3.12.71 \(\int \frac {1}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx\) [1171]

Optimal. Leaf size=354 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} (c-i d)^{5/2} f}-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d) (c+i d)^3 f (c+d \tan (e+f x))^{3/2}}+\frac {(c-3 i d) d \left (3 c^2+22 i c d+13 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d)^2 (c+i d)^4 f \sqrt {c+d \tan (e+f x)}} \]

[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)^
(5/2)/f*2^(1/2)+1/6*(c-3*I*d)*d*(3*c^2+22*I*c*d+13*d^2)*(a+I*a*tan(f*x+e))^(1/2)/a^2/(c-I*d)^2/(c+I*d)^4/f/(c+
d*tan(f*x+e))^(1/2)+1/2*(I*c-5*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2)+1/6*d*(3*c^2+1
4*I*c*d+21*d^2)*(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))^(3/2)-1/3/(I*c-d)/f/(a+I*a*t
an(f*x+e))^(3/2)/(c+d*tan(f*x+e))^(3/2)

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Rubi [A]
time = 0.85, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3640, 3677, 3679, 12, 3625, 214} \begin {gather*} -\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)^{5/2}}+\frac {d (c-3 i d) \left (3 c^2+22 i c d+13 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 f (c-i d)^2 (c+i d)^4 \sqrt {c+d \tan (e+f x)}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^{3/2}}+\frac {-5 d+i c}{2 a f (c+i d)^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac {1}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-1/2*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]])])/(Sqr
t[2]*a^(3/2)*(c - I*d)^(5/2)*f) - 1/(3*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)) +
(I*c - 5*d)/(2*a*(c + I*d)^2*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (d*(3*c^2 + (14*I)*c*d
+ 21*d^2)*Sqrt[a + I*a*Tan[e + f*x]])/(6*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^(3/2)) + ((c - (3*I
)*d)*d*(3*c^2 + (22*I)*c*d + 13*d^2)*Sqrt[a + I*a*Tan[e + f*x]])/(6*a^2*(c - I*d)^2*(c + I*d)^4*f*Sqrt[c + d*T
an[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 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]

