3.1170 \(\int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx\)
Optimal. Leaf size=277 \[ \frac{d (5 d+3 i c) \sqrt{a+i a \tan (e+f x)}}{3 a f (-d+i c) \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d (3 c-i d) (c-7 i d) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i d)^2 (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{1}{f (-d+i c) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{\sqrt{2} \sqrt{a} f (c-i d)^{5/2}} \]
[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)^(5/2)*f) - 1/((I*c - d)*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (d*((3*I
)*c + 5*d)*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*(I*c - d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + ((3*c - I*d)
*(c - (7*I)*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*(c - I*d)^2*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])
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Rubi [A] time = 0.859103, antiderivative size = 277, normalized size of antiderivative = 1.,
number of steps used = 6, 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} \[ \frac{d (5 d+3 i c) \sqrt{a+i a \tan (e+f x)}}{3 a f (-d+i c) \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d (3 c-i d) (c-7 i d) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i d)^2 (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{1}{f (-d+i c) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{\sqrt{2} \sqrt{a} f (c-i d)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)),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)^(5/2)*f) - 1/((I*c - d)*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (d*((3*I
)*c + 5*d)*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*(I*c - d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + ((3*c - I*d)
*(c - (7*I)*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*(c - I*d)^2*(c + I*d)^3*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))^{5/2}} \, dx &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{\sqrt{a+i a \tan (e+f x)} \left (-\frac{1}{2} a (i c-5 d)-2 i a d \tan (e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx}{a^2 (i c-d)}\\ &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-5 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{2 \int \frac{\sqrt{a+i a \tan (e+f x)} \left (\frac{1}{4} a^2 \left (12 c d-i \left (3 c^2+7 d^2\right )\right )-\frac{1}{2} a^2 d (3 i c+5 d) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 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)} (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-5 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-i d) (c-7 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}+\frac{4 \int \frac{3 a^3 (i c-d)^3 \sqrt{a+i a \tan (e+f x)}}{8 \sqrt{c+d \tan (e+f x)}} \, dx}{3 a^4 (i c-d)^3 (c-i d)^2}\\ &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-5 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-i d) (c-7 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d)^2 (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}{2 a (c-i d)^2}\\ &=-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-5 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-i d) (c-7 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}-\frac{(i 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 )}{(c-i d)^2 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)^{5/2} f}-\frac{1}{(i c-d) f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-5 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}+\frac{(3 c-i d) (c-7 i d) d \sqrt{a+i a \tan (e+f x)}}{3 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 8.15596, size = 353, normalized size = 1.27 \[ \frac{\sqrt{\sec (e+f x)} \left (\frac{\sec ^{\frac{3}{2}}(e+f x) \left (-2 d \left (-15 c^2 d-3 i c^3+13 i c d^2+d^3\right ) \sin (2 (e+f x))+\left (-24 i c^2 d^2+6 c^3 d+3 i c^4-26 c d^3+5 i d^4\right ) \cos (2 (e+f x))+3 i \left (-6 c^2 d^2-2 i c^3 d+c^4-6 i c d^3-3 d^4\right )\right )}{3 (c-i d)^2 (c+i d)^3 (c+d \tan (e+f x))^{3/2}}-\frac{i \sqrt{2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left (2 \left (\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )}{(c-i d)^{5/2}}\right )}{2 f \sqrt{a+i a \tan (e+f x)}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)),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)^(5/2) + (Sec[e + f*x]^(3/2)*((3*I)*(c^4 - (2*I)*c^3*d - 6*c^2*d^2
