Integrand size = 28, antiderivative size = 98 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=-\frac {2 i a (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {2 a (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f} \]
-2*I*a*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+2*a*( I*c+d)*(c+d*tan(f*x+e))^(1/2)/f+2/3*I*a*(c+d*tan(f*x+e))^(3/2)/f
Time = 0.64 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\frac {2 a \left (-3 i (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+(4 i c+3 d+i d \tan (e+f x)) \sqrt {c+d \tan (e+f x)}\right )}{3 f} \]
(2*a*((-3*I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d ]] + ((4*I)*c + 3*d + I*d*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(3*f)
Time = 0.70 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 4011, 3042, 4011, 3042, 4020, 25, 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.
\(\displaystyle \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}dx\) |
\(\Big \downarrow \) 4011 |
\(\displaystyle \int \sqrt {c+d \tan (e+f x)} (a (c-i d)+a (i c+d) \tan (e+f x))dx+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {c+d \tan (e+f x)} (a (c-i d)+a (i c+d) \tan (e+f x))dx+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}\) |
\(\Big \downarrow \) 4011 |
\(\displaystyle \int \frac {a (c-i d)^2+i a \tan (e+f x) (c-i d)^2}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a (c-i d)^2+i a \tan (e+f x) (c-i d)^2}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle \frac {i a^2 (c-i d)^4 \int -\frac {1}{a (c-i d)^2 \left (a (c-i d)^2-i a (c-i d)^2 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}d\left (i a (c-i d)^2 \tan (e+f x)\right )}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i a^2 (c-i d)^4 \int \frac {1}{a (c-i d)^2 \left (a (c-i d)^2-i a (c-i d)^2 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}d\left (i a (c-i d)^2 \tan (e+f x)\right )}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {i a (c-i d)^2 \int \frac {1}{\left (a (c-i d)^2-i a (c-i d)^2 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}d\left (i a (c-i d)^2 \tan (e+f x)\right )}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 a^2 (c-i d)^4 \int \frac {1}{\frac {i a^3 (c-i d)^6 \tan ^2(e+f x)}{d}-\frac {a (i c+d)^3}{d}}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 a (c-i d)^{3/2} \arctan \left (a (c-i d)^{3/2} \tan (e+f x)\right )}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f}\) |
(2*a*(c - I*d)^(3/2)*ArcTan[a*(c - I*d)^(3/2)*Tan[e + f*x]])/f + (2*a*(I*c + d)*Sqrt[c + d*Tan[e + f*x]])/f + (((2*I)/3)*a*(c + d*Tan[e + f*x])^(3/2 ))/f
3.12.9.3.1 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_), 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (81 ) = 162\).
Time = 0.66 (sec) , antiderivative size = 852, normalized size of antiderivative = 8.69
method | result | size |
derivativedivides | \(\frac {a \left (\frac {2 i \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i c \sqrt {c +d \tan \left (f x +e \right )}+2 d \sqrt {c +d \tan \left (f x +e \right )}+\frac {\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -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^{3}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}-\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -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}}}{\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {-\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )}{f}\) | \(852\) |
default | \(\frac {a \left (\frac {2 i \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i c \sqrt {c +d \tan \left (f x +e \right )}+2 d \sqrt {c +d \tan \left (f x +e \right )}+\frac {\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -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^{3}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}-\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -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}}}{\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {-\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )}{f}\) | \(852\) |
parts | \(\text {Expression too large to display}\) | \(1659\) |
1/f*a*(2/3*I*(c+d*tan(f*x+e))^(3/2)+2*I*c*(c+d*tan(f*x+e))^(1/2)+2*d*(c+d* tan(f*x+e))^(1/2)+1/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(1/2*(-I *c^2*(c^2+d^2)^(1/2)+I*d^2*(c^2+d^2)^(1/2)-I*c^3-I*c*d^2-2*c*d*(c^2+d^2)^( 1/2)-c^2*d-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^3-I*(2 *(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d-(2*( c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-1/2*(-I*c^2*(c^2+d^2)^(1/2)+I*d^2*(c^2+d^2)^ (1/2)-I*c^3-I*c*d^2-2*c*d*(c^2+d^2)^(1/2)-c^2*d-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*(c^2+d^ 2)^(1/2)+2*c)^(1/2)/(c^2+d^2)^(1/2)*(-1/2*(-I*c^2*(c^2+d^2)^(1/2)+I*d^2*(c ^2+d^2)^(1/2)-I*c^3-I*c*d^2-2*c*d*(c^2+d^2)^(1/2)-c^2*d-d^3)*ln((c+d*tan(f *x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2)) +2*(I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+I*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c* d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+ 1/2*(-I*c^2*(c^2+d^2)^(1/2)+I*d^2*(c^2+d^2)^(1/2)-I*c^3-I*c*d^2-2*c*d*(c^2 +d^2)^(1/2)-c^2*d-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^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/ (2*(c^2+d^2)^(1/2)-2*c)^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (76) = 152\).
