3.1122 \(\int \frac {1}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx\)
Optimal. Leaf size=155 \[ -\frac {\sqrt {c+d \tan (e+f x)}}{2 f (-d+i c) (a+i a \tan (e+f x))}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f \sqrt {c-i d}}+\frac {(-2 d+i c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f (c+i d)^{3/2}} \]
[Out]
1/2*(I*c-2*d)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a/(c+I*d)^(3/2)/f-1/2*I*arctanh((c+d*tan(f*x+e))^(
1/2)/(c-I*d)^(1/2))/a/f/(c-I*d)^(1/2)-1/2*(c+d*tan(f*x+e))^(1/2)/(I*c-d)/f/(a+I*a*tan(f*x+e))
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Rubi [A] time = 0.30, antiderivative size = 155, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used =
{3552, 3539, 3537, 63, 208} \[ -\frac {\sqrt {c+d \tan (e+f x)}}{2 f (-d+i c) (a+i a \tan (e+f x))}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f \sqrt {c-i d}}+\frac {(-2 d+i c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f (c+i d)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]
[Out]
((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*Sqrt[c - I*d]*f) + ((I*c - 2*d)*ArcTanh[Sqrt[c + d
*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(3/2)*f) - Sqrt[c + d*Tan[e + f*x]]/(2*(I*c - d)*f*(a + I*a*Tan[
e + f*x]))
Rule 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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]
Rule 3539
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rule 3552
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(a
*(c + d*Tan[e + f*x])^(n + 1))/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c +
d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
&& NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx &=-\frac {\sqrt {c+d \tan (e+f x)}}{2 (i c-d) f (a+i a \tan (e+f x))}+\frac {\int \frac {\frac {1}{2} a (2 i c-3 d)+\frac {1}{2} i a d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 a^2 (i c-d)}\\ &=-\frac {\sqrt {c+d \tan (e+f x)}}{2 (i c-d) f (a+i a \tan (e+f x))}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a}+\frac {(c+2 i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a (c+i d)}\\ &=-\frac {\sqrt {c+d \tan (e+f x)}}{2 (i c-d) f (a+i a \tan (e+f x))}+\frac {i \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{4 a f}-\frac {(i (c+2 i d)) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{4 a (c+i d) f}\\ &=-\frac {\sqrt {c+d \tan (e+f x)}}{2 (i c-d) f (a+i a \tan (e+f x))}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{2 a d f}-\frac {(c+2 i d) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{2 a (c+i d) d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a \sqrt {c-i d} f}+\frac {(i c-2 d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a (c+i d)^{3/2} f}-\frac {\sqrt {c+d \tan (e+f x)}}{2 (i c-d) f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 222, normalized size = 1.43 \[ \frac {\sec (e+f x) (\cos (f x)+i \sin (f x)) \left (\frac {2 \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt {c+d \tan (e+f x)}}{c+i d}-\frac {2 (\cos (e)+i \sin (e)) \left (\sqrt {-c+i d} (2 d-i c) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )-i (-c-i d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right )}{(-c-i d)^{3/2} \sqrt {-c+i d}}\right )}{4 f (a+i a \tan (e+f x))} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]
[Out]
(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-2*(Sqrt[-c + I*d]*((-I)*c + 2*d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt
[-c - I*d]] - I*(-c - I*d)^(3/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*Sin[e]))/((-c -
I*d)^(3/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*Sqrt[c + d*Tan[e + f*x]])/(c + I*d)))/(4*
f*(a + I*a*Tan[e + f*x]))
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fricas [B] time = 0.55, size = 984, normalized size = 6.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x, algorithm="fricas")
[Out]
-1/4*((I*a*c - a*d)*f*sqrt(1/4*I/((-I*a^2*c - a^2*d)*f^2))*e^(2*I*f*x + 2*I*e)*log(-2*(2*((I*a*c + a*d)*f*e^(2
*I*f*x + 2*I*e) + (I*a*c + a*d)*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))*s
qrt(1/4*I/((-I*a^2*c - a^2*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + (-I*a*c + a*d
)*f*sqrt(1/4*I/((-I*a^2*c - a^2*d)*f^2))*e^(2*I*f*x + 2*I*e)*log(-2*(2*((-I*a*c - a*d)*f*e^(2*I*f*x + 2*I*e) +
(-I*a*c - a*d)*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(1/4*I/((-I*a
^2*c - a^2*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + (I*a*c - a*d)*f*sqrt((-I*c^2
+ 4*c*d + 4*I*d^2)/((4*I*a^2*c^3 - 12*a^2*c^2*d - 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2))*e^(2*I*f*x + 2*I*e)*log(-1
/2*(-I*c^2 + 3*c*d + 2*I*d^2 + 2*((a*c^2 + 2*I*a*c*d - a*d^2)*f*e^(2*I*f*x + 2*I*e) + (a*c^2 + 2*I*a*c*d - a*d
^2)*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((-I*c^2 + 4*c*d + 4*I*d^
2)/((4*I*a^2*c^3 - 12*a^2*c^2*d - 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2)) + (-I*c^2 + 2*c*d)*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)*f)) + (-I*a*c + a*d)*f*sqrt((-I*c^2 + 4*c*d + 4*I*d^2)/((4*I*a
^2*c^3 - 12*a^2*c^2*d - 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2))*e^(2*I*f*x + 2*I*e)*log(-1/2*(-I*c^2 + 3*c*d + 2*I*d
^2 - 2*((a*c^2 + 2*I*a*c*d - a*d^2)*f*e^(2*I*f*x + 2*I*e) + (a*c^2 + 2*I*a*c*d - a*d^2)*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((-I*c^2 + 4*c*d + 4*I*d^2)/((4*I*a^2*c^3 - 12*a^2*
c^2*d - 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2)) + (-I*c^2 + 2*c*d)*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)*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))*(e^(2*I*
