3.1130 \(\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=368 \[ \frac{d \left (9 i c^2 d+2 c^3-17 c d^2+60 i d^3\right )}{16 a^3 f (c-i d) (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}+\frac{\left (-12 c^2 d+2 i c^3-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 f (c+i d)^{9/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 f (c-i d)^{3/2}}+\frac{-10 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}} \]
[Out]
((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(3/2)*f) + (((2*I)*c^3 - 12*c^2*d - (3
3*I)*c*d^2 + 58*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(9/2)*f) + (d*(2*c^3 +
(9*I)*c^2*d - 17*c*d^2 + (60*I)*d^3))/(16*a^3*(c - I*d)*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]]) - 1/(6*(I*c -
d)*f*(a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]) + ((3*I)*c - 10*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e
+ f*x])^2*Sqrt[c + d*Tan[e + f*x]]) + (6*c^2 + (27*I)*c*d - 56*d^2)/(48*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*
x])*Sqrt[c + d*Tan[e + f*x]])
________________________________________________________________________________________
Rubi [A] time = 1.20081, antiderivative size = 368, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used
= {3559, 3596, 3529, 3539, 3537, 63, 208} \[ \frac{d \left (9 i c^2 d+2 c^3-17 c d^2+60 i d^3\right )}{16 a^3 f (c-i d) (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}+\frac{\left (-12 c^2 d+2 i c^3-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 f (c+i d)^{9/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 f (c-i d)^{3/2}}+\frac{-10 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)),x]
[Out]
((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(3/2)*f) + (((2*I)*c^3 - 12*c^2*d - (3
3*I)*c*d^2 + 58*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(9/2)*f) + (d*(2*c^3 +
(9*I)*c^2*d - 17*c*d^2 + (60*I)*d^3))/(16*a^3*(c - I*d)*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]]) - 1/(6*(I*c -
d)*f*(a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]) + ((3*I)*c - 10*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e
+ f*x])^2*Sqrt[c + d*Tan[e + f*x]]) + (6*c^2 + (27*I)*c*d - 56*d^2)/(48*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*
x])*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 3596
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 3529
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
Rule 3539
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 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 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}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{-\frac{1}{2} a (6 i c-13 d)-\frac{7}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx}{6 a^2 (i c-d)}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{-\frac{1}{2} a^2 \left (12 c^2+39 i c d-62 d^2\right )-\frac{5}{2} a^2 (3 c+10 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{\frac{3}{2} a^3 \left (4 i c^3-18 c^2 d-39 i c d^2+60 d^3\right )+\frac{3}{2} a^3 d \left (6 i c^2-27 c d-56 i d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac{d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{\frac{3}{2} a^3 \left (4 i c^4-18 c^3 d-33 i c^2 d^2+33 c d^3-56 i d^4\right )+\frac{3}{2} a^3 d \left (2 i c^3-9 c^2 d-17 i c d^2-60 d^3\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )}\\ &=\frac{d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}+\frac{\int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{16 a^3 (c-i d)}+\frac{\left (2 c^3+12 i c^2 d-33 c d^2-58 i d^3\right ) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)^4}\\ &=\frac{d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{16 a^3 (i c+d) f}-\frac{\left (2 i c^3-12 c^2 d-33 i c d^2+58 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 (c+i d)^4 f}\\ &=\frac{d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-\frac{i c}{d}+\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{8 a^3 (c-i d) d f}-\frac{\left (2 c^3+12 i c^2 d-33 c d^2-58 i d^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i c}{d}-\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{16 a^3 (c+i d)^4 d f}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{8 a^3 (c-i d)^{3/2} f}+\frac{\left (2 i c^3-12 c^2 d-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{16 a^3 (c+i d)^{9/2} f}+\frac{d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}+\frac{3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}+\frac{6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 7.32581, size = 468, normalized size = 1.27 \[ \frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac{\cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c+d \tan (e+f x)} \left (i \left (\left (-57 c^2 d^2+40 i c^3 d+9 c^4+40 i c d^3-66 d^4\right ) \sin (e+f x)+\left (-3 c^2 d^2+28 i c^3 d+9 c^4+142 i c d^3+294 d^4\right ) \sin (3 (e+f x))\right )+\left (-71 c^2 d^2+90 i c^3 d+27 c^4+90 i c d^3-98 d^4\right ) \cos (e+f x)+\left (-3 c^2 d^2+36 i c^3 d+13 c^4+150 i c d^3+290 d^4\right ) \cos (3 (e+f x))\right )}{3 (c-i d) (c+i d)^4 (c \cos (e+f x)+d \sin (e+f x))}-\frac{2 (\cos (3 e)+i \sin (3 e)) \left (\sqrt{-c+i d} \left (21 i c^2 d^2+10 c^3 d-2 i c^4-25 c d^3+58 i d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right )+2 i (-c-i d)^{9/2} \tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right )\right )}{(-c-i d)^{9/2} (-c+i d)^{3/2}}\right )}{32 f (a+i a \tan (e+f x))^3} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)),x]
