3.1165 \(\int \frac {1}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=269 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f (c-i d)^{3/2}}+\frac {d (3 c-5 i d) (c+5 i d) \sqrt {a+i a \tan (e+f x)}}{6 a^2 f (c-i d) (c+i d)^3 \sqrt {c+d \tan (e+f x)}}+\frac {-11 d+3 i c}{6 a f (c+i d)^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {1}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}} \]

[Out]

-1/4*I*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2)/(a+I*a*tan(f*x+e))^(1/2))/a^(3/2)/(c-I*d)^
(3/2)/f*2^(1/2)+1/6*(3*I*c-11*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(1/2)+1/6*(3*c-5*I*d)
*(c+5*I*d)*d*(a+I*a*tan(f*x+e))^(1/2)/a^2/(c-I*d)/(c+I*d)^3/f/(c+d*tan(f*x+e))^(1/2)-1/3/(I*c-d)/f/(c+d*tan(f*
x+e))^(1/2)/(a+I*a*tan(f*x+e))^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.84, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3559, 3596, 3598, 12, 3544, 208} \[ \frac {d (3 c-5 i d) (c+5 i d) \sqrt {a+i a \tan (e+f x)}}{6 a^2 f (c-i d) (c+i d)^3 \sqrt {c+d \tan (e+f x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f (c-i d)^{3/2}}+\frac {-11 d+3 i c}{6 a f (c+i d)^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {1}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)),x]

[Out]

((-I/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[
2]*a^(3/2)*(c - I*d)^(3/2)*f) - 1/(3*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]) + ((3*
I)*c - 11*d)/(6*a*(c + I*d)^2*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + ((3*c - (5*I)*d)*(c + (
5*I)*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(6*a^2*(c - I*d)*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 3544

Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
(-2*a*b)/f, Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

Rule 3559

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 3596

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]

Rule 3598

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*d - B*c)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f
*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a (3 i c-7 d)-2 i a d \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{3 a^2 (i c-d)}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-11 d}{6 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (-\frac {1}{4} a^2 \left (3 c^2+12 i c d-25 d^2\right )-\frac {1}{2} a^2 (3 c+11 i d) d \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 a^4 (c+i d)^2}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-11 d}{6 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {(3 c-5 i d) (c+5 i d) d \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d) (c+i d)^3 f \sqrt {c+d \tan (e+f x)}}-\frac {2 \int -\frac {3 a^3 (c+i d)^3 \sqrt {a+i a \tan (e+f x)}}{8 \sqrt {c+d \tan (e+f x)}} \, dx}{3 a^5 (c+i d)^2 \left (c^2+d^2\right )}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-11 d}{6 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {(3 c-5 i d) (c+5 i d) d \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d) (c+i d)^3 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a^2 (c-i d)}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-11 d}{6 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {(3 c-5 i d) (c+5 i d) d \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d) (c+i d)^3 f \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{2 (i c+d) f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} (c-i d)^{3/2} f}-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-11 d}{6 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {(3 c-5 i d) (c+5 i d) d \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d) (c+i d)^3 f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 7.07, size = 333, normalized size = 1.24 \[ \frac {\sec ^{\frac {3}{2}}(e+f x) \left (\frac {\sqrt {\sec (e+f x)} \left (5 i c^3-13 c^2 d-\left (3 c^3+5 i c^2 d+23 c d^2-39 i d^3\right ) \sin (2 (e+f x))+\left (5 i c^3-7 c^2 d+25 i c d^2+37 d^3\right ) \cos (2 (e+f x))+5 i c d^2-13 d^3\right )}{3 (c-i d) (c+i d)^3 \sqrt {c+d \tan (e+f x)}}-\frac {i \sqrt {2} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{3/2} \left (1+e^{2 i (e+f x)}\right )^{3/2} \log \left (2 \left (\sqrt {c-i d} e^{i (e+f x)}+\sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )}{(c-i d)^{3/2}}\right )}{4 f (a+i a \tan (e+f x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)),x]

[Out]

