\(\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx\) [1115]

Optimal result

Integrand size = 28, antiderivative size = 131 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=-\frac {2 i a (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 a (i c+d) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f} \] Output:

-2*I*a*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/f+2*I*a 
*(c-I*d)^2*(c+d*tan(f*x+e))^(1/2)/f+2/3*a*(I*c+d)*(c+d*tan(f*x+e))^(3/2)/f 
+2/5*I*a*(c+d*tan(f*x+e))^(5/2)/f
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=\frac {2 a \left (-15 i (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} \left (23 i c^2+35 c d-15 i d^2+d (11 i c+5 d) \tan (e+f x)+3 i d^2 \tan ^2(e+f x)\right )\right )}{15 f} \] Input:

Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(2*a*((-15*I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I* 
d]] + Sqrt[c + d*Tan[e + f*x]]*((23*I)*c^2 + 35*c*d - (15*I)*d^2 + d*((11* 
I)*c + 5*d)*Tan[e + f*x] + (3*I)*d^2*Tan[e + f*x]^2)))/(15*f)
 

Rubi [A] (warning: unable to verify)

Time = 0.79 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {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.

\(\displaystyle \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}dx\)

\(\Big \downarrow \) 4011

\(\displaystyle \int (c+d \tan (e+f x))^{3/2} (a (c-i d)+a (i c+d) \tan (e+f x))dx+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d \tan (e+f x))^{3/2} (a (c-i d)+a (i c+d) \tan (e+f x))dx+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}\)

\(\Big \downarrow \) 4011

\(\displaystyle \int \left (a (c-i d)^2+i a \tan (e+f x) (c-i d)^2\right ) \sqrt {c+d \tan (e+f x)}dx+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a (c-i d)^2+i a \tan (e+f x) (c-i d)^2\right ) \sqrt {c+d \tan (e+f x)}dx+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\)

\(\Big \downarrow \) 4011

\(\displaystyle \int \frac {a (c-i d)^3-a (i c+d)^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a (c-i d)^3-a (i c+d)^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {i a^2 (c-i d)^6 \int \frac {1}{a \sqrt {c+d \tan (e+f x)} \left (a (i c+d)^6-a (c-i d)^3 (i c+d)^3 \tan (e+f x)\right )}d\left (-a (i c+d)^3 \tan (e+f x)\right )}{f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i a (c-i d)^6 \int \frac {1}{\sqrt {c+d \tan (e+f x)} \left (a (i c+d)^6-a (c-i d)^3 (i c+d)^3 \tan (e+f x)\right )}d\left (-a (i c+d)^3 \tan (e+f x)\right )}{f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {2 i a^2 (d+i c)^3 (c-i d)^6 \int \frac {1}{\frac {a (i c+d)^7}{d}+\frac {i a^3 (c-i d)^6 \tan ^2(e+f x) (i c+d)^6}{d}}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 a (d+i c)^3 \text {arctanh}\left (\frac {a (d+i c)^3 \tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 i a (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{f}\)

Input:

Int[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(-2*a*(I*c + d)^3*ArcTanh[(a*(I*c + d)^3*Tan[e + f*x])/Sqrt[c - I*d]])/(Sq 
rt[c - I*d]*f) + ((2*I)*a*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/f + (2*a*( 
I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (((2*I)/5)*a*(c + d*Tan[e + f 
*x])^(5/2))/f
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4011

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]
 

rule 4020

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (108 ) = 216\).

Time = 0.38 (sec) , antiderivative size = 970, normalized size of antiderivative = 7.40

method	result	size
derivativedivides	\(\frac {a \left (\frac {2 i \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i c \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}-2 i \sqrt {c +d \tan \left (f x +e \right )}\, d^{2}+\frac {2 \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} d}{3}+4 \sqrt {c +d \tan \left (f x +e \right )}\, c d +\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 (\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^{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 {\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}}}+\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}}}{\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )}{f}\)	\(970\)
default	\(\frac {a \left (\frac {2 i \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i c \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}-2 i \sqrt {c +d \tan \left (f x +e \right )}\, d^{2}+\frac {2 \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} d}{3}+4 \sqrt {c +d \tan \left (f x +e \right )}\, c d +\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 (\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^{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 {\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}}}+\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}}}{\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )}{f}\)	\(970\)
parts	\(\text {Expression too large to display}\)	\(2413\)

