3.1153 \(\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=257 \[ \frac{2 \sqrt [4]{-1} d^{5/2} \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right )}{a^{3/2} f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{2 \sqrt{2} a^{3/2} f}+\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{(c+i d) (3 d+i c) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}} \]
[Out]
(2*(-1)^(1/4)*d^(5/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]
])])/(a^(3/2)*f) - ((I/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sq
rt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) + ((c + I*d)*(I*c + 3*d)*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*Sqrt
[a + I*a*Tan[e + f*x]]) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))
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Rubi [A] time = 0.995615, antiderivative size = 257, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used =
{3558, 3595, 3601, 3544, 208, 3599, 63, 217, 206} \[ \frac{2 \sqrt [4]{-1} d^{5/2} \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right )}{a^{3/2} f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{2 \sqrt{2} a^{3/2} f}+\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{(c+i d) (3 d+i c) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2),x]
[Out]
(2*(-1)^(1/4)*d^(5/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]
])])/(a^(3/2)*f) - ((I/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sq
rt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) + ((c + I*d)*(I*c + 3*d)*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*Sqrt
[a + I*a*Tan[e + f*x]]) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))
Rule 3558
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[((b*c - a*d)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a
+ b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1))
- d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], 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] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m,
2*n])
Rule 3595
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*b - a*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(2*a*f*
m), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d*n
) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]
Rule 3601
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A*b + a*B)/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x]
, x] - Dist[B/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[e + f*x]), x], x] /; FreeQ[{a, b
, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
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]
Rule 3599
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*B)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x
]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B,
0]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx &=\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac{\int \frac{\sqrt{c+d \tan (e+f x)} \left (-\frac{3}{2} a \left (c^2-2 i c d+d^2\right )+3 i a d^2 \tan (e+f x)\right )}{\sqrt{a+i a \tan (e+f x)}} \, dx}{3 a^2}\\ &=\frac{(c+i d) (i c+3 d) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \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^3-3 i c^2 d-3 c d^2-3 i d^3\right )-3 a^2 d^3 \tan (e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{3 a^4}\\ &=\frac{(c+i d) (i c+3 d) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{(c-i d)^3 \int \frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}} \, dx}{4 a^2}-\frac{\left (i d^3\right ) \int \frac{(a-i a \tan (e+f x)) \sqrt{a+i a \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}} \, dx}{a^3}\\ &=\frac{(c+i d) (i c+3 d) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac{\left (i d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{a f}+\frac{(i c+d)^3 \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 )}{2 f}\\ &=-\frac{i (c-i d)^{5/2} \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}+\frac{(c+i d) (i c+3 d) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+i d-\frac{i d x^2}{a}}} \, dx,x,\sqrt{a+i a \tan (e+f x)}\right )}{a^2 f}\\ &=-\frac{i (c-i d)^{5/2} \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}+\frac{(c+i d) (i c+3 d) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{i d x^2}{a}} \, dx,x,\frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{a^2 f}\\ &=\frac{2 \sqrt [4]{-1} d^{5/2} \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right )}{a^{3/2} f}-\frac{i (c-i d)^{5/2} \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}+\frac{(c+i d) (i c+3 d) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [B] time = 7.44013, size = 560, normalized size = 2.18 \[ \frac{\sec (e+f x) (\cos (f x)+i \sin (f x))^2 \left (\frac{(\cos (2 e)+i \sin (2 e)) \left ((2+2 i) d^{5/2} \log \left (\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) e^{\frac{i e}{2}} \left ((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}-i c e^{i (e+f x)}+c-d e^{i (e+f x)}+i d\right )}{d^{7/2} \left (e^{i (e+f x)}+i\right )}\right )-(2+2 i) d^{5/2} \log \left (-\frac{\left (\frac{1}{4}+\frac{i}{4}\right ) e^{\frac{i e}{2}} \left ((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right )}{d^{7/2} \left (e^{i (e+f x)}-i\right )}\right )-i (c-i d)^{5/2} \log \left (2 \left (i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right )\right )\right )}{\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}+\frac{1}{3} (c+i d) (\sin (2 f x)+i \cos (2 f x)) \sqrt{c+d \tan (e+f x)} ((11 d+3 i c) \sin (e+f x)+(5 c-9 i d) \cos (e+f x))\right )}{2 f (a+i a \tan (e+f x))^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2),x]
[Out]
(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*((((2 + 2*I)*d^(5/2)*Log[((1/4 + I/4)*E^((I/2)*e)*(c + I*d - I*c*E^(I*
(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*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)))]))/(d^(7/2)*(I + E^(I*(e + f*x))))] - (2 + 2*I)*d^(5/2)*Log[((-1/4 - I/4)
*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*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)))]))/(d^(7/2)*(-I + E^(I*(e + f*x))))] - I
*(c - I*d)^(5/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)]
+ I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] + I*Sin[2*e]))/Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2