- (6*I)*c*d^3 - 3*d^4) + ((3*I)*c^4 + 6*c^3*d - (24*I)*c^2*d^2 - 26*c*d^3 + (5*I)*d^4)*Cos[2*(e + f*x)] - 2*d*
((-3*I)*c^3 - 15*c^2*d + (13*I)*c*d^2 + d^3)*Sin[2*(e + f*x)]))/(3*(c - I*d)^2*(c + I*d)^3*(c + d*Tan[e + f*x]
)^(3/2))))/(2*f*Sqrt[a + I*a*Tan[e + f*x]])
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Maple [B] time = 0.121, size = 4889, normalized size = 17.7 \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))^(5/2),x)
[Out]
-1/12/f*(-24*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^6*(-a*(I*d-c))^(1/2)+3*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^6*d*(-a*(I*d-c))^(1/2)+75*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^3*(-a*(I*d-c))^(1/2)-27*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^5*(-a*(I*d-c))
^(1/2)+3*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)^4*c^5*d^2*(-a*(I*d-c))^(1/2)-48*c^5*d^2*(a*(c+d*tan(
f*x+e))*(1+I*tan(f*x+e)))^(1/2)+28*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*d^7-84*(a*(c+d*tan
(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^3*d^4-24*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^6-8*I*(a*(c+d*tan(
f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^7-3*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^7*(-a*(I*d-c))^(1/2)-36*I*(a*(c+
d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*d^7+12*I*tan(f*x+e)*c^7*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)
))^(1/2)-12*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^6*d-80*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/
2)*c^4*d^3-76*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d^5+108*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^
(1/2)*tan(f*x+e)^2*c^5*d^2+264*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^3*d^4+156*(a*(c+d*ta
n(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c*d^6+76*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)
^3*c^4*d^3+104*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*c^2*d^5+36*(a*(c+d*tan(f*x+e))*(1+I*ta
n(f*x+e)))^(1/2)*tan(f*x+e)*c^6*d+12*I*tan(f*x+e)^3*c^5*d^2*(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^4+60*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/
2)*tan(f*x+e)^3*c*d^6+24*I*tan(f*x+e)^2*c^6*d*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)-36*I*(a*(c+d*tan(f*x
+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^4*d^3-96*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^
2*c^2*d^5-144*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^5*d^2-276*I*(a*(c+d*tan(f*x+e))*(1+I*
tan(f*x+e)))^(1/2)*tan(f*x+e)*c^3*d^4-120*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c*d^6+30*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*d^2*(-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*d^7*(-a*(I*d-c))^(1/2)+3*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^7*(-a*(I*d-c))^(1/2)-15*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^4*(-a*(I*d-c))^(1/2)+120*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)
*tan(f*x+e)*c^4*d^3+84*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^2*d^5+12*(a*(c+d*tan(f*x+e))*(
1+I*tan(f*x+e)))^(1/2)*c^7+30*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*d^2*(-a*(I*d-c))^(1/2)+30
*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^4*(-a*(I*d-c))^(1/2)-6*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*d^6*(-a*(I*d-c))^(1/2)+15*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)^4*c^4*d^3
*(-a*(I*d-c))^(1/2)-30*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)^4*c^2*d^5*(-a*(I*d-c))^(1/2)+24*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^5*d^2*(-a*(I*d-c))^(1/2)-3*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^5*(-a*(I*d-c))^(1/2)-30*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)^4*c^3*d^4*(-a*(I*d-c))^
(1/2)+15*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)^4*c*d^6*(-a*(I*d-c))^(1/2)+6*2^(1/2)*ln((3*a*c+I*a*t