f*x + 2*I*e) + 1))*e^(-2*I*f*x - 2*I*e)/((I*a*c - a*d)*f)
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giac [B] time = 0.75, size = 376, normalized size = 2.43 \[ \frac {\sqrt {d \tan \left (f x + e\right ) + c} d}{2 \, {\left (a c f + i \, a d f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}} + \frac {4 \, {\left (c + 2 i \, d\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (2 i \, a c f - 2 \, a d f\right )} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 i \, \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{a \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x, algorithm="giac")
[Out]
1/2*sqrt(d*tan(f*x + e) + c)*d/((a*c*f + I*a*d*f)*(d*tan(f*x + e) - I*d)) + 4*(c + 2*I*d)*arctan(4*(sqrt(d*tan
(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c +
8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/((2*I*a*c*f - 2*a*d*f)*sqrt(-8*c + 8*
sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 2*I*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)
*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) - I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 +
d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/(a*sqrt(-8*c + 8*sqrt(c^2 + d^2))*f*(-I*d/(c - sqrt(c^2 + d^2)) + 1))
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maple [A] time = 0.38, size = 191, normalized size = 1.23 \[ \frac {d \sqrt {c +d \tan \left (f x +e \right )}}{2 f a \left (i d +c \right ) \left (d \tan \left (f x +e \right )-i d \right )}-\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c}{2 f a \left (i d +c \right ) \sqrt {-i d -c}}+\frac {d \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{f a \left (i d +c \right ) \sqrt {-i d -c}}+\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{2 f a \sqrt {i d -c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x)
[Out]
1/2/f/a*d/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(d*tan(f*x+e)-I*d)-1/2*I/f/a/(c+I*d)/(-I*d-c)^(1/2)*arctan((c+d*tan(f
*x+e))^(1/2)/(-I*d-c)^(1/2))*c+1/f/a*d/(c+I*d)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))+1/
2*I/f/a/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.
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mupad [B] time = 8.31, size = 12379, normalized size = 79.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + a*tan(e + f*x)*1i)*(c + d*tan(e + f*x))^(1/2)),x)
[Out]
log(a*d^6*f*1i - ((-(c*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i - a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*
c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4
*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a
^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c
^2*d^4))^(1/2)*1i + a^2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c
^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((
4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4
*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i + 2*a^2*c*d*f^2*((((48*c^2*d^7
- 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5
*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4
*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(2
56*d^6 + 256*c^2*d^4))^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2)*(
24*a^3*d^7*f^3 + a^3*c*d^6*f^3*16i + 32*a^3*c^2*d^5*f^3 + a^3*c^3*d^4*f^3*16i + 8*a^3*c^4*d^3*f^3 - 2*(c + d*t
an(e + f*x))^(1/2)*(a^2*c^2*d^3*f^2*64i - 32*a^2*c*d^4*f^2 + 32*a^2*c^3*d^2*f^2)*(a^2*d^2*f^2 - a^2*c^2*f^2 +
a^2*c*d*f^2*2i)*(-(c*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i - a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^
4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f
^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4
*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2
*d^4))^(1/2)*1i + a^2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2
*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*
d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c
^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i + 2*a^2*c*d*f^2*((((48*c^2*d^7 -
48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d
^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d
^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256
*d^6 + 256*c^2*d^4))^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2)) -
2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(c*d^3*6i - 5*d^4 + 2*c^2*d^2))*(-(c
*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i - a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*
f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2
*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*
c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i + a^
2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d
^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^
4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4
+ 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i + 2*a^2*c*d*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)
*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a
^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d
^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))
^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2) - (3*a*c*d^5*f)/2 - (a*
c^3*d^3*f)/2)*(-(c*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i - a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*
d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2
+ a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c
^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d
^4))^(1/2)*1i + a^2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d
^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^
8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4
*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i + 2*a^2*c*d*f^2*((((48*c^2*d^7 - 48
*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4
)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4
*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d
^6 + 256*c^2*d^4))^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2) + log
(a*d^6*f*1i - ((-(c*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i + a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4
*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^
2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*
c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*
d^4))^(1/2)*1i - a^2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*
d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d
^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^
4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i - 2*a^2*c*d*f^2*((((48*c^2*d^7 - 4
8*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^
4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^
4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*
d^6 + 256*c^2*d^4))^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2)*(24*
a^3*d^7*f^3 + a^3*c*d^6*f^3*16i + 32*a^3*c^2*d^5*f^3 + a^3*c^3*d^4*f^3*16i + 8*a^3*c^4*d^3*f^3 - 2*(c + d*tan(
e + f*x))^(1/2)*(a^2*c^2*d^3*f^2*64i - 32*a^2*c*d^4*f^2 + 32*a^2*c^3*d^2*f^2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2
*c*d*f^2*2i)*(-(c*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i + a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d
^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2
+ a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^
2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^
4))^(1/2)*1i - a^2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^
2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8
+ 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*
f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i - 2*a^2*c*d*f^2*((((48*c^2*d^7 - 48*
d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)
/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*
f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^
6 + 256*c^2*d^4))^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2)) - 2*(
c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(c*d^3*6i - 5*d^4 + 2*c^2*d^2))*(-(c*d^
6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i + a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2
+ a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^
2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d
^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i - a^2*d
^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8
+ 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d
^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 +
2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i - 2*a^2*c*d*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i
)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*
d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*
f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1
/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2) - (3*a*c*d^5*f)/2 - (a*c^3
*d^3*f)/2)*(-(c*d^6*48i + 48*d^7 + 96*c^2*d^5 - c^3*d^4*32i + a^2*c^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5
)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 +
a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*
d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4)
)^(1/2)*1i - a^2*d^2*f^2*((((48*c^2*d^7 - 48*d^9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*
f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 +
3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^
4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i - 2*a^2*c*d*f^2*((((48*c^2*d^7 - 48*d^
9 + 32*c^4*d^5)*1i)/(a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2) + (144*c*d^8 + 112*c^3*d^6 + 32*c^5*d^4)/(
a^2*c^4*f^2 + a^2*d^4*f^2 + 2*a^2*c^2*d^2*f^2))^2 - 4*((4*d^8 + 3*c^2*d^6 + c^4*d^4)/(a^4*c^4*f^4 + a^4*d^4*f^
4 + 2*a^4*c^2*d^2*f^4) + ((4*c*d^7 + 2*c^3*d^5)*1i)/(a^4*c^4*f^4 + a^4*d^4*f^4 + 2*a^4*c^2*d^2*f^4))*(256*d^6
+ 256*c^2*d^4))^(1/2))/(512*(d^6 + c^2*d^4)*(a^2*d^2*f^2*1i - a^2*c^2*f^2*1i + 2*a^2*c*d*f^2)))^(1/2) - log(a*
d^6*f*1i - (-(144*c*d^8 - d^9*48i + c^2*d^7*48i + 112*c^3*d^6 + c^4*d^5*32i + 32*c^5*d^4 + a^2*c^4*f^2*(-(c*d^
17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^
4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) + a^2*d^4*f^2*(-(c
*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/
(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) + 2*a^2*c^2*d^2
*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c
^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2))/(512*
a^2*d^10*f^2 + 1536*a^2*c^2*d^8*f^2 + 1536*a^2*c^4*d^6*f^2 + 512*a^2*c^6*d^4*f^2))^(1/2)*((-(144*c*d^8 - d^9*4
8i + c^2*d^7*48i + 112*c^3*d^6 + c^4*d^5*32i + 32*c^5*d^4 + a^2*c^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^
2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4
*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) + a^2*d^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984
*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*
a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) + 2*a^2*c^2*d^2*f^2*(-(c*d^17*17920i + 6400*d^
18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8
*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2))/(512*a^2*d^10*f^2 + 1536*a^2*c^2*d^8
*f^2 + 1536*a^2*c^4*d^6*f^2 + 512*a^2*c^6*d^4*f^2))^(1/2)*(24*a^3*d^7*f^3 + a^3*c*d^6*f^3*16i + 32*a^3*c^2*d^5
*f^3 + a^3*c^3*d^4*f^3*16i + 8*a^3*c^4*d^3*f^3 + 2*(-(144*c*d^8 - d^9*48i + c^2*d^7*48i + 112*c^3*d^6 + c^4*d^
5*32i + 32*c^5*d^4 + a^2*c^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^
14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^
4*c^6*d^2*f^4))^(1/2) + a^2*d^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4
*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4
*a^4*c^6*d^2*f^4))^(1/2) + 2*a^2*c^2*d^2*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i -
10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^
4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2))/(512*a^2*d^10*f^2 + 1536*a^2*c^2*d^8*f^2 + 1536*a^2*c^4*d^6*f^2 + 512*a^2*c
^6*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(a^2*c^2*d^3*f^2*64i - 32*a^2*c*d^4*f^2 + 32*a^2*c^3*d^2*f^2)*(a
^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)) + 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*
d*f^2*2i)*(c*d^3*6i - 5*d^4 + 2*c^2*d^2)) - (3*a*c*d^5*f)/2 - (a*c^3*d^3*f)/2)*(-(144*c*d^8 - d^9*48i + c^2*d^
7*48i + 112*c^3*d^6 + c^4*d^5*32i + 32*c^5*d^4 + a^2*c^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^
3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^
4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) + a^2*d^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 +
c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6
*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) + 2*a^2*c^2*d^2*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c
^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^
4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2))/(512*a^2*d^10*f^2 + 1536*a^2*c^2*d^8*f^2 + 1536
*a^2*c^4*d^6*f^2 + 512*a^2*c^6*d^4*f^2))^(1/2) - log(a*d^6*f*1i - (-(144*c*d^8 - d^9*48i + c^2*d^7*48i + 112*c
^3*d^6 + c^4*d^5*32i + 32*c^5*d^4 - a^2*c^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i
- 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4
*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) - a^2*d^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*112
64i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*
c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) - 2*a^2*c^2*d^2*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3
*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4
+ 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2))/(512*a^2*d^10*f^2 + 1536*a^2*c^2*d^8*f^2 + 1536*a^2*c^4*d^6*
f^2 + 512*a^2*c^6*d^4*f^2))^(1/2)*((-(144*c*d^8 - d^9*48i + c^2*d^7*48i + 112*c^3*d^6 + c^4*d^5*32i + 32*c^5*d
^4 - a^2*c^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*15
36i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^
(1/2) - a^2*d^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13
*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4
))^(1/2) - 2*a^2*c^2*d^2*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 +
c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^
6*d^2*f^4))^(1/2))/(512*a^2*d^10*f^2 + 1536*a^2*c^2*d^8*f^2 + 1536*a^2*c^4*d^6*f^2 + 512*a^2*c^6*d^4*f^2))^(1/
2)*(24*a^3*d^7*f^3 + a^3*c*d^6*f^3*16i + 32*a^3*c^2*d^5*f^3 + a^3*c^3*d^4*f^3*16i + 8*a^3*c^4*d^3*f^3 + 2*(-(1
44*c*d^8 - d^9*48i + c^2*d^7*48i + 112*c^3*d^6 + c^4*d^5*32i + 32*c^5*d^4 - a^2*c^4*f^2*(-(c*d^17*17920i + 640
0*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4
*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) - a^2*d^4*f^2*(-(c*d^17*17920i +
6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c^8*f^4 +
a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) - 2*a^2*c^2*d^2*f^2*(-(c*d^17*
17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*c^6*d^12)/(a^4*c
^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2))/(512*a^2*d^10*f^2 +
1536*a^2*c^2*d^8*f^2 + 1536*a^2*c^4*d^6*f^2 + 512*a^2*c^6*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(a^2*c^2*
d^3*f^2*64i - 32*a^2*c*d^4*f^2 + 32*a^2*c^3*d^2*f^2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)) + 2*(c + d*
tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(c*d^3*6i - 5*d^4 + 2*c^2*d^2)) - (3*a*c*d^5*
f)/2 - (a*c^3*d^3*f)/2)*(-(144*c*d^8 - d^9*48i + c^2*d^7*48i + 112*c^3*d^6 + c^4*d^5*32i + 32*c^5*d^4 - a^2*c^
4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 2304*
c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) - a^2
*d^4*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1536i - 23
04*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))^(1/2) -
2*a^2*c^2*d^2*f^2*(-(c*d^17*17920i + 6400*d^18 - 9984*c^2*d^16 + c^3*d^15*11264i - 10496*c^4*d^14 + c^5*d^13*1
536i - 2304*c^6*d^12)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4))
^(1/2))/(512*a^2*d^10*f^2 + 1536*a^2*c^2*d^8*f^2 + 1536*a^2*c^4*d^6*f^2 + 512*a^2*c^6*d^4*f^2))^(1/2) - (d*(c
+ d*tan(e + f*x))^(1/2)*1i)/(2*a*f*(d*1i - d*tan(e + f*x))*(c*1i - d))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {1}{\sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} - i \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e)),x)
[Out]
-I*Integral(1/(sqrt(c + d*tan(e + f*x))*tan(e + f*x) - I*sqrt(c + d*tan(e + f*x))), x)/a
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