(Sec[e + f*x]^(3/2)*(((-I)*Sqrt[2]*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2)*(1 + E^((2*I)*(e + f*x)))
^(3/2)*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e +
f*x))))/(1 + E^((2*I)*(e + f*x)))])])/(c - I*d)^(3/2) + (Sqrt[Sec[e + f*x]]*((5*I)*c^3 - 13*c^2*d + (5*I)*c*d^
2 - 13*d^3 + ((5*I)*c^3 - 7*c^2*d + (25*I)*c*d^2 + 37*d^3)*Cos[2*(e + f*x)] - (3*c^3 + (5*I)*c^2*d + 23*c*d^2
- (39*I)*d^3)*Sin[2*(e + f*x)]))/(3*(c - I*d)*(c + I*d)^3*Sqrt[c + d*Tan[e + f*x]])))/(4*f*(a + I*a*Tan[e + f*
x])^(3/2))

________________________________________________________________________________________

fricas [B]  time = 0.57, size = 979, normalized size = 3.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^(3/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

(sqrt(2)*(c^3 + I*c^2*d + c*d^2 + I*d^3 + (4*c^3 + 6*I*c^2*d + 24*c*d^2 - 38*I*d^3)*e^(6*I*f*x + 6*I*e) + (9*c
^3 + 19*I*c^2*d + 29*c*d^2 - 25*I*d^3)*e^(4*I*f*x + 4*I*e) + (6*c^3 + 14*I*c^2*d + 6*c*d^2 + 14*I*d^3)*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))*sqrt(a/(e^(2*I*f*x + 2
*I*e) + 1)) + ((-3*I*a^2*c^5 + 3*a^2*c^4*d - 6*I*a^2*c^3*d^2 + 6*a^2*c^2*d^3 - 3*I*a^2*c*d^4 + 3*a^2*d^5)*f*e^
(5*I*f*x + 5*I*e) + (-3*I*a^2*c^5 + 9*a^2*c^4*d + 6*I*a^2*c^3*d^2 + 6*a^2*c^2*d^3 + 9*I*a^2*c*d^4 - 3*a^2*d^5)
*f*e^(3*I*f*x + 3*I*e))*sqrt(I/((-2*I*a^3*c^3 - 6*a^3*c^2*d + 6*I*a^3*c*d^2 + 2*a^3*d^3)*f^2))*log((2*I*a^2*c^
2 + 4*a^2*c*d - 2*I*a^2*d^2)*f*sqrt(I/((-2*I*a^3*c^3 - 6*a^3*c^2*d + 6*I*a^3*c*d^2 + 2*a^3*d^3)*f^2))*e^(I*f*x
+ I*e) + sqrt(2)*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/(e^(2*I*f*x
+ 2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 1)) + ((3*I*a^2*c^5 - 3*a^2*c^4*d + 6*I*a^2*c^3*d^2 - 6*a^2*c^2*d^3 + 3
*I*a^2*c*d^4 - 3*a^2*d^5)*f*e^(5*I*f*x + 5*I*e) + (3*I*a^2*c^5 - 9*a^2*c^4*d - 6*I*a^2*c^3*d^2 - 6*a^2*c^2*d^3
- 9*I*a^2*c*d^4 + 3*a^2*d^5)*f*e^(3*I*f*x + 3*I*e))*sqrt(I/((-2*I*a^3*c^3 - 6*a^3*c^2*d + 6*I*a^3*c*d^2 + 2*a
^3*d^3)*f^2))*log((-2*I*a^2*c^2 - 4*a^2*c*d + 2*I*a^2*d^2)*f*sqrt(I/((-2*I*a^3*c^3 - 6*a^3*c^2*d + 6*I*a^3*c*d
^2 + 2*a^3*d^3)*f^2))*e^(I*f*x + I*e) + sqrt(2)*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/(e^(2*I*f*x + 2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 1)))/((-12*I*a^2*c^5 + 12*a^2*c^4*d - 24*
I*a^2*c^3*d^2 + 24*a^2*c^2*d^3 - 12*I*a^2*c*d^4 + 12*a^2*d^5)*f*e^(5*I*f*x + 5*I*e) + (-12*I*a^2*c^5 + 36*a^2*
c^4*d + 24*I*a^2*c^3*d^2 + 24*a^2*c^2*d^3 + 36*I*a^2*c*d^4 - 12*a^2*d^5)*f*e^(3*I*f*x + 3*I*e))

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^(3/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Evaluation time: 3.08Error: Bad Argument Type

