, dx + \int c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )}\, dx + \int \left (- 2 i c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 i d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx + \int 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 4 i c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \] Input:
integrate((a+I*a*tan(f*x+e))**2*(c+d*tan(f*x+e))**(5/2),x)
Output:
-a**2*(Integral(-c**2*sqrt(c + d*tan(e + f*x)), x) + Integral(c**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2, x) + Integral(-d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2, x) + Integral(d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**4, x) + Integral(-2*I*c**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x), x ) + Integral(-2*I*d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3, x) + Inte gral(-2*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x), x) + Integral(2*c*d*sqr t(c + d*tan(e + f*x))*tan(e + f*x)**3, x) + Integral(-4*I*c*d*sqrt(c + d*t an(e + f*x))*tan(e + f*x)**2, x))
\[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+I*a*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima ")
Output:
integrate((I*a*tan(f*x + e) + a)^2*(d*tan(f*x + e) + c)^(5/2), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (132) = 264\).
Time = 0.76 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.90 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=-\frac {2 \, a^{2} {\left (\frac {210 \, \sqrt {2} {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2} c \sqrt {-c + \sqrt {c^{2} + d^{2}}} - i \, \sqrt {2} \sqrt {-c + \sqrt {c^{2} + d^{2}}} d - \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {-c + \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-c + \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {15 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} d^{6} - 42 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{7} - 70 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{7} - 210 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{7} - 70 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{8} - 420 \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{8} + 210 i \, \sqrt {d \tan \left (f x + e\right ) + c} d^{9}}{d^{7}}\right )}}{105 \, f} \] Input:
integrate((a+I*a*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
Output:
-2/105*a^2*(210*sqrt(2)*(-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*arctan(2*(sqr t(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(sqrt( 2)*c*sqrt(-c + sqrt(c^2 + d^2)) - I*sqrt(2)*sqrt(-c + sqrt(c^2 + d^2))*d - sqrt(2)*sqrt(c^2 + d^2)*sqrt(-c + sqrt(c^2 + d^2))))/(sqrt(-c + sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + (15*(d*tan(f*x + e) + c)^(7/2) *d^6 - 42*I*(d*tan(f*x + e) + c)^(5/2)*d^7 - 70*I*(d*tan(f*x + e) + c)^(3/ 2)*c*d^7 - 210*I*sqrt(d*tan(f*x + e) + c)*c^2*d^7 - 70*(d*tan(f*x + e) + c )^(3/2)*d^8 - 420*sqrt(d*tan(f*x + e) + c)*c*d^8 + 210*I*sqrt(d*tan(f*x + e) + c)*d^9)/d^7)/f
Time = 16.71 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.55 \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=-\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{5\,d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{5\,d\,f}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}-{\left (c-d\,1{}\mathrm {i}\right )}^2\,\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{d\,f}\right )\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}-\frac {\left (c-d\,1{}\mathrm {i}\right )\,\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{d\,f}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3}-\frac {2\,a^2\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{7\,d\,f}-\frac {\sqrt {4{}\mathrm {i}}\,a^2\,\mathrm {atan}\left (\frac {\sqrt {4{}\mathrm {i}}\,{\left (d+c\,1{}\mathrm {i}\right )}^{5/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{2\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )}\right )\,{\left (d+c\,1{}\mathrm {i}\right )}^{5/2}\,2{}\mathrm {i}}{f} \] Input:
int((a + a*tan(e + f*x)*1i)^2*(c + d*tan(e + f*x))^(5/2),x)
Output:
- ((2*a^2*(c - d*1i))/(5*d*f) - (2*a^2*(c + d*1i))/(5*d*f))*(c + d*tan(e + f*x))^(5/2) - (c - d*1i)^2*((2*a^2*(c - d*1i))/(d*f) - (2*a^2*(c + d*1i)) /(d*f))*(c + d*tan(e + f*x))^(1/2) - ((c - d*1i)*((2*a^2*(c - d*1i))/(d*f) - (2*a^2*(c + d*1i))/(d*f))*(c + d*tan(e + f*x))^(3/2))/3 - (2*a^2*(c + d *tan(e + f*x))^(7/2))/(7*d*f) - (4i^(1/2)*a^2*atan((4i^(1/2)*(c*1i + d)^(5 /2)*(c + d*tan(e + f*x))^(1/2)*1i)/(2*(c*d^2*3i - 3*c^2*d - c^3*1i + d^3)) )*(c*1i + d)^(5/2)*2i)/f
\[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx=a^{2} \left (\left (\int \sqrt {d \tan \left (f x +e \right )+c}d x \right ) c^{2}-\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{4}d x \right ) d^{2}-2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}d x \right ) c d +2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}d x \right ) d^{2} i -\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) c^{2}+4 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) c d i +\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c^{2} i +2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c d \right ) \] Input:
int((a+I*a*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2),x)
Output:
a**2*(int(sqrt(tan(e + f*x)*d + c),x)*c**2 - int(sqrt(tan(e + f*x)*d + c)* tan(e + f*x)**4,x)*d**2 - 2*int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3,x )*c*d + 2*int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3,x)*d**2*i - int(sqr t(tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*c**2 + 4*int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*c*d*i + int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)* *2,x)*d**2 + 2*int(sqrt(tan(e + f*x)*d + c)*tan(e + f*x),x)*c**2*i + 2*int (sqrt(tan(e + f*x)*d + c)*tan(e + f*x),x)*c*d)