Integrand size = 28, antiderivative size = 109 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {2 i a \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {2 a}{3 (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i a}{(c-i d)^2 f \sqrt {c+d \tan (e+f x)}} \]
-2*I*a*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f+2*I*a /(c-I*d)^2/f/(c+d*tan(f*x+e))^(1/2)-2/3*a/(I*c+d)/f/(c+d*tan(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.56 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.56 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {2 i a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{3 (c-i d) f (c+d \tan (e+f x))^{3/2}} \]
(((2*I)/3)*a*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I* d)])/((c - I*d)*f*(c + d*Tan[e + f*x])^(3/2))
Time = 0.66 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 4012, 3042, 4012, 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.
\(\displaystyle \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {\int \frac {a (c+i d)+a (i c-d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (c+i d)+a (i c-d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {\frac {\int \frac {a (c+i d)^2+i a \tan (e+f x) (c+i d)^2}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {a (c+i d)^2+i a \tan (e+f x) (c+i d)^2}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle \frac {\frac {i a^2 (c+i d)^4 \int -\frac {1}{a (c+i d)^2 \left (a (c+i d)^2-i a (c+i d)^2 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}d\left (i a (c+i d)^2 \tan (e+f x)\right )}{f \left (c^2+d^2\right )}+\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}-\frac {i a^2 (c+i d)^4 \int \frac {1}{a (c+i d)^2 \left (a (c+i d)^2-i a (c+i d)^2 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}d\left (i a (c+i d)^2 \tan (e+f x)\right )}{f \left (c^2+d^2\right )}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}-\frac {i a (c+i d)^2 \int \frac {1}{\left (a (c+i d)^2-i a (c+i d)^2 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}d\left (i a (c+i d)^2 \tan (e+f x)\right )}{f \left (c^2+d^2\right )}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 a^2 (c+i d)^4 \int \frac {1}{\frac {i a^3 \tan ^2(e+f x) (c+i d)^6}{d}+\frac {a (i c+d) (c+i d)^2}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (c^2+d^2\right )}+\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 a (c+i d)^2 \arctan \left (\frac {a (c+i d)^2 \tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d} \left (c^2+d^2\right )}+\frac {2 a (-d+i c)}{f (c-i d) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}\) |
(-2*a)/(3*(I*c + d)*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*a*(c + I*d)^2*ArcT an[(a*(c + I*d)^2*Tan[e + f*x])/Sqrt[c - I*d]])/(Sqrt[c - I*d]*(c^2 + d^2) *f) + (2*a*(I*c - d))/((c - I*d)*f*Sqrt[c + d*Tan[e + f*x]]))/(c^2 + d^2)
3.12.33.3.1 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[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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. 2596 vs. \(2 (91 ) = 182\).
Time = 0.68 (sec) , antiderivative size = 2597, normalized size of antiderivative = 23.83
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2597\) |
default | \(\text {Expression too large to display}\) | \(2597\) |
parts | \(\text {Expression too large to display}\) | \(4475\) |
2/f*a/(c^2+d^2)^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))*c*d+I /f*a/(c^2+d^2)^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))*c^2-2/ f*a/(c^2+d^2)^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))*c*d+1/f *a/(c^2+d^2)^2/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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))*c*d-1/f*a/(c^2+ d^2)^2/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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))*c*d+I/f*a/(c^2+d^2)^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))*d^2-2/3/f*a/(c^2+d^2)/(c +d*tan(f*x+e))^(3/2)*d+3*I/f*a/(c^2+d^2)^(5/2)/(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))*c*d^2-3/2*I/f*a/(c^2+d^2)^(5/2)/(2*(c^2+d^2)^(1/2) +2*c)^(1/2)*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))*c*d^2-3*I/f*a/(c^2+d^2)^(5/2)/(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))*c*d^2+3/2*I/f*a/(c^2+d^2)^(5/2)/(2*(c^2+d ^2)^(1/2)+2*c)^(1/2)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (85) = 170\).