________________________________________________________________________________________

maple [B]  time = 0.32, size = 4835, normalized size = 17.97 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24/f*(75*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c)
)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)*c^4*d^2-15*2^(1/2)*(-a*(I*d-c)
)^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+
I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)*c^2*d^4+3*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)
*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e
)+I))*tan(f*x+e)^4*c^5*d-30*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(
1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^4*c^3*d^3+15*2
^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c
+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^4*c*d^5+15*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3
*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e))
)^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^3*c^4*d^2-75*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3
*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(
f*x+e)^3*c^2*d^4+36*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a
*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^2*c^5*d-36*2^(1/2)*(-a
*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x
+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^2*c*d^5+12*I*tan(f*x+e)^2*c^6*(a*(c+d*tan(f*x+e))*(1+
I*tan(f*x+e)))^(1/2)+256*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*d^6-40*I*(a*(c+d*tan(f*x+e
))*(1+I*tan(f*x+e)))^(1/2)*c^4*d^2-68*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d^4+30*2^(1/2)*(-a*(I*
d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))
*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^3*d^3-3*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*
a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c
*d^5-20*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^5*d+104*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)
))^(1/2)*tan(f*x+e)^2*c^3*d^3+124*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c*d^5+296*(a*(c+d*t
an(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^2*d^4-52*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e
)^3*c^4*d^2-152*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*c^2*d^4+124*(a*(c+d*tan(f*x+e))*(1+I*
tan(f*x+e)))^(1/2)*tan(f*x+e)*c^4*d^2+15*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*
d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^4*c^4*d^2*(-a*(I*d
-c))^(1/2)-30*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x
+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^4*c^2*d^4*(-a*(I*d-c))^(1/2)+6*I*ln((3*c*a+I*
a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)
)/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^3*c^5*d*(-a*(I*d-c))^(1/2)+60*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(
f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan
(f*x+e)^3*c^3*d^3*(-a*(I*d-c))^(1/2)-42*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d
-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^3*c*d^5*(-a*(I*d-c)
)^(1/2)+45*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e)
)*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^2*c^4*d^2*(-a*(I*d-c))^(1/2)+45*I*ln((3*c*a+I*a*
tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/
(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^2*c^2*d^4*(-a*(I*d-c))^(1/2)-42*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(
f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan
(f*x+e)*c^5*d*(-a*(I*d-c))^(1/2)+60*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))
^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)*c^3*d^3*(-a*(I*d-c))^(1
/2)+6*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+
I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)*c*d^5*(-a*(I*d-c))^(1/2)+3*I*ln((3*c*a+I*a*tan(f*x+e)
*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e
)+I))*2^(1/2)*c^6*(-a*(I*d-c))^(1/2)+136*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^4*d^2+38
0*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^2*d^4+92*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e))
)^(1/2)*tan(f*x+e)*c^5*d+40*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^3*d^3-52*I*(a*(c+d*tan(
f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c*d^5+12*I*tan(f*x+e)^3*c^5*d*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))
^(1/2)+72*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*c^3*d^3-100*(a*(c+d*tan(f*x+e))*(1+I*tan(
f*x+e)))^(1/2)*tan(f*x+e)^3*d^6+204*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*d^6+60*(a*(c+d*tan(
f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^5*d+72*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^3*d^3+12*(a*(c+d*tan(f*
x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^5+32*tan(f*x+e)*c^6*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)+3*I*ln((3*c*
a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(
1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^4*d^6*(-a*(I*d-c))^(1/2)-9*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan
(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*ta
n(f*x+e)^2*c^6*(-a*(I*d-c))^(1/2)-9*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))
^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*tan(f*x+e)^2*d^6*(-a*(I*d-c))^(1/2
)-30*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I
*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(1/2)*c^4*d^2*(-a*(I*d-c))^(1/2)+15*I*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+
3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*2^(
1/2)*c^2*d^4*(-a*(I*d-c))^(1/2)-20*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^6-48*I*(a*(c+d*tan(f*x+e))*
(1+I*tan(f*x+e)))^(1/2)*d^6+3*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2
^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^3*c^6+9*2^(1
/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*
tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^3*d^6-9*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I
*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2
))/(tan(f*x+e)+I))*tan(f*x+e)*c^6-3*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)
*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)*d^6-15
*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*
(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^5*d+60*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/
2)*tan(f*x+e)^3*c*d^5)/a^2*(a*(1+I*tan(f*x+e)))^(1/2)/(-tan(f*x+e)+I)^3/(I*c-d)/(c+I*d)^4/(I*d-c)^2/(a*(c+d*ta
n(f*x+e))*(1+I*tan(f*x+e)))^(1/2)/(c+d*tan(f*x+e))^(1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))**(3/2)/(c+d*tan(f*x+e))**(3/2),x)

[Out]

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

________________________________________________________________________________________