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]
-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 = 0.86, 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 =
{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 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 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}{\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*}
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Mathematica [B] time = 9.08, size = 687, normalized size = 2.48 \[ \frac {\sec (e+f x) (\cos (f x)+i \sin (f x)) \left (\frac {\left (\frac {\cos (e)}{6}+\frac {1}{6} i \sin (e)\right ) \left (3 i c^3 \cos (e)+3 i c^2 d \sin (e)+6 c^2 d \cos (e)+6 c d^2 \sin (e)-39 i c d^2 \cos (e)+i d^3 \sin (e)-8 d^3 \cos (e)\right )}{(c-i d)^2 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac {\frac {2}{3} d^4 \sin (e)-\frac {2}{3} i d^4 \cos (e)}{(c-i d)^2 (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))^2}+\frac {4 \left (-\frac {5}{2} i c d^3 \sin (e-f x)+\frac {5}{2} i c d^3 \sin (e+f x)-\frac {5}{2} c d^3 \cos (e-f x)+\frac {5}{2} c d^3 \cos (e+f x)-\frac {1}{2} d^4 \sin (e-f x)+\frac {1}{2} d^4 \sin (e+f x)+\frac {1}{2} i d^4 \cos (e-f x)-\frac {1}{2} i d^4 \cos (e+f x)\right )}{3 (c-i d)^2 (c+i d)^3 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac {\left (\frac {\sin (e)}{2}+\frac {1}{2} i \cos (e)\right ) \cos (2 f x)}{(c+i d)^3}+\frac {\left (\frac {\cos (e)}{2}-\frac {1}{2} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^3}\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {i e^{i e} \sqrt {e^{i f x}} \sqrt {\sec (e+f x)} \sqrt {\cos (f x)+i \sin (f x)} \log \left (2 \left (\sqrt {c-i d} e^{i (e+f x)}+\sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )}{\sqrt {2} f (c-i d)^{5/2} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} \sqrt {a+i a \tan (e+f x)}} \]
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]
((-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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fricas [B] time = 0.64, size = 1304, normalized size = 4.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+I*a*tan(f*x+e))^(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)) - ((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^(5*I*f*x + 5*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^(3*I*f*x + 3*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
^(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))*l
og((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)) - ((-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^(5*I*f*x + 5*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^(3*I*f*x + 3*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^(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)))/((12*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^(5*I*f*x + 5*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^(3*I*f*x + 3*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^(I*f*x + I*e))
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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))^(1/2)/(c+d*tan(f*x+e))^(5/2),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: 2.57Error: Bad Argument Type
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maple [B] time = 0.37, size = 4889, normalized size = 17.65 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(5/2),x)
[Out]
-1/12/f*(-30*2^(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*ta
n(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)-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+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*ta
n(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*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+10
4*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*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^2-3*2^(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^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)-1
2*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+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*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c*d^6+36*(a*(c+d*tan(f*x+e))*(1+
I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^6*d+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^5-30*2^(1/2)*ln((3*c*a+I*a*tan(f*x+e)*c-I*d*a+3*a*t
an(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*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^6*(-a*(I*d-c))^(1/2)
+6*2^(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*d*(-a*(I*d-c))^(1/2)-30*2^(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^3*(-a*(I*d-c))^(1/2)+27*2^(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^
2*(-a*(I*d-c))^(1/2)-75*2^(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^3*d^4*(-a*(I*d-c))^(1/2)-3*2^(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^6*(-a*(I*d-c))^(1/2)+24*2^(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*d*(-a*(I*d-c))^(1/2)-24*2^(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^5*(-a*(I*d-c))^
(1/2)+3*I*2^(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*d^7*(-a*(I*d-c))^(1/2)-3*I*2^(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*d^7*(-a*(I*d-c))^(1/2)-6*I*2^(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^7*(
-a*(I*d-c))^(1/2)-15*I*2^(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^6*d*(-a*(I*d-c))^(1/2)+30*I*2^(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^4*d^3*(-a*(I*d-c))^(1/2)-3*I*2^(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^2*d^5*(-a*(I*d-c))^(
1/2)+3*2^(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^2*(-a*(I*d-c))^(1/2)+12*(a*(c+d*tan(f*x+e))*(1
+I*tan(f*x+e)))^(1/2)*c^7+30*2^(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^2*(-a*(I*d-c))^(1/2)-15*2^(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^4*(-a*(I*d-c))^(1/2)+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)-24
*I*2^(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*d^6*(-a*(I*d-c))^(1/2)+3*I*2^(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^6*d*(-a*(I*d-c))^(1/2)+75*I*2^(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^4*d
^3*(-a*(I*d-c))^(1/2)-27*I*2^(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^2*d^5*(-a*(I*d-c))^(1/2)+30*I*
2^(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^5*d^2*(-a*(I*d-c))^(1/2)+30*I*2^(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^3*d^4*(-a*(I*d-c))^(1/2)-6*I*2^(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*d^6*(-a*
(I*d-c))^(1/2)+15*I*2^(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^4*d^3*(-a*(I*d-c))^(1/2)-30*I*2^(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^2*d^5*(-a*(I*d-c))^(1/2)+24*I*2^(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^5*d^2*(-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*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+6*2^(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^7*(-a*(I*d-c))^(1/2)+3*2^(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^7*(-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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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