3.12.23 \(\int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx\) [1123]

Optimal. Leaf size=221 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 \sqrt {c-i d} f}+\frac {\left (2 i c^2-6 c d-7 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{5/2} f}+\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2} \]

[Out]

1/8*(2*I*c^2-6*c*d-7*I*d^2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^2/(c+I*d)^(5/2)/f-1/4*I*arctanh((c
+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^2/f/(c-I*d)^(1/2)+1/8*(2*I*c-5*d)*(c+d*tan(f*x+e))^(1/2)/a^2/(c+I*d)^2/f
/(1+I*tan(f*x+e))-1/4*(c+d*tan(f*x+e))^(1/2)/(I*c-d)/f/(a+I*a*tan(f*x+e))^2

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Rubi [A]
time = 0.42, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3640, 3677, 3620, 3618, 65, 214} \begin {gather*} \frac {\left (2 i c^2-6 c d-7 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 f (c+i d)^{5/2}}+\frac {(-5 d+2 i c) \sqrt {c+d \tan (e+f x)}}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 f \sqrt {c-i d}}-\frac {\sqrt {c+d \tan (e+f x)}}{4 f (-d+i c) (a+i a \tan (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]),x]

[Out]

((-1/4*I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*Sqrt[c - I*d]*f) + (((2*I)*c^2 - 6*c*d - (7*I)
*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(5/2)*f) + (((2*I)*c - 5*d)*Sqrt[c + d
*Tan[e + f*x]])/(8*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])) - Sqrt[c + d*Tan[e + f*x]]/(4*(I*c - d)*f*(a + I*a*
Tan[e + f*x])^2)

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 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]

Rule 3620

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 3640

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*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] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3677

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx &=-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a (4 i c-7 d)-\frac {3}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{4 a^2 (i c-d)}\\ &=\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^2+10 i c d-9 d^2\right )-\frac {1}{2} a^2 (2 c+5 i d) d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^4 (c+i d)^2}\\ &=\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^2}+\frac {\left (2 c^2+6 i c d-7 d^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^2 (c+i d)^2}\\ &=\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}+\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{8 a^2 f}-\frac {\left (i \left (2 c^2+6 i c d-7 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{16 a^2 (c+i d)^2 f}\\ &=\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\text {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 )}{4 a^2 d f}-\frac {\left (2 c^2+6 i c d-7 d^2\right ) \text {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 )}{8 a^2 (c+i d)^2 d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 \sqrt {c-i d} f}-\frac {\left (6 c d-i \left (2 c^2-7 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{5/2} f}+\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}\\ \end {align*}

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Mathematica [A]
time = 2.24, size = 275, normalized size = 1.24 \begin {gather*} \frac {\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (\frac {2 \left (\sqrt {-c+i d} \left (-2 i c^2+6 c d+7 i d^2\right ) \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )+2 i (-c-i d)^{5/2} \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right ) (\cos (2 e)+i \sin (2 e))}{(-c-i d)^{5/2} \sqrt {-c+i d}}+\frac {2 \cos (e+f x) (i \cos (2 f x)+\sin (2 f x)) ((4 c+7 i d) \cos (e+f x)+(2 i c-5 d) \sin (e+f x)) \sqrt {c+d \tan (e+f x)}}{(c+i d)^2}\right )}{16 f (a+i a \tan (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]),x]

[Out]

(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*(Sqrt[-c + I*d]*((-2*I)*c^2 + 6*c*d + (7*I)*d^2)*ArcTan[Sqrt[c +
d*Tan[e + f*x]]/Sqrt[-c - I*d]] + (2*I)*(-c - I*d)^(5/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Co
s[2*e] + I*Sin[2*e]))/((-c - I*d)^(5/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[2*f*x] + Sin[2*f*x])*((4*c +
(7*I)*d)*Cos[e + f*x] + ((2*I)*c - 5*d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(c + I*d)^2))/(16*f*(a + I*a*T
an[e + f*x])^2)