Time = 0.26 (sec) , antiderivative size = 504, normalized size of antiderivative = 5.14 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\frac {3 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} \log \left (-\frac {2 \, {\left (-i \, a c^{2} - a c d + {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} + {\left (-i \, a c^{2} - 2 \, a c d + i \, a d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{i \, a c + a d}\right ) - 3 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} \log \left (-\frac {2 \, {\left (-i \, a c^{2} - a c d - {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} + {\left (-i \, a c^{2} - 2 \, a c d + i \, a d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{i \, a c + a d}\right ) - 8 \, {\left (-2 i \, a c - a d + 2 \, {\left (-i \, a c - a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
1/6*(3*(f*e^(2*I*f*x + 2*I*e) + f)*sqrt(-(a^2*c^3 - 3*I*a^2*c^2*d - 3*a^2* c*d^2 + I*a^2*d^3)/f^2)*log(-2*(-I*a*c^2 - a*c*d + (f*e^(2*I*f*x + 2*I*e) + 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^2*c^3 - 3*I*a^2*c^2*d - 3*a^2*c*d^2 + I*a^2*d^3)/f^2) + (-I* a*c^2 - 2*a*c*d + I*a*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(I*a* c + a*d)) - 3*(f*e^(2*I*f*x + 2*I*e) + f)*sqrt(-(a^2*c^3 - 3*I*a^2*c^2*d - 3*a^2*c*d^2 + I*a^2*d^3)/f^2)*log(-2*(-I*a*c^2 - a*c*d - (f*e^(2*I*f*x + 2*I*e) + 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^2*c^3 - 3*I*a^2*c^2*d - 3*a^2*c*d^2 + I*a^2*d^3)/f^2) + (-I*a*c^2 - 2*a*c*d + I*a*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e )/(I*a*c + a*d)) - 8*(-2*I*a*c - a*d + 2*(-I*a*c - a*d)*e^(2*I*f*x + 2*I*e ))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1 )))/(f*e^(2*I*f*x + 2*I*e) + f)
\[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=i a \left (\int \left (- i c \sqrt {c + d \tan {\left (e + f x \right )}}\right )\, dx + \int c \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\, dx + \int d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- i d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx\right ) \]
I*a*(Integral(-I*c*sqrt(c + d*tan(e + f*x)), x) + Integral(c*sqrt(c + d*ta n(e + f*x))*tan(e + f*x), x) + Integral(d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2, x) + Integral(-I*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x), x))
Timed out. \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (76) = 152\).
Time = 0.51 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.34 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\frac {2}{3} i \, a {\left (\frac {6 \, {\left (c^{2} - 2 i \, c d - d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} f^{2} + 3 \, \sqrt {d \tan \left (f x + e\right ) + c} c f^{2} - 3 i \, \sqrt {d \tan \left (f x + e\right ) + c} d f^{2}}{f^{3}}\right )} \]
2/3*I*a*(6*(c^2 - 2*I*c*d - d^2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sq rt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt( c^2 + d^2))))/(sqrt(-2*c + 2*sqrt(c^2 + d^2))*f*(-I*d/(c - sqrt(c^2 + d^2) ) + 1)) + ((d*tan(f*x + e) + c)^(3/2)*f^2 + 3*sqrt(d*tan(f*x + e) + c)*c*f ^2 - 3*I*sqrt(d*tan(f*x + e) + c)*d*f^2)/f^3)
Time = 17.91 (sec) , antiderivative size = 2869, normalized size of antiderivative = 29.28 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\text {Too large to display} \]
log(((((-a^4*d^2*f^4*(3*c^2 - d^2)^2)^(1/2) - a^2*c^3*f^2 + 3*a^2*c*d^2*f^ 2)/f^4)^(1/2)*((16*c*d^2*(((-a^4*d^2*f^4*(3*c^2 - d^2)^2)^(1/2) - a^2*c^3* f^2 + 3*a^2*c*d^2*f^2)/f^4)^(1/2)*(a*c^2*1i + a*d^2*1i - f*(((-a^4*d^2*f^4 *(3*c^2 - d^2)^2)^(1/2) - a^2*c^3*f^2 + 3*a^2*c*d^2*f^2)/f^4)^(1/2)*(c + d *tan(e + f*x))^(1/2)))/f - (16*a^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^4 + d ^4 - 6*c^2*d^2))/f^2))/2 - (a^3*d^2*(c^2 - d^2)*(c^2*1i + d^2*1i)^2*8i)/f^ 3)*((6*a^4*c^2*d^4*f^4 - a^4*d^6*f^4 - 9*a^4*c^4*d^2*f^4)^(1/2)/(4*f^4) - (a^2*c^3)/(4*f^2) + (3*a^2*c*d^2)/(4*f^2))^(1/2) - log(((((-a^4*d^2*f^4*(3 *c^2 - d^2)^2)^(1/2) - a^2*c^3*f^2 + 3*a^2*c*d^2*f^2)/f^4)^(1/2)*((16*c*d^ 2*(((-a^4*d^2*f^4*(3*c^2 - d^2)^2)^(1/2) - a^2*c^3*f^2 + 3*a^2*c*d^2*f^2)/ f^4)^(1/2)*(a*c^2*1i + a*d^2*1i + f*(((-a^4*d^2*f^4*(3*c^2 - d^2)^2)^(1/2) - a^2*c^3*f^2 + 3*a^2*c*d^2*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/ f + (16*a^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^4 + d^4 - 6*c^2*d^2))/f^2))/ 2 - (a^3*d^2*(c^2 - d^2)*(c^2*1i + d^2*1i)^2*8i)/f^3)*(((6*a^4*c^2*d^4*f^4 - a^4*d^6*f^4 - 9*a^4*c^4*d^2*f^4)^(1/2) - a^2*c^3*f^2 + 3*a^2*c*d^2*f^2) /(4*f^4))^(1/2) - log(((-((-a^4*d^2*f^4*(3*c^2 - d^2)^2)^(1/2) + a^2*c^3*f ^2 - 3*a^2*c*d^2*f^2)/f^4)^(1/2)*((16*c*d^2*(-((-a^4*d^2*f^4*(3*c^2 - d^2) ^2)^(1/2) + a^2*c^3*f^2 - 3*a^2*c*d^2*f^2)/f^4)^(1/2)*(a*c^2*1i + a*d^2*1i + f*(-((-a^4*d^2*f^4*(3*c^2 - d^2)^2)^(1/2) + a^2*c^3*f^2 - 3*a^2*c*d^2*f ^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/f + (16*a^2*d^2*(c + d*tan(...