3.12.0 ∫ a+ia tan(e+fx) ___ (c+d tan(e+fx))5∕2 dx [1133]

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)-

Rubi [A] (veriﬁed)

Time = 0.30 (sec) , antiderivative size = 109, 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 = {3610, 3618, 65, 214} \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {2 i a \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}+\frac {2 i a}{f (c-i d)^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}} \]

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

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

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

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,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{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] + Dist[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] :> 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 a}{3 (i c+d) f (c+d \tan (e+f x))^{3/2}}+\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 (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)}}+\frac {\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}{\left (c^2+d^2\right )^2} \\ & = -\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)}}+\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 )}{\left (c^2+d^2\right )^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)}}-\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 \left (c^2+d^2\right )^2 f} \\ & = -\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)}} \\ \end{align*}

Mathematica [C] (veriﬁed)

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 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {2 i a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{3 (c-i d) f (c+d \tan (e+f x))^{3/2}} \]

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

(((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

Maple [B] (veriﬁed)

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	\(\text {Expression too large to display}\)	\(4475\)

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

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)*a

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

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

n((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))*d^2-1/2*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^2)^(1/2)+2*c)^(1/2)+(c

^2+d^2)^(1/2))*c^3+1/2*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^3-I/f*a/(c^2+d^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arcta

n(((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^2-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))*d^2-1/2*I/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^2+3/2/f*a/(c^2+d^2)^(5/2)/(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*l

n(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^2*d-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^3-3/2/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^2*d+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^3-3/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^2*d+3/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^2*d

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. 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 )}} \]

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

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 - 1

0*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) - 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 + 2*I*e) + (c^4 + 2*c^2*d^2 + d^4)*f)

Sympy [F]

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

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

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

*tan(e + f*x))*tan(e + f*x) + d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2), x))

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]

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

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(((-(2*c*d^4)/((c^2-d^2)^2>0)',

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. 224 vs. \(2 (85) = 170\).

Time = 0.91 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.06 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-2 \, a {\left (\frac {3 \, d \tan \left (f x + e\right ) + 4 \, c - i \, d}{-3 \, {\left (-i \, c^{2} f - 2 \, c d f + i \, d^{2} f\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} - \frac {2 i \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (c^{2} f - 2 i \, c d f - d^{2} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}}\right )} \]

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

-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*arct

an(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*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)))

Mupad [B] (veriﬁcation not implemented)

Time = 26.52 (sec) , antiderivative size = 7068, normalized size of antiderivative = 64.84 \[ \int \frac {a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]

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

((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 + 2880*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 - 128*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^10*d^11*f^4 + 4480*a*c^12*d^9*f^4 + 2432*a*c^14*d^7*f^4 + 736*

a*c^16*d^5*f^4 + 96*a*c^18*d^3*f^4))/4))/4 + 16*a^3*c*d^15*f^2 + 96*a^3*c^3*d^13*f^2 + 240*a^3*c^5*d^11*f^2 +

320*a^3*c^7*d^9*f^2 + 240*a^3*c^9*d^7*f^2 + 96*a^3*c^11*d^5*f^2 + 16*a^3*c^13*d^3*f^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))/4 + (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)*(((-((3

20*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 + 288

0*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^1

5*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 -

128*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^10*d^11*f^4 + 4480*a*c^12*d^9*f^4 + 2

432*a*c^14*d^7*f^4 + 736*a*c^16*d^5*f^4 + 96*a*c^18*d^3*f^4))/4))/4 + 16*a^3*c*d^15*f^2 + 96*a^3*c^3*d^13*f^2

+ 240*a^3*c^5*d^11*f^2 + 320*a^3*c^7*d^9*f^2 + 240*a^3*c^9*d^7*f^2 + 96*a^3*c^11*d^5*f^2 + 16*a^3*c^13*d^3*f^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))/4 - log(16*a^3*c*d^15*f^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)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f

^4 + 80*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) + (((3

20*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)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c

^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(896*a*c^6*d^15*f^4 - (((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)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 +

80*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 + 2880*c^5*d^18*f^5 + 768

0*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^1

7*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 160*a*c^2*d^19*f^4 - 128*a*c^4*d^17*f^4 - 32*a*d^21*f^4 + 31