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}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {3}{2} a (i c-3 d)-3 i a d \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx}{3 a^2 (i c-d)}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 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 {3}{4} a^2 \left (c^2+6 i c d-21 d^2\right )-3 a^2 (c+5 i d) d \tan (e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx}{3 a^4 (c+i d)^2}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \int \frac {\sqrt {a+i a \tan (e+f x)} \left (-\frac {3}{8} a^3 \left (3 c^3+15 i c^2 d-37 c d^2+39 i d^3\right )-\frac {3}{4} a^3 d \left (3 c^2+14 i c d+21 d^2\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{9 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} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {(c-3 i d) d \left (3 c^2+22 i c d+13 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d)^2 (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {4 \int -\frac {9 a^4 (c+i d)^4 \sqrt {a+i a \tan (e+f x)}}{16 \sqrt {c+d \tan (e+f x)}} \, dx}{9 a^6 (c+i d)^2 \left (c^2+d^2\right )^2}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {(c-3 i d) d \left (3 c^2+22 i c d+13 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d)^2 (c+i d)^4 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)^2}\\ &=-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {(c-3 i d) d \left (3 c^2+22 i c d+13 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d)^2 (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {i \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 )}{2 (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 )}{2 \sqrt {2} a^{3/2} (c-i d)^{5/2} f}-\frac {1}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {i c-5 d}{2 a (c+i d)^2 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (3 c^2+14 i c d+21 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {(c-3 i d) d \left (3 c^2+22 i c d+13 d^2\right ) \sqrt {a+i a \tan (e+f x)}}{6 a^2 (c-i d)^2 (c+i d)^4 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. \(803\) vs. \(2(354)=708\).
time = 9.37, size = 803, normalized size = 2.27 \begin {gather*} -\frac {i e^{2 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 ) \sec ^{\frac {3}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{3/2}}{2 \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 (a+i a \tan (e+f x))^{3/2}}+\frac {\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac {i (5 c+21 i d) \cos (2 f x)}{12 (c+i d)^4}+\frac {\left (i c^4 \cos (e)-3 c^3 d \cos (e)+9 i c^2 d^2 \cos (e)+29 c d^3 \cos (e)-10 i d^4 \cos (e)+i c^3 d \sin (e)-3 c^2 d^2 \sin (e)+9 i c d^3 \sin (e)+3 d^4 \sin (e)\right ) \left (\frac {1}{3} \cos (2 e)+\frac {1}{3} i \sin (2 e)\right )}{(c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e))}+\frac {\cos (4 f x) \left (\frac {1}{12} i \cos (2 e)+\frac {1}{12} \sin (2 e)\right )}{(c+i d)^3}+\frac {(5 c+21 i d) \sin (2 f x)}{12 (c+i d)^4}+\frac {\left (\frac {1}{12} \cos (2 e)-\frac {1}{12} i \sin (2 e)\right ) \sin (4 f x)}{(c+i d)^3}+\frac {\frac {2}{3} d^5 \cos (2 e)+\frac {2}{3} i d^5 \sin (2 e)}{(c-i d)^2 (c+i d)^4 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac {2 \left (\frac {13}{2} i c d^4 \cos (2 e-f x)+\frac {5}{2} d^5 \cos (2 e-f x)-\frac {13}{2} i c d^4 \cos (2 e+f x)-\frac {5}{2} d^5 \cos (2 e+f x)-\frac {13}{2} c d^4 \sin (2 e-f x)+\frac {5}{2} i d^5 \sin (2 e-f x)+\frac {13}{2} c d^4 \sin (2 e+f x)-\frac {5}{2} i d^5 \sin (2 e+f x)\right )}{3 (c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right )}{f (a+i a \tan (e+f x))^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-1/2*I)*E^((2*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))]*Sqr
t[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]*Sec[e + f*x]^(3/2)*(Cos[f*x] + I*Sin[f*x])
^(3/2))/(Sqrt[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*(a + I*a*Tan[e + f*x])^(3/2)) + (Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x]
+ d*Sin[e + f*x])]*(((I/12)*(5*c + (21*I)*d)*Cos[2*f*x])/(c + I*d)^4 + ((I*c^4*Cos[e] - 3*c^3*d*Cos[e] + (9*I
)*c^2*d^2*Cos[e] + 29*c*d^3*Cos[e] - (10*I)*d^4*Cos[e] + I*c^3*d*Sin[e] - 3*c^2*d^2*Sin[e] + (9*I)*c*d^3*Sin[e
] + 3*d^4*Sin[e])*(Cos[2*e]/3 + (I/3)*Sin[2*e]))/((c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + (Cos[4*f*x]
*((I/12)*Cos[2*e] + Sin[2*e]/12))/(c + I*d)^3 + ((5*c + (21*I)*d)*Sin[2*f*x])/(12*(c + I*d)^4) + ((Cos[2*e]/12
- (I/12)*Sin[2*e])*Sin[4*f*x])/(c + I*d)^3 + ((2*d^5*Cos[2*e])/3 + ((2*I)/3)*d^5*Sin[2*e])/((c - I*d)^2*(c +
I*d)^4*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) - (2*(((13*I)/2)*c*d^4*Cos[2*e - f*x] + (5*d^5*Cos[2*e - f*x])/2 -
((13*I)/2)*c*d^4*Cos[2*e + f*x] - (5*d^5*Cos[2*e + f*x])/2 - (13*c*d^4*Sin[2*e - f*x])/2 + ((5*I)/2)*d^5*Sin[
2*e - f*x] + (13*c*d^4*Sin[2*e + f*x])/2 - ((5*I)/2)*d^5*Sin[2*e + f*x]))/(3*(c - I*d)^2*(c + I*d)^4*(c*Cos[e]
+ d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^(3/2))

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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. 7060 vs. \(2 (294 ) = 588\).
time = 0.65, size = 7061, normalized size = 19.95

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(7061\)
default	\(\text {Expression too large to display}\)	\(7061\)

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))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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))^(3/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