Input:

int((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/f*a*(2/5*I*(c+d*tan(f*x+e))^(5/2)+2/3*I*c*(c+d*tan(f*x+e))^(3/2)+2*I*(c+ 
d*tan(f*x+e))^(1/2)*c^2-2*I*(c+d*tan(f*x+e))^(1/2)*d^2+2/3*(c+d*tan(f*x+e) 
)^(3/2)*d+4*(c+d*tan(f*x+e))^(1/2)*c*d+1/(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((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^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^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)))+1 
/(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/...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (101) = 202\).

Time = 0.17 (sec) , antiderivative size = 732, normalized size of antiderivative = 5.59 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")
 

Output:

-1/30*(15*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*sqrt(-(a^2 
*c^5 - 5*I*a^2*c^4*d - 10*a^2*c^3*d^2 + 10*I*a^2*c^2*d^3 + 5*a^2*c*d^4 - I 
*a^2*d^5)/f^2)*log(2*(a*c^3 - 2*I*a*c^2*d - a*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^2*c^5 - 5*I*a^2*c^4*d - 10*a^2*c^3*d^2 + 10*I*a^2*c^2 
*d^3 + 5*a^2*c*d^4 - I*a^2*d^5)/f^2) + (a*c^3 - 3*I*a*c^2*d - 3*a*c*d^2 + 
I*a*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a*c^2 - 2*I*a*c*d - a* 
d^2)) - 15*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*sqrt(-(a^ 
2*c^5 - 5*I*a^2*c^4*d - 10*a^2*c^3*d^2 + 10*I*a^2*c^2*d^3 + 5*a^2*c*d^4 - 
I*a^2*d^5)/f^2)*log(2*(a*c^3 - 2*I*a*c^2*d - a*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^2*c^5 - 5*I*a^2*c^4*d - 10*a^2*c^3*d^2 + 10*I*a^2*c 
^2*d^3 + 5*a^2*c*d^4 - I*a^2*d^5)/f^2) + (a*c^3 - 3*I*a*c^2*d - 3*a*c*d^2 
+ I*a*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a*c^2 - 2*I*a*c*d - 
a*d^2)) + 4*(-23*I*a*c^2 - 24*a*c*d + 13*I*a*d^2 + 23*(-I*a*c^2 - 2*a*c*d 
+ I*a*d^2)*e^(4*I*f*x + 4*I*e) + 2*(-23*I*a*c^2 - 35*a*c*d + 12*I*a*d^2)*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^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)
 

Sympy [F]

\[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=i a \left (\int \left (- i c^{2} \sqrt {c + d \tan {\left (e + f x \right )}}\right )\, dx + \int c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- i d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx\right ) \] Input:

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))**(5/2),x)
 

Output:

I*a*(Integral(-I*c**2*sqrt(c + d*tan(e + f*x)), x) + Integral(c**2*sqrt(c 
+ d*tan(e + f*x))*tan(e + f*x), x) + Integral(d**2*sqrt(c + d*tan(e + f*x) 
)*tan(e + f*x)**3, x) + Integral(-I*d**2*sqrt(c + d*tan(e + f*x))*tan(e + 
f*x)**2, x) + Integral(2*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2, x) 
+ Integral(-2*I*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x), x))
 

Maxima [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (101) = 202\).

Time = 0.55 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.12 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (-3 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} - 5 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c - 15 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} - 5 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d - 30 \, \sqrt {d \tan \left (f x + e\right ) + c} c d + 15 i \, \sqrt {d \tan \left (f x + e\right ) + c} d^{2} + \frac {15 \, \sqrt {2} {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2} c \sqrt {-c + \sqrt {c^{2} + d^{2}}} - i \, \sqrt {2} \sqrt {-c + \sqrt {c^{2} + d^{2}}} d - \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {-c + \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-c + \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}}\right )} a}{15 \, f} \] Input:

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

-2/15*(-3*I*(d*tan(f*x + e) + c)^(5/2) - 5*I*(d*tan(f*x + e) + c)^(3/2)*c 
- 15*I*sqrt(d*tan(f*x + e) + c)*c^2 - 5*(d*tan(f*x + e) + c)^(3/2)*d - 30* 
sqrt(d*tan(f*x + e) + c)*c*d + 15*I*sqrt(d*tan(f*x + e) + c)*d^2 + 15*sqrt 
(2)*(-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*arctan(2*(sqrt(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 - s 
qrt(c^2 + d^2)) + 1)))*a/f
 

Mupad [B] (verification not implemented)

Time = 25.43 (sec) , antiderivative size = 3899, normalized size of antiderivative = 29.76 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:

int((a + a*tan(e + f*x)*1i)*(c + d*tan(e + f*x))^(5/2),x)
 

Output:

log(((((-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - a^2*c^5*f^2 - 5 
*a^2*c*d^4*f^2 + 10*a^2*c^3*d^2*f^2)/f^4)^(1/2)*(((((-a^4*d^2*f^4*(5*c^4 + 
 d^4 - 10*c^2*d^2)^2)^(1/2) - a^2*c^5*f^2 - 5*a^2*c*d^4*f^2 + 10*a^2*c^3*d 
^2*f^2)/f^4)^(1/2)*(64*a*c^3*d^3 + 64*a*c*d^5 - 32*c*d^2*f*(((-a^4*d^2*f^4 
*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - a^2*c^5*f^2 - 5*a^2*c*d^4*f^2 + 10* 
a^2*c^3*d^2*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1/2)))/(2*f) - (16*a^2*d 
^2*(c + d*tan(e + f*x))^(1/2)*(c^6 - d^6 + 15*c^2*d^4 - 15*c^4*d^2))/f^2)) 
/2 - (8*a^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3)*((20*a^4*c^2*d^8*f^4 - a 
^4*d^10*f^4 - 110*a^4*c^4*d^6*f^4 + 100*a^4*c^6*d^4*f^4 - 25*a^4*c^8*d^2*f 
^4)^(1/2)/(4*f^4) - (a^2*c^5)/(4*f^2) - (5*a^2*c*d^4)/(4*f^2) + (5*a^2*c^3 
*d^2)/(2*f^2))^(1/2) - log(((((-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^ 
(1/2) - a^2*c^5*f^2 - 5*a^2*c*d^4*f^2 + 10*a^2*c^3*d^2*f^2)/f^4)^(1/2)*((( 
((-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - a^2*c^5*f^2 - 5*a^2*c 
*d^4*f^2 + 10*a^2*c^3*d^2*f^2)/f^4)^(1/2)*(64*a*c^3*d^3 + 64*a*c*d^5 + 32* 
c*d^2*f*(((-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(1/2) - a^2*c^5*f^2 
- 5*a^2*c*d^4*f^2 + 10*a^2*c^3*d^2*f^2)/f^4)^(1/2)*(c + d*tan(e + f*x))^(1 
/2)))/(2*f) + (16*a^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^6 - d^6 + 15*c^2*d 
^4 - 15*c^4*d^2))/f^2))/2 - (8*a^3*d^3*(3*c^2 - d^2)*(c^2 + d^2)^3)/f^3)*( 
((20*a^4*c^2*d^8*f^4 - a^4*d^10*f^4 - 110*a^4*c^4*d^6*f^4 + 100*a^4*c^6*d^ 
4*f^4 - 25*a^4*c^8*d^2*f^4)^(1/2) - a^2*c^5*f^2 - 5*a^2*c*d^4*f^2 + 10*...
 

Reduce [F]

\[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx=a \left (\left (\int \sqrt {d \tan \left (f x +e \right )+c}d x \right ) c^{2}+\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}d x \right ) d^{2} i +2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) c d i +\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) d^{2}+\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c^{2} i +2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c d \right ) \] Input:

int((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(5/2),x)
 

Output:

a*(int(sqrt(tan(e + f*x)*d + c),x)*c**2 + int(sqrt(tan(e + f*x)*d + c)*tan 
(e + f*x)**3,x)*d**2*i + 2*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 + int(sqrt(t 
an(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)