3.12.70 \(\int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx\) [1170]

Optimal. Leaf size=277 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {2} \sqrt {a} (c-i d)^{5/2} f}-\frac {1}{(i c-d) f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d (3 i c+5 d) \sqrt {a+i a \tan (e+f x)}}{3 a (i c-d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {(3 c-i d) (c-7 i d) d \sqrt {a+i a \tan (e+f x)}}{3 a (c-i d)^2 (c+i d)^3 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

-1/2*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))/(c-I*d)^(5/2)/f*
2^(1/2)/a^(1/2)+1/3*(3*c-I*d)*(c-7*I*d)*d*(a+I*a*tan(f*x+e))^(1/2)/a/(c-I*d)^2/(c+I*d)^3/f/(c+d*tan(f*x+e))^(1
/2)-1/(I*c-d)/f/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2)+1/3*d*(3*I*c+5*d)*(a+I*a*tan(f*x+e))^(1/2)/a/(
I*c-d)/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)

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Rubi [A]
time = 0.59, antiderivative size = 277, normalized size of antiderivative = 1.00, 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 = {3640, 3679, 12, 3625, 214} \begin {gather*} \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}} \end {gather*}

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 3625

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 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 3679

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}{\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) \text {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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(687\) vs. \(2(277)=554\).
time = 8.97, size = 687, normalized size = 2.48 \begin {gather*} -\frac {i e^{i e} \sqrt {e^{i 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 ) \sqrt {\sec (e+f x)} \sqrt {\cos (f x)+i \sin (f x)}}{\sqrt {2} (c-i d)^{5/2} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} f \sqrt {a+i a \tan (e+f x)}}+\frac {\sec (e+f x) (\cos (f x)+i \sin (f x)) \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac {\cos (2 f x) \left (\frac {1}{2} i \cos (e)+\frac {\sin (e)}{2}\right )}{(c+i d)^3}+\frac {\left (\frac {\cos (e)}{6}+\frac {1}{6} i \sin (e)\right ) \left (3 i c^3 \cos (e)+6 c^2 d \cos (e)-39 i c d^2 \cos (e)-8 d^3 \cos (e)+3 i c^2 d \sin (e)+6 c d^2 \sin (e)+i d^3 \sin (e)\right )}{(c-i d)^2 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac {\left (\frac {\cos (e)}{2}-\frac {1}{2} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^3}+\frac {-\frac {2}{3} i d^4 \cos (e)+\frac {2}{3} d^4 \sin (e)}{(c-i d)^2 (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))^2}+\frac {4 \left (-\frac {5}{2} c d^3 \cos (e-f x)+\frac {1}{2} i d^4 \cos (e-f x)+\frac {5}{2} c d^3 \cos (e+f x)-\frac {1}{2} i d^4 \cos (e+f x)-\frac {5}{2} i c d^3 \sin (e-f x)-\frac {1}{2} d^4 \sin (e-f x)+\frac {5}{2} i c d^3 \sin (e+f x)+\frac {1}{2} d^4 \sin (e+f x)\right )}{3 (c-i d)^2 (c+i d)^3 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right )}{f \sqrt {a+i a \tan (e+f x)}} \end {gather*}

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]

((-I)*E^(I*e)*Sqrt[E^(I*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)))])]*Sqrt[Sec[e + f*x]]*Sqrt[Cos[f*x] + I*Sin[f*x]])/(S
qrt[2]*(c - I*d)^(5/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[a
+ I*a*Tan[e + f*x]]) + (Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x
])]*((Cos[2*f*x]*((I/2)*Cos[e] + Sin[e]/2))/(c + I*d)^3 + ((Cos[e]/6 + (I/6)*Sin[e])*((3*I)*c^3*Cos[e] + 6*c^2
*d*Cos[e] - (39*I)*c*d^2*Cos[e] - 8*d^3*Cos[e] + (3*I)*c^2*d*Sin[e] + 6*c*d^2*Sin[e] + I*d^3*Sin[e]))/((c - I*
d)^2*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + ((Cos[e]/2 - (I/2)*Sin[e])*Sin[2*f*x])/(c + I*d)^3 + (((-2*I)/3)*d^4
*Cos[e] + (2*d^4*Sin[e])/3)/((c - I*d)^2*(c + I*d)^3*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + (4*((-5*c*d^3*Cos[
e - f*x])/2 + (I/2)*d^4*Cos[e - f*x] + (5*c*d^3*Cos[e + f*x])/2 - (I/2)*d^4*Cos[e + f*x] - ((5*I)/2)*c*d^3*Sin
[e - f*x] - (d^4*Sin[e - f*x])/2 + ((5*I)/2)*c*d^3*Sin[e + f*x] + (d^4*Sin[e + f*x])/2))/(3*(c - I*d)^2*(c + I
*d)^3*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*Sqrt[a + I*a*Tan[e + f*x]])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4888 vs. \(2 (231 ) = 462\).
time = 0.71, size = 4889, normalized size = 17.65

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4889\)
default	\(\text {Expression too large to display}\)	\(4889\)