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

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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. 1572 vs. \(2 (286) = 572\).
time = 1.27, size = 1572, normalized size = 4.44 \begin {gather*} \frac {\sqrt {2} {\left (i \, c^{5} - c^{4} d + 2 i \, c^{3} d^{2} - 2 \, c^{2} d^{3} + i \, c d^{4} - d^{5} - 4 \, {\left (-i \, c^{5} + c^{4} d - 14 i \, c^{3} d^{2} - 50 \, c^{2} d^{3} + 51 i \, c d^{4} + 13 \, d^{5}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (13 i \, c^{5} - 25 \, c^{4} d + 134 i \, c^{3} d^{2} + 314 \, c^{2} d^{3} - 87 i \, c d^{4} + 35 \, d^{5}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 3 \, {\left (-5 i \, c^{5} + 13 \, c^{4} d - 30 i \, c^{3} d^{2} - 26 \, c^{2} d^{3} - 41 i \, c d^{4} - 23 \, d^{5}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (7 i \, c^{5} - 19 \, c^{4} d + 14 i \, c^{3} d^{2} - 38 \, c^{2} d^{3} + 7 i \, c d^{4} - 19 \, d^{5}\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 (a^{2} c^{8} + 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} + 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right )} f e^{\left (7 i \, f x + 7 i \, e\right )} + 2 \, {\left (a^{2} c^{8} + 2 i \, a^{2} c^{7} d + 2 \, a^{2} c^{6} d^{2} + 6 i \, a^{2} c^{5} d^{3} + 6 i \, a^{2} c^{3} d^{5} - 2 \, a^{2} c^{2} d^{6} + 2 i \, a^{2} c d^{7} - a^{2} d^{8}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (a^{2} c^{8} + 4 i \, a^{2} c^{7} d - 4 \, a^{2} c^{6} d^{2} + 4 i \, a^{2} c^{5} d^{3} - 10 \, a^{2} c^{4} d^{4} - 4 i \, a^{2} c^{3} d^{5} - 4 \, a^{2} c^{2} d^{6} - 4 i \, a^{2} c d^{7} + a^{2} d^{8}\right )} f e^{\left (3 i \, f x + 3 i \, e\right )}\right )} \sqrt {\frac {i}{2 \, {\left (-i \, a^{3} c^{5} - 5 \, a^{3} c^{4} d + 10 i \, a^{3} c^{3} d^{2} + 10 \, a^{3} c^{2} d^{3} - 5 i \, a^{3} c d^{4} - a^{3} d^{5}\right )} f^{2}}} \log \left (-2 \, {\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 \sqrt {\frac {i}{2 \, {\left (-i \, a^{3} c^{5} - 5 \, a^{3} c^{4} d + 10 i \, a^{3} c^{3} d^{2} + 10 \, a^{3} c^{2} d^{3} - 5 i \, a^{3} c d^{4} - a^{3} 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 (a^{2} c^{8} + 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} + 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right )} f e^{\left (7 i \, f x + 7 i \, e\right )} + 2 \, {\left (a^{2} c^{8} + 2 i \, a^{2} c^{7} d + 2 \, a^{2} c^{6} d^{2} + 6 i \, a^{2} c^{5} d^{3} + 6 i \, a^{2} c^{3} d^{5} - 2 \, a^{2} c^{2} d^{6} + 2 i \, a^{2} c d^{7} - a^{2} d^{8}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (a^{2} c^{8} + 4 i \, a^{2} c^{7} d - 4 \, a^{2} c^{6} d^{2} + 4 i \, a^{2} c^{5} d^{3} - 10 \, a^{2} c^{4} d^{4} - 4 i \, a^{2} c^{3} d^{5} - 4 \, a^{2} c^{2} d^{6} - 4 i \, a^{2} c d^{7} + a^{2} d^{8}\right )} f e^{\left (3 i \, f x + 3 i \, e\right )}\right )} \sqrt {\frac {i}{2 \, {\left (-i \, a^{3} c^{5} - 5 \, a^{3} c^{4} d + 10 i \, a^{3} c^{3} d^{2} + 10 \, a^{3} c^{2} d^{3} - 5 i \, a^{3} c d^{4} - a^{3} d^{5}\right )} f^{2}}} \log \left (-2 \, {\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 \sqrt {\frac {i}{2 \, {\left (-i \, a^{3} c^{5} - 5 \, a^{3} c^{4} d + 10 i \, a^{3} c^{3} d^{2} + 10 \, a^{3} c^{2} d^{3} - 5 i \, a^{3} c d^{4} - a^{3} 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 (a^{2} c^{8} + 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} + 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right )} f e^{\left (7 i \, f x + 7 i \, e\right )} + 2 \, {\left (a^{2} c^{8} + 2 i \, a^{2} c^{7} d + 2 \, a^{2} c^{6} d^{2} + 6 i \, a^{2} c^{5} d^{3} + 6 i \, a^{2} c^{3} d^{5} - 2 \, a^{2} c^{2} d^{6} + 2 i \, a^{2} c d^{7} - a^{2} d^{8}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (a^{2} c^{8} + 4 i \, a^{2} c^{7} d - 4 \, a^{2} c^{6} d^{2} + 4 i \, a^{2} c^{5} d^{3} - 10 \, a^{2} c^{4} d^{4} - 4 i \, a^{2} c^{3} d^{5} - 4 \, a^{2} c^{2} d^{6} - 4 i \, a^{2} c d^{7} + a^{2} d^{8}\right )} f e^{\left (3 i \, f x + 3 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))^(3/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/12*(sqrt(2)*(I*c^5 - c^4*d + 2*I*c^3*d^2 - 2*c^2*d^3 + I*c*d^4 - d^5 - 4*(-I*c^5 + c^4*d - 14*I*c^3*d^2 - 50
*c^2*d^3 + 51*I*c*d^4 + 13*d^5)*e^(8*I*f*x + 8*I*e) + (13*I*c^5 - 25*c^4*d + 134*I*c^3*d^2 + 314*c^2*d^3 - 87*
I*c*d^4 + 35*d^5)*e^(6*I*f*x + 6*I*e) - 3*(-5*I*c^5 + 13*c^4*d - 30*I*c^3*d^2 - 26*c^2*d^3 - 41*I*c*d^4 - 23*d
^5)*e^(4*I*f*x + 4*I*e) + (7*I*c^5 - 19*c^4*d + 14*I*c^3*d^2 - 38*c^2*d^3 + 7*I*c*d^4 - 19*d^5)*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*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(7*I*f*x + 7*I*e) + 2*(a^2*c
^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*d
^8)*f*e^(5*I*f*x + 5*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I*
a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c*d^7 + a^2*d^8)*f*e^(3*I*f*x + 3*I*e))*sqrt(1/2*I/((-I*a^3*c^5 - 5*a^3*
c^4*d + 10*I*a^3*c^3*d^2 + 10*a^3*c^2*d^3 - 5*I*a^3*c*d^4 - a^3*d^5)*f^2))*log(-2*(I*a^2*c^3 + 3*a^2*c^2*d - 3
*I*a^2*c*d^2 - a^2*d^3)*f*sqrt(1/2*I/((-I*a^3*c^5 - 5*a^3*c^4*d + 10*I*a^3*c^3*d^2 + 10*a^3*c^2*d^3 - 5*I*a^3*
c*d^4 - a^3*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)) + 3*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^
2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(7*I*f*x + 7*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*
a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*d^8)*f*e^(5*I*f*x + 5*I*e) + (a^2*c^8 + 4*
I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c*d
^7 + a^2*d^8)*f*e^(3*I*f*x + 3*I*e))*sqrt(1/2*I/((-I*a^3*c^5 - 5*a^3*c^4*d + 10*I*a^3*c^3*d^2 + 10*a^3*c^2*d^3
- 5*I*a^3*c*d^4 - a^3*d^5)*f^2))*log(-2*(-I*a^2*c^3 - 3*a^2*c^2*d + 3*I*a^2*c*d^2 + a^2*d^3)*f*sqrt(1/2*I/((-
I*a^3*c^5 - 5*a^3*c^4*d + 10*I*a^3*c^3*d^2 + 10*a^3*c^2*d^3 - 5*I*a^3*c*d^4 - a^3*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)))/((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(7
*I*f*x + 7*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d
^6 + 2*I*a^2*c*d^7 - a^2*d^8)*f*e^(5*I*f*x + 5*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d
^3 - 10*a^2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c*d^7 + a^2*d^8)*f*e^(3*I*f*x + 3*I*e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {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))**(3/2)/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Integral(1/((I*a*(tan(e + f*x) - I))**(3/2)*(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))^(3/2)/(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}{{\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 )}^{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)^(3/2)*(c + d*tan(e + f*x))^(5/2)),x)

[Out]

int(1/((a + a*tan(e + f*x)*1i)^(3/2)*(c + d*tan(e + f*x))^(5/2)), x)

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