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Maple [A]
time = 0.35, size = 232, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 d^{3} \sqrt {i d -c}}+\frac {\frac {\frac {d \left (5 i d +2 c \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4 i c d +2 c^{2}-2 d^{2}}-\frac {d \left (9 i c d +2 c^{2}-7 d^{2}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (2 i c d +c^{2}-d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (2 i c^{2}-7 i d^{2}-6 c d \right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}}{8 d^{3}}\right )}{f \,a^{2}}\)	\(232\)
default	\(\frac {2 d^{3} \left (\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 d^{3} \sqrt {i d -c}}+\frac {\frac {\frac {d \left (5 i d +2 c \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4 i c d +2 c^{2}-2 d^{2}}-\frac {d \left (9 i c d +2 c^{2}-7 d^{2}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (2 i c d +c^{2}-d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (2 i c^{2}-7 i d^{2}-6 c d \right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}}{8 d^{3}}\right )}{f \,a^{2}}\)	\(232\)

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))^2,x,method=_RETURNVERBOSE)

[Out]

2/f/a^2*d^3*(1/8*I/d^3/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))+1/8/d^3*((1/2*d*(5*I*d+2*c)/
(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(3/2)-1/2*d*(9*I*c*d+2*c^2-7*d^2)/(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(1/2))
/(-d*tan(f*x+e)+I*d)^2-1/2*(2*I*c^2-6*c*d-7*I*d^2)/(2*I*c*d+c^2-d^2)/(-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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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))^2,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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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1429 vs. \(2 (179) = 358\).
time = 1.57, size = 1429, normalized size = 6.47 \begin {gather*} -\frac {{\left (8 \, {\left (a^{2} c^{2} + 2 i \, a^{2} c d - a^{2} d^{2}\right )} f \sqrt {\frac {i}{16 \, {\left (-i \, a^{4} c - a^{4} d\right )} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-2 \, {\left (4 \, {\left ({\left (i \, a^{2} c + a^{2} d\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, a^{2} c + a^{2} d\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 {i}{16 \, {\left (-i \, a^{4} c - a^{4} d\right )} f^{2}}} - {\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - c\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) - 8 \, {\left (a^{2} c^{2} + 2 i \, a^{2} c d - a^{2} d^{2}\right )} f \sqrt {\frac {i}{16 \, {\left (-i \, a^{4} c - a^{4} d\right )} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-2 \, {\left (4 \, {\left ({\left (-i \, a^{2} c - a^{2} d\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, a^{2} c - a^{2} d\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 {i}{16 \, {\left (-i \, a^{4} c - a^{4} d\right )} f^{2}}} - {\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - c\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) + {\left (a^{2} c^{2} + 2 i \, a^{2} c d - a^{2} d^{2}\right )} f \sqrt {-\frac {4 i \, c^{4} - 24 \, c^{3} d - 64 i \, c^{2} d^{2} + 84 \, c d^{3} + 49 i \, d^{4}}{{\left (i \, a^{4} c^{5} - 5 \, a^{4} c^{4} d - 10 i \, a^{4} c^{3} d^{2} + 10 \, a^{4} c^{2} d^{3} + 5 i \, a^{4} c d^{4} - a^{4} d^{5}\right )} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (2 \, c^{3} + 8 i \, c^{2} d - 13 \, c d^{2} - 7 i \, d^{3} + {\left ({\left (i \, a^{2} c^{3} - 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} + a^{2} d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, a^{2} c^{3} - 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} + a^{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 \, c^{4} - 24 \, c^{3} d - 64 i \, c^{2} d^{2} + 84 \, c d^{3} + 49 i \, d^{4}}{{\left (i \, a^{4} c^{5} - 5 \, a^{4} c^{4} d - 10 i \, a^{4} c^{3} d^{2} + 10 \, a^{4} c^{2} d^{3} + 5 i \, a^{4} c d^{4} - a^{4} d^{5}\right )} f^{2}}} + {\left (2 \, c^{3} + 6 i \, c^{2} d - 7 \, c d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, {\left (-i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} f}\right ) - {\left (a^{2} c^{2} + 2 i \, a^{2} c d - a^{2} d^{2}\right )} f \sqrt {-\frac {4 i \, c^{4} - 24 \, c^{3} d - 64 i \, c^{2} d^{2} + 84 \, c d^{3} + 49 i \, d^{4}}{{\left (i \, a^{4} c^{5} - 5 \, a^{4} c^{4} d - 10 i \, a^{4} c^{3} d^{2} + 10 \, a^{4} c^{2} d^{3} + 5 i \, a^{4} c d^{4} - a^{4} d^{5}\right )} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (2 \, c^{3} + 8 i \, c^{2} d - 13 \, c d^{2} - 7 i \, d^{3} + {\left ({\left (-i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 i \, a^{2} c d^{2} - a^{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 \, c^{4} - 24 \, c^{3} d - 64 i \, c^{2} d^{2} + 84 \, c d^{3} + 49 i \, d^{4}}{{\left (i \, a^{4} c^{5} - 5 \, a^{4} c^{4} d - 10 i \, a^{4} c^{3} d^{2} + 10 \, a^{4} c^{2} d^{3} + 5 i \, a^{4} c d^{4} - a^{4} d^{5}\right )} f^{2}}} + {\left (2 \, c^{3} + 6 i \, c^{2} d - 7 \, c d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, {\left (-i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} f}\right ) - 2 \, {\left (3 \, {\left (i \, c - 2 \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - {\left (-4 i \, c + 7 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d\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}}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{32 \, {\left (a^{2} c^{2} + 2 i \, a^{2} c d - a^{2} d^{2}\right )} f} \end {gather*}

