3.1164 \(\int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=194 \[ \frac{d (c-3 i d) \sqrt{a+i a \tan (e+f x)}}{a f (c-i d) (c+i d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{f (-d+i c) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{\sqrt{2} \sqrt{a} f (c-i d)^{3/2}} \]
[Out]
((-I)*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]
*Sqrt[a]*(c - I*d)^(3/2)*f) - 1/((I*c - d)*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + ((c - (3*I
)*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(a*(c - I*d)*(c + I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])
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Rubi [A] time = 0.473327, antiderivative size = 194, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used =
{3559, 3598, 12, 3544, 208} \[ \frac{d (c-3 i d) \sqrt{a+i a \tan (e+f x)}}{a f (c-i d) (c+i d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{f (-d+i c) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{\sqrt{2} \sqrt{a} f (c-i d)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)),x]
[Out]
((-I)*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]
*Sqrt[a]*(c - I*d)^(3/2)*f) - 1/((I*c - d)*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + ((c - (3*I
)*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(a*(c - I*d)*(c + I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])
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 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]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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]
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{1}{(i c-d) 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}{2} a (i c-3 d)-i a d \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{a^2 (i c-d)}\\ &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{(c-3 i d) d \sqrt{a+i a \tan (e+f x)}}{a (c-i d) (c+i d)^2 f \sqrt{c+d \tan (e+f x)}}-\frac{2 \int -\frac{i a^2 (c+i d)^2 \sqrt{a+i a \tan (e+f x)}}{4 \sqrt{c+d \tan (e+f x)}} \, dx}{a^3 (i c-d) \left (c^2+d^2\right )}\\ &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{(c-3 i d) d \sqrt{a+i a \tan (e+f x)}}{a (c-i d) (c+i d)^2 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}{2 a (c-i d)}\\ &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{(c-3 i d) d \sqrt{a+i a \tan (e+f x)}}{a (c-i d) (c+i d)^2 f \sqrt{c+d \tan (e+f x)}}+\frac{a \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 )}{(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 )}{\sqrt{2} \sqrt{a} (c-i d)^{3/2} f}-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{(c-3 i d) d \sqrt{a+i a \tan (e+f x)}}{a (c-i d) (c+i d)^2 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.32285, size = 267, normalized size = 1.38 \[ \frac{\sqrt{\sec (e+f x)} \left (\frac{2 i c^2+2 d (3 d+i c) \tan (e+f x)+2 c d-4 i d^2}{(c-i d) (c+i d)^2 \sqrt{\sec (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \sqrt{2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left (2 \left (\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )}{(c-i d)^{3/2}}\right )}{2 f \sqrt{a+i a \tan (e+f x)}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)),x]
[Out]
(Sqrt[Sec[e + f*x]]*(((-I)*Sqrt[2]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x)
)]*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) + ((2*I)*c^2 + 2*c*d - (4*I)*d^2 + 2*d*(I*c + 3*d)*Tan[e + f
*x])/((c - I*d)*(c + I*d)^2*Sqrt[Sec[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])))/(2*f*Sqrt[a + I*a*Tan[e + f*x]])
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Maple [B] time = 0.121, size = 3196, normalized size = 16.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2),x)
[Out]
1/4/f*(2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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*(-a*(I*d-c))^(1/2)+20*(a*(c+d*tan(f*x+e))*(1+I
*tan(f*x+e)))^(1/2)*tan(f*x+e)*d^5-8*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^5-4*c*d^4*(a*(c+d*tan(f*x
+e))*(1+I*tan(f*x+e)))^(1/2)+2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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*(-a*(I*d-c))^(1/2)-2^(1/2)
*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^5*(-a*(I*d-c))^(1/2)+6*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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
^2*(-a*(I*d-c))^(1/2)-2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^4*(-a*(I*d-c))^(1/2)+4*I*tan(f*x+e)^2*c^4*d*(a
*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)+16*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^2*d^
3-8*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^3*d^2-12*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e))