an(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^6*d*(-a*(I*d-c))^(1/2)-30*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^3*(-a*(I*d-c))^(1/2)-30*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^5*(-a*(I*d-c))^(1/2)+27*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*d^2*(-a*(I*d-c))^(1/2)-75*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^3*d^4*(-a*(I*d-c))^(1/2)-3*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^6*(-a*
(I*d-c))^(1/2)+24*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^6*d*(-a*(I*d-c))^(1/2)-24*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^5*(-a*(I*d-c))^(1/2)+3*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)^4*d^7*(-a*(I*d-c))^(1/2)-3*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^7*(-a*(I*d-c))^(1/2)-6*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^7*(-a*(I*d-c))^(1/2)-15*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^6*d*(-a*(I*d-c))^(1/2)+30*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^3*(-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)/(c+I*d)^4/(I*d-c)^3/(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)
/(c+d*tan(f*x+e))^(3/2)
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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{5}{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))^(5/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(I*a*tan(f*x + e) + a)*(d*tan(f*x + e) + c)^(5/2)), x)
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Fricas [B] time = 2.54529, size = 3386, normalized size = 12.22 \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))^(5/2),x, algorithm="fricas")
[Out]
-(sqrt(2)*(6*c^4 + 12*c^2*d^2 + 6*d^4 + (6*c^4 - 24*I*c^3*d - 108*c^2*d^2 + 104*I*c*d^3 + 14*d^4)*e^(6*I*f*x +
6*I*e) + (18*c^4 - 48*I*c^3*d - 180*c^2*d^2 + 32*I*c*d^3 - 22*d^4)*e^(4*I*f*x + 4*I*e) + (18*c^4 - 24*I*c^3*d
- 60*c^2*d^2 - 72*I*c*d^3 - 30*d^4)*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) - ((3*I*a*c^7 + 3*a*c^6*d + 9*I*a*c^5*
d^2 + 9*a*c^4*d^3 + 9*I*a*c^3*d^4 + 9*a*c^2*d^5 + 3*I*a*c*d^6 + 3*a*d^7)*f*e^(6*I*f*x + 6*I*e) + (6*I*a*c^7 -
6*a*c^6*d + 18*I*a*c^5*d^2 - 18*a*c^4*d^3 + 18*I*a*c^3*d^4 - 18*a*c^2*d^5 + 6*I*a*c*d^6 - 6*a*d^7)*f*e^(4*I*f*
x + 4*I*e) + (3*I*a*c^7 - 9*a*c^6*d - 3*I*a*c^5*d^2 - 15*a*c^4*d^3 - 15*I*a*c^3*d^4 - 3*a*c^2*d^5 - 9*I*a*c*d^
6 + 3*a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt(-2*I/((I*a*c^5 + 5*a*c^4*d - 10*I*a*c^3*d^2 - 10*a*c^2*d^3 + 5*I*a*c*
d^4 + a*d^5)*f^2))*log(((I*a*c^3 + 3*a*c^2*d - 3*I*a*c*d^2 - a*d^3)*f*sqrt(-2*I/((I*a*c^5 + 5*a*c^4*d - 10*I*a
*c^3*d^2 - 10*a*c^2*d^3 + 5*I*a*c*d^4 + a*d^5)*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)) - ((-3*I*a*c^7 - 3*a*c^6*d - 9*I*a*c^5*d^2 - 9*a*c^4*d^3 - 9*I*a*c^3*d^4 - 9*a
*c^2*d^5 - 3*I*a*c*d^6 - 3*a*d^7)*f*e^(6*I*f*x + 6*I*e) + (-6*I*a*c^7 + 6*a*c^6*d - 18*I*a*c^5*d^2 + 18*a*c^4*
d^3 - 18*I*a*c^3*d^4 + 18*a*c^2*d^5 - 6*I*a*c*d^6 + 6*a*d^7)*f*e^(4*I*f*x + 4*I*e) + (-3*I*a*c^7 + 9*a*c^6*d +
3*I*a*c^5*d^2 + 15*a*c^4*d^3 + 15*I*a*c^3*d^4 + 3*a*c^2*d^5 + 9*I*a*c*d^6 - 3*a*d^7)*f*e^(2*I*f*x + 2*I*e))*s
qrt(-2*I/((I*a*c^5 + 5*a*c^4*d - 10*I*a*c^3*d^2 - 10*a*c^2*d^3 + 5*I*a*c*d^4 + a*d^5)*f^2))*log(((-I*a*c^3 - 3
*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f*sqrt(-2*I/((I*a*c^5 + 5*a*c^4*d - 10*I*a*c^3*d^2 - 10*a*c^2*d^3 + 5*I*a*c*d^
4 + a*d^5)*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)))/((1
2*I*a*c^7 + 12*a*c^6*d + 36*I*a*c^5*d^2 + 36*a*c^4*d^3 + 36*I*a*c^3*d^4 + 36*a*c^2*d^5 + 12*I*a*c*d^6 + 12*a*d
^7)*f*e^(6*I*f*x + 6*I*e) + (24*I*a*c^7 - 24*a*c^6*d + 72*I*a*c^5*d^2 - 72*a*c^4*d^3 + 72*I*a*c^3*d^4 - 72*a*c
^2*d^5 + 24*I*a*c*d^6 - 24*a*d^7)*f*e^(4*I*f*x + 4*I*e) + (12*I*a*c^7 - 36*a*c^6*d - 12*I*a*c^5*d^2 - 60*a*c^4
*d^3 - 60*I*a*c^3*d^4 - 12*a*c^2*d^5 - 36*I*a*c*d^6 + 12*a*d^7)*f*e^(2*I*f*x + 2*I*e))
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Sympy [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))**(5/2),x)
[Out]
Timed out
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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))^(5/2),x, algorithm="giac")
[Out]
Timed out