36*a*c^8*d^13*f^4 + 4928*a*c^10*d^11*f^4 + 4480*a*c^12*d^9*f^4 + 2432*a*c^14*d^7*f^4 + 736*a*c^16*d^5*f^4 + 96

*a*c^18*d^3*f^4)) + 96*a^3*c^3*d^13*f^2 + 240*a^3*c^5*d^11*f^2 + 320*a^3*c^7*d^9*f^2 + 240*a^3*c^9*d^7*f^2 + 9

6*a^3*c^11*d^5*f^2 + 16*a^3*c^13*d^3*f^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 + 16

00*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)/(16*c

^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - log(16*a

^3*c*d^15*f^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)/(16*c^10*f^4 + 16*d^10*f^4 + 8

0*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*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)/(16

*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(896*a*c

^6*d^15*f^4 - (-((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)/(16*c^10*f^4 + 16*d^10*f^4 + 80*

c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*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 + 2880*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 + 1

3440*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) - 160*a*c^2*d

^19*f^4 - 128*a*c^4*d^17*f^4 - 32*a*d^21*f^4 + 3136*a*c^8*d^13*f^4 + 4928*a*c^10*d^11*f^4 + 4480*a*c^12*d^9*f^

4 + 2432*a*c^14*d^7*f^4 + 736*a*c^16*d^5*f^4 + 96*a*c^18*d^3*f^4)) + 96*a^3*c^3*d^13*f^2 + 240*a^3*c^5*d^11*f^

2 + 320*a^3*c^7*d^9*f^2 + 240*a^3*c^9*d^7*f^2 + 96*a^3*c^11*d^5*f^2 + 16*a^3*c^13*d^3*f^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)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 1

60*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) + (log((8*a^3*d^2*(c^2*1i - d^2*1i))/(f^3*(c^2 + d^2)^4) - ((((16*c*d^

2*(c + d*tan(e + f*x))^(1/2)*((4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2) - (32*a*c*d^2*(c^2*1i - d^2*3i))/(f*(c^2 + d^2)^2))*(

(4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/

(f^4*(c^2 + d^2)^5))^(1/2))/4 + (16*a^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^4 + d^4 - 6*c^2*d^2))/(f^2*(c^2 + d^

2)^4))*((4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2))/4)*(((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))/4 + (log((8*a^3*d^2*(c

^2*1i - d^2*1i))/(f^3*(c^2 + d^2)^4) - ((((16*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(4*(-a^4*d^2*f^4*(5*c^4 + d^4

- 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2) -

(32*a*c*d^2*(c^2*1i - d^2*3i))/(f*(c^2 + d^2)^2))*(-(4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2))/4 + (16*a^2*d^2*(c + d*tan(e +

f*x))^(1/2)*(c^4 + d^4 - 6*c^2*d^2))/(f^2*(c^2 + d^2)^4))*(-(4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2))/4)*(-((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))/4 - log(((((16*c*d^2*(c + d*tan(e + f*x))^(1/2)*((4*(-a^4*d^2*f^4*(5*c^4 + d^4

- 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2) +

(32*a*c*d^2*(c^2*1i - d^2*3i))/(f*(c^2 + d^2)^2))*((4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2))/4 + (16*a^2*d^2*(c + d*tan(e +

f*x))^(1/2)*(c^4 + d^4 - 6*c^2*d^2))/(f^2*(c^2 + d^2)^4))*((4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2))/4 + (8*a^3*d^2*(c^2*1i

- d^2*1i))/(f^3*(c^2 + d^2)^4))*(((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)/(16*c^10*f^4 +

16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - log(((((16*c*d^2*(

c + d*tan(e + f*x))^(1/2)*(-(4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2) + (32*a*c*d^2*(c^2*1i - d^2*3i))/(f*(c^2 + d^2)^2))*(-(

4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(

f^4*(c^2 + d^2)^5))^(1/2))/4 + (16*a^2*d^2*(c + d*tan(e + f*x))^(1/2)*(c^4 + d^4 - 6*c^2*d^2))/(f^2*(c^2 + d^2

)^4))*(-(4*(-a^4*d^2*f^4*(5*c^4 + d^4 - 10*c^2*d^2)^2)^(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)/(f^4*(c^2 + d^2)^5))^(1/2))/4 + (8*a^3*d^2*(c^2*1i - d^2*1i))/(f^3*(c^2 + d^2)^4))*(-((320*a^4*c^2*d^8