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,method=_RETURNVERBOSE)

[Out]

1/12/f*(-28*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^7*tan(f*x+e)^3+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+48*c^5*d^2*(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)*d^7-30*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*
d+3*a*d*tan(f*x+e)+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)+15*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)-12*I*c^5*d^2*(a*
(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3+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)*c^5*
d^2*tan(f*x+e)^2-264*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^3*d^4*tan(f*x+e)^2-156*(a*(c+d*tan(f*x+e))*
(1+I*tan(f*x+e)))^(1/2)*c*d^6*tan(f*x+e)^2-36*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^6*d*tan(f*x+e)-120
*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^4*d^3*tan(f*x+e)-84*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)
*c^2*d^5*tan(f*x+e)-76*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^4*d^3*tan(f*x+e)^3-104*(a*(c+d*tan(f*x+e)
)*(1+I*tan(f*x+e)))^(1/2)*c^2*d^5*tan(f*x+e)^3+3*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*d^7*tan(f*x+e)^2-12*I*c^7*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)
))^(1/2)*tan(f*x+e)+12*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^6*d-72*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f
*x+e)))^(1/2)*c^3*d^4*tan(f*x+e)^3-60*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^6*tan(f*x+e)^3-24*I*c^
6*d*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2+36*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*
c^4*d^3*tan(f*x+e)^2+96*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d^5*tan(f*x+e)^2+144*I*(a*(c+d*tan(f
*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^5*d^2*tan(f*x+e)+276*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^3*d^4*ta
n(f*x+e)+120*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^6*tan(f*x+e)-6*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)
*c-I*a*d+3*a*d*tan(f*x+e)+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))*d^7*(-a*(I*d-c))^(1/2)*tan(f*x+e)^3-3*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*tan(f
*x+e)^2-12*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^7-15*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d
*tan(f*x+e)+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)*tan(f*x+e)^4+30*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*tan(f*x+
e)^4+3*I*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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*d*tan(f*x+e)+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)*tan(f*x+e)^4-15*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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^6*(-a*(I*d-c))^(1/2)
*tan(f*x+e)^4-6*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*tan(f*x+e)^3+30*2^(1/2)*ln((3*a
*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*tan(f*x+e)^3+30*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a
*d*tan(f*x+e)+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)*tan(f*x+e)^3-27*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*tan(f*x+
e)^2+75*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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)*tan(f*x+e)^2+3*2^(1/2)*ln((3*a*c+I*a*
tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+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^6*(-a*(I*d-c))^(1/2)*tan(f*x+e)^2-24*2^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*
x+e)+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)*tan(f*x+e)+24*2^(1/2)*ln((3*a*c+I*a*t...

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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/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found %i

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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. 1325 vs. \(2 (225) = 450\).
time = 1.39, size = 1325, normalized size = 4.78 \begin {gather*} \frac {2 \, \sqrt {2} {\left (3 \, c^{4} + 6 \, c^{2} d^{2} + 3 \, d^{4} + {\left (3 \, c^{4} - 12 i \, c^{3} d - 54 \, c^{2} d^{2} + 52 i \, c d^{3} + 7 \, d^{4}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (9 \, c^{4} - 24 i \, c^{3} d - 90 \, c^{2} d^{2} + 16 i \, c d^{3} - 11 \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (3 \, c^{4} - 4 i \, c^{3} d - 10 \, c^{2} d^{2} - 12 i \, c d^{3} - 5 \, d^{4}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\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 {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + 3 \, {\left ({\left (-i \, a c^{7} - a c^{6} d - 3 i \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} - 3 i \, a c^{3} d^{4} - 3 \, a c^{2} d^{5} - i \, a c d^{6} - a d^{7}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )} + 2 \, {\left (-i \, a c^{7} + a c^{6} d - 3 i \, a c^{5} d^{2} + 3 \, a c^{4} d^{3} - 3 i \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} - i \, a c d^{6} + a d^{7}\right )} f e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, a c^{7} + 3 \, a c^{6} d + i \, a c^{5} d^{2} + 5 \, a c^{4} d^{3} + 5 i \, a c^{3} d^{4} + a c^{2} d^{5} + 3 i \, a c d^{6} - a d^{7}\right )} f e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {-\frac {2 i}{{\left (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}\right )} f^{2}}} \log \left ({\left (i \, a c^{3} + 3 \, a c^{2} d - 3 i \, a c d^{2} - a d^{3}\right )} f \sqrt {-\frac {2 i}{{\left (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}\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \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 {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) + 3 \, {\left ({\left (i \, a c^{7} + a c^{6} d + 3 i \, a c^{5} d^{2} + 3 \, a c^{4} d^{3} + 3 i \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} + i \, a c d^{6} + a d^{7}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )} + 2 \, {\left (i \, a c^{7} - a c^{6} d + 3 i \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} + 3 i \, a c^{3} d^{4} - 3 \, a c^{2} d^{5} + i \, a c d^{6} - a d^{7}\right )} f e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (i \, a c^{7} - 3 \, a c^{6} d - i \, a c^{5} d^{2} - 5 \, a c^{4} d^{3} - 5 i \, a c^{3} d^{4} - a c^{2} d^{5} - 3 i \, a c d^{6} + a d^{7}\right )} f e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {-\frac {2 i}{{\left (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}\right )} f^{2}}} \log \left ({\left (-i \, a c^{3} - 3 \, a c^{2} d + 3 i \, a c d^{2} + a d^{3}\right )} f \sqrt {-\frac {2 i}{{\left (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}\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \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 {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right )}{12 \, {\left ({\left (-i \, a c^{7} - a c^{6} d - 3 i \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} - 3 i \, a c^{3} d^{4} - 3 \, a c^{2} d^{5} - i \, a c d^{6} - a d^{7}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )} + 2 \, {\left (-i \, a c^{7} + a c^{6} d - 3 i \, a c^{5} d^{2} + 3 \, a c^{4} d^{3} - 3 i \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} - i \, a c d^{6} + a d^{7}\right )} f e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, a c^{7} + 3 \, a c^{6} d + i \, a c^{5} d^{2} + 5 \, a c^{4} d^{3} + 5 i \, a c^{3} d^{4} + a c^{2} d^{5} + 3 i \, a c d^{6} - a d^{7}\right )} f e^{\left (i \, f x + i \, e\right )}\right )}} \end {gather*}