Time = 0.33 (sec) , antiderivative size = 823, normalized size of antiderivative = 7.55 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {3 \, {\left ({\left (c^{4} - 4 i \, c^{3} d - 6 \, c^{2} d^{2} + 4 i \, c d^{3} + d^{4}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{4} - 2 i \, c^{3} d - 2 i \, c d^{3} - d^{4}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{4} + 2 \, c^{2} d^{2} + d^{4}\right )} f\right )} \sqrt {-\frac {4 i \, a^{2}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{2}}} \log \left (\frac {{\left (2 \, a c + {\left ({\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}\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 {4 i \, a^{2}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{2}}} + 2 \, {\left (a c - i \, a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a}\right ) - 3 \, {\left ({\left (c^{4} - 4 i \, c^{3} d - 6 \, c^{2} d^{2} + 4 i \, c d^{3} + d^{4}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{4} - 2 i \, c^{3} d - 2 i \, c d^{3} - d^{4}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{4} + 2 \, c^{2} d^{2} + d^{4}\right )} f\right )} \sqrt {-\frac {4 i \, a^{2}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{2}}} \log \left (\frac {{\left (2 \, a c + {\left ({\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\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 {4 i \, a^{2}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{2}}} + 2 \, {\left (a c - i \, a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a}\right ) - 16 \, {\left (-2 i \, a c + a d + 2 \, {\left (-i \, a c - a d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-4 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}}}{12 \, {\left ({\left (c^{4} - 4 i \, c^{3} d - 6 \, c^{2} d^{2} + 4 i \, c d^{3} + d^{4}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{4} - 2 i \, c^{3} d - 2 i \, c d^{3} - d^{4}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{4} + 2 \, c^{2} d^{2} + d^{4}\right )} f\right )}} \]
1/12*(3*((c^4 - 4*I*c^3*d - 6*c^2*d^2 + 4*I*c*d^3 + d^4)*f*e^(4*I*f*x + 4* I*e) + 2*(c^4 - 2*I*c^3*d - 2*I*c*d^3 - d^4)*f*e^(2*I*f*x + 2*I*e) + (c^4 + 2*c^2*d^2 + d^4)*f)*sqrt(-4*I*a^2/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10* c^2*d^3 + 5*I*c*d^4 + d^5)*f^2))*log((2*a*c + ((I*c^3 + 3*c^2*d - 3*I*c*d^ 2 - d^3)*f*e^(2*I*f*x + 2*I*e) + (I*c^3 + 3*c^2*d - 3*I*c*d^2 - d^3)*f)*sq rt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sq rt(-4*I*a^2/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^2*d^3 + 5*I*c*d^4 + d^ 5)*f^2)) + 2*(a*c - I*a*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a) - 3*((c^4 - 4*I*c^3*d - 6*c^2*d^2 + 4*I*c*d^3 + d^4)*f*e^(4*I*f*x + 4*I*e) + 2*(c^4 - 2*I*c^3*d - 2*I*c*d^3 - d^4)*f*e^(2*I*f*x + 2*I*e) + (c^4 + 2*c^ 2*d^2 + d^4)*f)*sqrt(-4*I*a^2/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^2*d^ 3 + 5*I*c*d^4 + d^5)*f^2))*log((2*a*c + ((-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d ^3)*f*e^(2*I*f*x + 2*I*e) + (-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*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(- 4*I*a^2/((I*c^5 + 5*c^4*d - 10*I*c^3*d^2 - 10*c^2*d^3 + 5*I*c*d^4 + d^5)*f ^2)) + 2*(a*c - I*a*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a) - 16*( -2*I*a*c + a*d + 2*(-I*a*c - a*d)*e^(4*I*f*x + 4*I*e) + (-4*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)))/((c^4 - 4*I*c^3*d - 6*c^2*d^2 + 4*I*c*d^3 + d^4)*f*e^ (4*I*f*x + 4*I*e) + 2*(c^4 - 2*I*c^3*d - 2*I*c*d^3 - d^4)*f*e^(2*I*f*x ...