)^(1/2)*tan(f*x+e)*c*d^4+2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^5*(-a*(I*d-c))^(1/2)+4*(a*(c+d*t
an(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^4*d+24*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*
c^2*d^3+8*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^3*d^2+8*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+
e)))^(1/2)*tan(f*x+e)^2*c*d^4-2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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*(-a*(I*d-c))^(1/2)+12*I*(a*(c+d*tan(f*
x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*d^5+4*I*tan(f*x+e)*c^5*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)-
8*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d^3+4*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^5-2*I*
2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^5*(-a*(I*d-c))^(1/2)-4*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c
-I*a*d+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^4*d*(-a*(I*d-c))^(1/2)+4*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^2*d^3*(-a*(I*d-c))^(1/2)-6*2^(1/2)*ln
((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^3*(-a*(I*d-c))^(1/2)+2*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d
+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))*ta
n(f*x+e)^2*c^3*d^2*(-a*(I*d-c))^(1/2)-7*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^4*(-a*(I*d-c))^
(1/2)+7*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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*(-a*(I*d-c))^(1/2)-2*2^(1/2)*ln((3*a*c+I*a*tan(
f*x+e)*c-I*a*d+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^3*(-a*(I*d-c))^(1/2)-2*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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*d^5*(
-a*(I*d-c))^(1/2)-4*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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*d^4*(-a*(I*d-c))^(1/2)+2*I*2^(1/2)*
ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^4*d*(-a*(I*d-c))^(1/2)+8*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a
*d+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^2*d^3*(-a*(I*d-c))^(1/2)+8*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+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^3*d^2*(-a*(I*d-
c))^(1/2)+2*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*t
an(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)*c*d^4*(-a*(I*d-c))^(1/2)+4*I*2^(1/2)*ln((3*a*c+
I*a*tan(f*x+e)*c-I*a*d+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^3*d^2*(-a*(I*d-c))^(1/2))/a*(a*(1+I*tan(f*x+e)))^(1/2)/(-tan(f*x+e)+I)^2/(I
*c-d)/(I*d-c)^2/(c+I*d)^3/(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)/(c+d*tan(f*x+e))^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{i \, a \tan \left (f x + e\right ) + a}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(I*a*tan(f*x + e) + a)*(d*tan(f*x + e) + c)^(3/2)), x)
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Fricas [B] time = 1.80945, size = 1821, normalized size = 9.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
(sqrt(2)*(2*I*c^2 + 2*I*d^2 + (2*I*c^2 + 4*c*d - 10*I*d^2)*e^(4*I*f*x + 4*I*e) + (4*I*c^2 + 4*c*d - 8*I*d^2)*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))*e^(I*f*x + I*e) + ((a*c^4 + 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f*x + 4*I*e) + (a*c^4 + 2*I*a*c^3*
d + 2*I*a*c*d^3 - a*d^4)*f*e^(2*I*f*x + 2*I*e))*sqrt(2*I/((-I*a*c^3 - 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f^2))*l
og(((I*a*c^2 + 2*a*c*d - I*a*d^2)*f*sqrt(2*I/((-I*a*c^3 - 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f^2))*e^(2*I*f*x +
2*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)*e^(I*f*x + I*e))*e^(-I*f*x - I*e)) - ((a*c^4 + 2*a*c^2*d^2 + a*d^4)*f
*e^(4*I*f*x + 4*I*e) + (a*c^4 + 2*I*a*c^3*d + 2*I*a*c*d^3 - a*d^4)*f*e^(2*I*f*x + 2*I*e))*sqrt(2*I/((-I*a*c^3
- 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f^2))*log(((-I*a*c^2 - 2*a*c*d + I*a*d^2)*f*sqrt(2*I/((-I*a*c^3 - 3*a*c^2*d
+ 3*I*a*c*d^2 + a*d^3)*f^2))*e^(2*I*f*x + 2*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)*e^(I*f*x + I*e))*e^(-I*f*x
- I*e)))/(4*(a*c^4 + 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f*x + 4*I*e) + (4*a*c^4 + 8*I*a*c^3*d + 8*I*a*c*d^3 - 4*a*
d^4)*f*e^(2*I*f*x + 2*I*e))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \left (i \tan{\left (e + f x \right )} + 1\right )} \left (c + d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+I*a*tan(f*x+e))**(1/2)/(c+d*tan(f*x+e))**(3/2),x)
[Out]
Integral(1/(sqrt(a*(I*tan(e + f*x) + 1))*(c + d*tan(e + f*x))**(3/2)), x)
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
[Out]
Timed out