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]

1/12*(2*sqrt(2)*(3*c^4 + 6*c^2*d^2 + 3*d^4 + (3*c^4 - 12*I*c^3*d - 54*c^2*d^2 + 52*I*c*d^3 + 7*d^4)*e^(6*I*f*x
+ 6*I*e) + (9*c^4 - 24*I*c^3*d - 90*c^2*d^2 + 16*I*c*d^3 - 11*d^4)*e^(4*I*f*x + 4*I*e) + 3*(3*c^4 - 4*I*c^3*d
- 10*c^2*d^2 - 12*I*c*d^3 - 5*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)) + 3*((-I*a*c^7 - a*c^6*d - 3*I*a*c^5*d^2 - 3*a*c^4*d^3
- 3*I*a*c^3*d^4 - 3*a*c^2*d^5 - I*a*c*d^6 - a*d^7)*f*e^(5*I*f*x + 5*I*e) + 2*(-I*a*c^7 + a*c^6*d - 3*I*a*c^5*d
^2 + 3*a*c^4*d^3 - 3*I*a*c^3*d^4 + 3*a*c^2*d^5 - I*a*c*d^6 + a*d^7)*f*e^(3*I*f*x + 3*I*e) + (-I*a*c^7 + 3*a*c^
6*d + I*a*c^5*d^2 + 5*a*c^4*d^3 + 5*I*a*c^3*d^4 + a*c^2*d^5 + 3*I*a*c*d^6 - a*d^7)*f*e^(I*f*x + 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^(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))*sq
rt(a/(e^(2*I*f*x + 2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 1)) + 3*((I*a*c^7 + a*c^6*d + 3*I*a*c^5*d^2 + 3*a*c^4*d
^3 + 3*I*a*c^3*d^4 + 3*a*c^2*d^5 + I*a*c*d^6 + a*d^7)*f*e^(5*I*f*x + 5*I*e) + 2*(I*a*c^7 - a*c^6*d + 3*I*a*c^5
*d^2 - 3*a*c^4*d^3 + 3*I*a*c^3*d^4 - 3*a*c^2*d^5 + I*a*c*d^6 - a*d^7)*f*e^(3*I*f*x + 3*I*e) + (I*a*c^7 - 3*a*c
^6*d - I*a*c^5*d^2 - 5*a*c^4*d^3 - 5*I*a*c^3*d^4 - a*c^2*d^5 - 3*I*a*c*d^6 + a*d^7)*f*e^(I*f*x + 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^(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)))/((-I*a*c^7 - a*c^6*d - 3*I*a*c^5*d^2 - 3*a*c^4*d
^3 - 3*I*a*c^3*d^4 - 3*a*c^2*d^5 - I*a*c*d^6 - a*d^7)*f*e^(5*I*f*x + 5*I*e) + 2*(-I*a*c^7 + a*c^6*d - 3*I*a*c^
5*d^2 + 3*a*c^4*d^3 - 3*I*a*c^3*d^4 + 3*a*c^2*d^5 - I*a*c*d^6 + a*d^7)*f*e^(3*I*f*x + 3*I*e) + (-I*a*c^7 + 3*a
*c^6*d + I*a*c^5*d^2 + 5*a*c^4*d^3 + 5*I*a*c^3*d^4 + a*c^2*d^5 + 3*I*a*c*d^6 - a*d^7)*f*e^(I*f*x + I*e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

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]

Integral(1/(sqrt(I*a*(tan(e + f*x) - I))*(c + d*tan(e + f*x))**(5/2)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*tan(e + f*x)*1i)^(1/2)*(c + d*tan(e + f*x))^(5/2)),x)

[Out]

int(1/((a + a*tan(e + f*x)*1i)^(1/2)*(c + d*tan(e + f*x))^(5/2)), x)

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