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))^2,x, algorithm="fricas")

[Out]

-1/32*(8*(a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(1/16*I/((-I*a^4*c - a^4*d)*f^2))*e^(4*I*f*x + 4*I*e)*log(-2*
(4*((I*a^2*c + a^2*d)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c + a^2*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(1/16*I/((-I*a^4*c - a^4*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-
2*I*f*x - 2*I*e)) - 8*(a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(1/16*I/((-I*a^4*c - a^4*d)*f^2))*e^(4*I*f*x + 4
*I*e)*log(-2*(4*((-I*a^2*c - a^2*d)*f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c - a^2*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(1/16*I/((-I*a^4*c - a^4*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2
*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + (a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(-(4*I*c^4 - 24*c^3*d - 64*I*c^2*d^
2 + 84*c*d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d
^5)*f^2))*e^(4*I*f*x + 4*I*e)*log(1/8*(2*c^3 + 8*I*c^2*d - 13*c*d^2 - 7*I*d^3 + ((I*a^2*c^3 - 3*a^2*c^2*d - 3*
I*a^2*c*d^2 + a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c^3 - 3*a^2*c^2*d - 3*I*a^2*c*d^2 + a^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*c^4 - 24*c^3*d - 64*I*c^2*d^2 + 8
4*c*d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d^5)*f
^2)) + (2*c^3 + 6*I*c^2*d - 7*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-I*a^2*c^3 + 3*a^2*c^2*d + 3*
I*a^2*c*d^2 - a^2*d^3)*f)) - (a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(-(4*I*c^4 - 24*c^3*d - 64*I*c^2*d^2 + 84
*c*d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d^5)*f^
2))*e^(4*I*f*x + 4*I*e)*log(1/8*(2*c^3 + 8*I*c^2*d - 13*c*d^2 - 7*I*d^3 + ((-I*a^2*c^3 + 3*a^2*c^2*d + 3*I*a^2
*c*d^2 - a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c^3 + 3*a^2*c^2*d + 3*I*a^2*c*d^2 - a^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*c^4 - 24*c^3*d - 64*I*c^2*d^2 + 84*c*
d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d^5)*f^2))
+ (2*c^3 + 6*I*c^2*d - 7*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-I*a^2*c^3 + 3*a^2*c^2*d + 3*I*a^
2*c*d^2 - a^2*d^3)*f)) - 2*(3*(I*c - 2*d)*e^(4*I*f*x + 4*I*e) - (-4*I*c + 7*d)*e^(2*I*f*x + 2*I*e) + I*c - d)*
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^(-4*I*f*x - 4*I*e)/((a^2*c^2 + 2*
I*a^2*c*d - a^2*d^2)*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {1}{\sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 i \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} - \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx}{a^{2}} \end {gather*}