*(e + f*x)]] + ((c + I*d)*(I*Cos[2*f*x] + Sin[2*f*x])*((5*c - (9*I)*d)*Cos[e + f*x] + ((3*I)*c + 11*d)*Sin[e +
f*x])*Sqrt[c + d*Tan[e + f*x]])/3))/(2*f*(a + I*a*Tan[e + f*x])^(3/2))
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Maple [B] time = 0.056, size = 3161, normalized size = 12.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x)
[Out]
1/24*I/f*(c+d*tan(f*x+e))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^2*(-9*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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+24*I*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(
1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*tan(f*x+e)^3*a*c*d^3+9*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))*
(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*tan(f*x+e)^2*c*d^2-9*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))*(I*a*d)^(1/2)*2^(1
/2)*(-a*(I*d-c))^(1/2)*tan(f*x+e)*c^2*d-9*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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^2+3*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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^2+72*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*
a*d)^(1/2))*tan(f*x+e)^2*a*c*d^3-3*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))*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*c^3+
16*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*tan(f*x+e)*c^2*d-76*(a*(c+d*tan(f*x+e))*(1+I*tan(
f*x+e)))^(1/2)*(I*a*d)^(1/2)*tan(f*x+e)^2*c*d^2+72*I*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(1
+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*tan(f*x+e)^2*a*d^4-44*I*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+
e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*d^3+32*I*tan(f*x+e)*c^3*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*
x+e)))^(1/2)+4*I*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d+20*c^3*(I*a*d)^(1/2)*(a*(c+d*
tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)-12*tan(f*x+e)^2*c^3*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1
/2)+72*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*
d)^(1/2))*tan(f*x+e)*a*d^4-24*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(
I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c*d^3-80*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*tan(f*x+
e)*d^3+52*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*c*d^2-24*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+
2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*tan(f*x+e)^3*a*d^4-24*I*ln(1/2
*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d
^4+36*I*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^3+128*I*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*
(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c*d^2+3*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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^3-72*I*ln(1/2*(2*I*a*tan(f*x+e)*d+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*
d)^(1/2)+a*d)/(I*a*d)^(1/2))*tan(f*x+e)*a*c*d^3+3*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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))/(ta
n(f*x+e)+I))*d^3+20*I*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^2*d-3*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))*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*c*d^2+9*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))*(I*a*d)^(1
/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*tan(f*x+e)^2*c^3-9*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))*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d
-c))^(1/2)*tan(f*x+e)*d^3+3*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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^3-9*I*(I*a*d
)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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^3-9*I*(I*a*d)^(1/2)*2^(1/2)*(-
a*(I*d-c))^(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+3*I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(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)/(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)/(I*a*d)^(1/2)/(I*c-d)/(-tan(f*x+
e)+I)^3
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
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Fricas [B] time = 2.8302, size = 2830, normalized size = 11.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
-1/12*(3*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^3*f^2))*e^(4
*I*f*x + 4*I*e)*log(-(2*I*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5
)/(a^3*f^2))*e^(2*I*f*x + 2*I*e) - sqrt(2)*(c^2 - 2*I*c*d - d^2 + (c^2 - 2*I*c*d - d^2)*e^(2*I*f*x + 2*I*e))*s
qrt(((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))*e^(-I*f*x - I*e)/(c^2 - 2*I*c*d - d^2)) - 3*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2
+ 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^3*f^2))*e^(4*I*f*x + 4*I*e)*log(-(-2*I*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c
^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^3*f^2))*e^(2*I*f*x + 2*I*e) - sqrt(2)*(c^2 - 2*I*c*d -
d^2 + (c^2 - 2*I*c*d - 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))*e^(-I*f*x - I*e)/(c^2 - 2*I*c*d - d^2)) + 3*a^
2*f*sqrt(4*I*d^5/(a^3*f^2))*e^(4*I*f*x + 4*I*e)*log((8*sqrt(2)*(d^3*e^(2*I*f*x + 2*I*e) + d^3)*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) +
((2*a^2*c - 6*I*a^2*d)*f*e^(2*I*f*x + 2*I*e) + (2*a^2*c + 2*I*a^2*d)*f)*sqrt(4*I*d^5/(a^3*f^2)))/(I*c^3 + c^2
*d + I*c*d^2 + d^3 + (I*c^3 + c^2*d + I*c*d^2 + d^3)*e^(2*I*f*x + 2*I*e))) - 3*a^2*f*sqrt(4*I*d^5/(a^3*f^2))*e
^(4*I*f*x + 4*I*e)*log((8*sqrt(2)*(d^3*e^(2*I*f*x + 2*I*e) + d^3)*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) - ((2*a^2*c - 6*I*a^2*d)*f*e^(
2*I*f*x + 2*I*e) + (2*a^2*c + 2*I*a^2*d)*f)*sqrt(4*I*d^5/(a^3*f^2)))/(I*c^3 + c^2*d + I*c*d^2 + d^3 + (I*c^3 +
c^2*d + I*c*d^2 + d^3)*e^(2*I*f*x + 2*I*e))) - sqrt(2)*(I*c^2 - 2*c*d - I*d^2 + (4*I*c^2 + 6*c*d + 10*I*d^2)*
e^(4*I*f*x + 4*I*e) + (5*I*c^2 + 4*c*d + 9*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))*e^(-4*I*f*x - 4*I*e)/(a^
2*f)
________________________________________________________________________________________
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((c+d*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e))**(3/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError