\(\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx\) [1109]

Optimal result

Integrand size = 28, antiderivative size = 98 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=-\frac {2 i a (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {2 a (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3609, 3618, 65, 214} \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=-\frac {2 i a (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a (d+i c) \sqrt {c+d \tan (e+f x)}}{f} \]

[In]

Int[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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 3609

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 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\int \sqrt {c+d \tan (e+f x)} (a (c-i d)+a (i c+d) \tan (e+f x)) \, dx \\ & = \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}+\int \frac {a (c-i d)^2+i a (c-i d)^2 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \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}+\frac {\left (i a^2 (c-i d)^4\right ) \text {Subst}\left (\int \frac {1}{\left (-a^2 (c-i d)^4+a (c-i d)^2 x\right ) \sqrt {c-\frac {i d x}{a (c-i d)^2}}} \, dx,x,i a (c-i d)^2 \tan (e+f x)\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}-\frac {\left (2 a^3 (c-i d)^6\right ) \text {Subst}\left (\int \frac {1}{-a^2 (c-i d)^4-\frac {i a^2 c (c-i d)^4}{d}+\frac {i a^2 (c-i d)^4 x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\frac {2 a \left (-3 i (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+(4 i c+3 d+i d \tan (e+f x)) \sqrt {c+d \tan (e+f x)}\right )}{3 f} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2),x]

[Out]

(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)

Maple [B] (verified)

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	\(\text {Expression too large to display}\)	\(1659\)

[In]

int((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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))))

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. 504 vs. \(2 (76) = 152\).

Time = 0.26 (sec) , antiderivative size = 504, normalized size of antiderivative = 5.14 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\frac {3 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} \log \left (-\frac {2 \, {\left (-i \, a c^{2} - a c d + {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} + {\left (-i \, a c^{2} - 2 \, a c d + i \, a d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{i \, a c + a d}\right ) - 3 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} \log \left (-\frac {2 \, {\left (-i \, a c^{2} - a c d - {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {a^{2} c^{3} - 3 i \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} + i \, a^{2} d^{3}}{f^{2}}} + {\left (-i \, a c^{2} - 2 \, a c d + i \, a d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{i \, a c + a d}\right ) - 8 \, {\left (-2 i \, a c - a d + 2 \, {\left (-i \, a c - a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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)

Sympy [F]

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

[In]

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))**(3/2),x)

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

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. 229 vs. \(2 (76) = 152\).

Time = 0.51 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.34 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\frac {2}{3} i \, a {\left (\frac {6 \, {\left (c^{2} - 2 i \, c d - d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} f^{2} + 3 \, \sqrt {d \tan \left (f x + e\right ) + c} c f^{2} - 3 i \, \sqrt {d \tan \left (f x + e\right ) + c} d f^{2}}{f^{3}}\right )} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

2/3*I*a*(6*(c^2 - 2*I*c*d - d^2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(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*sqr
t(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)

Mupad [B] (verification not implemented)

Time = 17.91 (sec) , antiderivative size = 2869, normalized size of antiderivative = 29.28 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx=\text {Too large to display} \]

[In]

int((a + a*tan(e + f*x)*1i)*(c + d*tan(e + f*x))^(3/2),x)

[Out]

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)*((1
6*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*1
i + 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*1
i + 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*ta
n(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(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)*((3*a^2*c*d^2)/(4*f^2) - (a^2*c^3)/(4*f^2) - (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))^(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*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*d^3 + a*c^2*d + c*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 - (16*a^3*c*d^3*(c^2 + d^2)^2)/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*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*d^3 + a*c^2*d + c*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 - (16*a^3*c*d^3*(c^2 + d^2)^2)/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*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*d^3 + a*c^2*d - c*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 - (1
6*a^3*c*d^3*(c^2 + d^2)^2)/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*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*d^3 + a*c^2*d - c*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 -
(16*a^3*c*d^3*(c^2 + d^2)^2)/f^3)*((3*a^2*c*d^2)/(4*f^2) - (a^2*c^3)/(4*f^2) - (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))^(1/2) + (a*(c + d*tan(e + f*x))^(3/2)*2i)/(3*f) + (a*c*(c + d*tan(e + f*
x))^(1/2)*2i)/f + (2*a*d*(c + d*tan(e + f*x))^(1/2))/f