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))**2,x)

[Out]

-Integral(1/(sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2 - 2*I*sqrt(c + d*tan(e + f*x))*tan(e + f*x) - sqrt(c + d
*tan(e + f*x))), x)/a**2

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (179) = 358\).
time = 0.64, size = 494, normalized size = 2.24 \begin {gather*} -\frac {2 \, {\left (2 \, c^{2} + 6 i \, c d - 7 \, 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 )}{-8 \, {\left (i \, a^{2} c^{2} f - 2 \, a^{2} c d f - i \, a^{2} 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 )}} + \frac {2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d - 2 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d + 5 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{2} - 9 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{2} + 7 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{3}}{8 \, {\left (a^{2} c^{2} f + 2 i \, a^{2} c d f - a^{2} d^{2} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2}} - \frac {\arctan \left (-\frac {2 \, {\left (-i \, \sqrt {d \tan \left (f x + e\right ) + c} c - i \, \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} c - 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 )}{2 \, a^{2} \sqrt {2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c + \sqrt {c^{2} + d^{2}}} + 1\right )}} \end {gather*}

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))^2,x, algorithm="giac")

[Out]

-2*(2*c^2 + 6*I*c*d - 7*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*sqrt(c^
2 + d^2))))/((-8*I*a^2*c^2*f + 16*a^2*c*d*f + 8*I*a^2*d^2*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2
+ d^2)) + 1)) + 1/8*(2*(d*tan(f*x + e) + c)^(3/2)*c*d - 2*sqrt(d*tan(f*x + e) + c)*c^2*d + 5*I*(d*tan(f*x + e
) + c)^(3/2)*d^2 - 9*I*sqrt(d*tan(f*x + e) + c)*c*d^2 + 7*sqrt(d*tan(f*x + e) + c)*d^3)/((a^2*c^2*f + 2*I*a^2*
c*d*f - a^2*d^2*f)*(d*tan(f*x + e) - I*d)^2) - 1/2*arctan(-2*(-I*sqrt(d*tan(f*x + e) + c)*c - I*sqrt(c^2 + d^2
)*sqrt(d*tan(f*x + e) + c))/(sqrt(2*c + 2*sqrt(c^2 + d^2))*c - 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))))/(a^2*sqrt(2*c + 2*sqrt(c^2 + d^2))*f*(-I*d/(c + sqrt(c^2 + d^2)) + 1))

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Mupad [B]
time = 9.17, size = 2500, normalized size = 11.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a^2*d^10*f*35i - ((-(45*d^9 - c*d^8*15i + 60*c^2*d^7 - c^3*d^6*80i - 40*c^4*d^5 + c^5*d^4*8i + a^4*c^4*f^2
*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6
*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5
)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6
+ 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f
^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^
8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8
*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4
*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13
+ 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2
+ 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^
7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d
^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c
^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 +
73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a
^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f
^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4
+ (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8
*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4
)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a
^4*c*d^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4
*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7
+ 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 -
4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4
+ a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32
+ (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4
*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*
c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d
^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^
4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*
d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*
f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*
f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i
+ a^4*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2)*(1152*a^6*d^11*f^3 + a^6*c
*d^10*f^3*1664i + 8*(c + d*tan(e + f*x))^(1/2)*(-(45*d^9 - c*d^8*15i + 60*c^2*d^7 - c^3*d^6*80i - 40*c^4*d^5 +
c^5*d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^
4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9
+ 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*
d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)
/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11
*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 +
6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*
c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) +
((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^
6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/1
6 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8...

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