\[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=i a \left (\int \left (- \frac {i}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\right )\, dx + \int \frac {\tan {\left (e + f x \right )}}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx\right ) \]
I*a*(Integral(-I/(c**2*sqrt(c + d*tan(e + f*x)) + 2*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x) + d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2), x) + Integral(tan(e + f*x)/(c**2*sqrt(c + d*tan(e + f*x)) + 2*c*d*sqrt(c + d*t an(e + f*x))*tan(e + f*x) + d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2) , x))
Exception generated. \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((-(2*c*d^4)/((c^2-d^2)^2>0)', s ee `assume
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (85) = 170\).
Time = 0.91 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.06 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-2 \, a {\left (\frac {3 \, d \tan \left (f x + e\right ) + 4 \, c - i \, d}{-3 \, {\left (-i \, c^{2} f - 2 \, c d f + i \, d^{2} f\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} - \frac {2 i \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (c^{2} f - 2 i \, c d f - d^{2} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}}\right )} \]
-2*a*((3*d*tan(f*x + e) + 4*c - I*d)/((3*I*c^2*f + 6*c*d*f - 3*I*d^2*f)*(d *tan(f*x + e) + c)^(3/2)) - 2*I*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqr t(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))))/((c^2*f - 2*I*c*d*f - d^2*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*( -I*d/(c - sqrt(c^2 + d^2)) + 1)))
Time = 26.52 (sec) , antiderivative size = 7068, normalized size of antiderivative = 64.84 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
((a*c*2i)/(3*f*(c^2 + d^2)) + (a*(c^2 - d^2)*(c + d*tan(e + f*x))*2i)/(f*( c^2 + d^2)^2))/(c + d*tan(e + f*x))^(3/2) + (log(((((320*a^4*c^2*d^8*f^4 - 16*a^4*d^10*f^4 - 1760*a^4*c^4*d^6*f^4 + 1600*a^4*c^6*d^4*f^4 - 400*a^4*c ^8*d^2*f^4)^(1/2) - 4*a^2*c^5*f^2 - 20*a^2*c*d^4*f^2 + 40*a^2*c^3*d^2*f^2) /(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(320*a^2*c^4*d^14*f^3 - 16*a^2*d^18*f^3 + 1024*a^2*c^6*d^12*f^3 + 1440*a^2*c^8*d^10*f^3 + 1024*a^2 *c^10*d^8*f^3 + 320*a^2*c^12*d^6*f^3 - 16*a^2*c^16*d^2*f^3) - ((((320*a^4* c^2*d^8*f^4 - 16*a^4*d^10*f^4 - 1760*a^4*c^4*d^6*f^4 + 1600*a^4*c^6*d^4*f^ 4 - 400*a^4*c^8*d^2*f^4)^(1/2) - 4*a^2*c^5*f^2 - 20*a^2*c*d^4*f^2 + 40*a^2 *c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c ^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((((320*a^4*c^2*d^8*f^4 - 16*a^4*d^10* f^4 - 1760*a^4*c^4*d^6*f^4 + 1600*a^4*c^6*d^4*f^4 - 400*a^4*c^8*d^2*f^4)^( 1/2) - 4*a^2*c^5*f^2 - 20*a^2*c*d^4*f^2 + 40*a^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4 ))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 28 80*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12 *f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c ^19*d^4*f^5 + 64*c^21*d^2*f^5))/4 - 32*a*d^21*f^4 - 160*a*c^2*d^19*f^4 - 1 28*a*c^4*d^17*f^4 + 896*a*c^6*d^15*f^4 + 3136*a*c^8*d^13*f^4 + 4928*a*c...