[Out]
(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((-2*(Sqrt[-c + I*d]*((-2*I)*c^4 + 10*c^3*d + (21*I)*c^2*d^2 - 25*c*
d^3 + (58*I)*d^4)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + (2*I)*(-c - I*d)^(9/2)*ArcTan[Sqrt[c + d*T
an[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(9/2)*(-c + I*d)^(3/2)) + (Cos[e + f*x]*(I*
Cos[3*f*x] + Sin[3*f*x])*((27*c^4 + (90*I)*c^3*d - 71*c^2*d^2 + (90*I)*c*d^3 - 98*d^4)*Cos[e + f*x] + (13*c^4
+ (36*I)*c^3*d - 3*c^2*d^2 + (150*I)*c*d^3 + 290*d^4)*Cos[3*(e + f*x)] + I*((9*c^4 + (40*I)*c^3*d - 57*c^2*d^2
+ (40*I)*c*d^3 - 66*d^4)*Sin[e + f*x] + (9*c^4 + (28*I)*c^3*d - 3*c^2*d^2 + (142*I)*c*d^3 + 294*d^4)*Sin[3*(e
+ f*x)]))*Sqrt[c + d*Tan[e + f*x]])/(3*(c - I*d)*(c + I*d)^4*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(32*f*(a +
I*a*Tan[e + f*x])^3)
________________________________________________________________________________________
Maple [B] time = 0.088, size = 3053, normalized size = 8.3 \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))^3/(c+d*tan(f*x+e))^(3/2),x)
[Out]
-1/8/f/a^3*d/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^6
-15/16*I/f/a^3*d^2/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/
2)*c^5+13/4*I/f/a^3*d^4/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e)
)^(5/2)*c^3+67/16*I/f/a^3*d^6/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(
f*x+e))^(5/2)*c+9/4*I/f/a^3*d^2/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*ta
n(f*x+e))^(3/2)*c^6-59/4*I/f/a^3*d^4/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c
+d*tan(f*x+e))^(3/2)*c^4-155/12*I/f/a^3*d^6/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+
c^3)*(c+d*tan(f*x+e))^(3/2)*c^2-57/16*I/f/a^3*d^2/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1
/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^5+45/8*I/f/a^3*d^4/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*I*c
^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3+149/16*I/f/a^3*d^6/(I*d-c)/(c+I*d)^
4/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c-21/16*I/f/a^3*
d^2/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^7-2*I/f/a^
3*d^4/(I*c-d)/(I*c+d)/(c+I*d)^3/(c+d*tan(f*x+e))^(1/2)+3/4*I/f/a^3*d^2/(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan
(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2-29/8/f/a^3*d^7/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/
2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))-7/4/f/a^3*d^7/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3
-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)+5/2/f/a^3*d^9/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3
*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)+25/8/f/a^3*d^3/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*
c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^4-1/f/a^3*d/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I
*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^6+1/8*I/f/a^3/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*
I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^7+49/12*I/f/a^3*d^8/(I*d-c)/(c+I*d
)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)+3/2/f/a^3*d^5/(I*d-c)/(c+I*d)^
4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(5/2)*c^2+1/4/f/a^3*d/(I*d-c)/(c+I*d)^
4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^7-109/12/f/a^3*d^3/(I*d-c)/(c+
I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^5-127/8/f/a^3*d^7/(I*d-c)
/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^2+53/12/f/a^3*d^5/(I*
d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c^3+55/4/f/a^3*d^7/
(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(3/2)*c-1/8/f/a^3*d/(I
*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^8+51/8/f/a^3*d^3
/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^6-95/8/f/a^3*
d^5/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^4+15/2/f/a
^3*d^3/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^
(1/2))*c^4+39/8/f/a^3*d^5/(I*d-c)/(c+I*d)^4/(-I*d^3-3*c*d^2+3*I*c^2*d+c^3)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+
e))^(1/2)/(-I*d-c)^(1/2))*c^2+225/16*I/f/a^3*d^4/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2+3*I*c
^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^5+73/16*I/f/a^3*d^6/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-3*c*d^2
+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c^3-173/16*I/f/a^3*d^8/(I*d-c)/(c+I*d)^4/(-I*d+d*tan(f*x+e))^3/(-I*d^3-
3*c*d^2+3*I*c^2*d+c^3)*(c+d*tan(f*x+e))^(1/2)*c-1/2/f/a^3*d^3/(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^
(1/2)/(I*d-c)^(1/2))*c+1/2/f/a^3*d/(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^3-1/
8*I/f/a^3/(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^4-1/8*I/f/a^3*d^4/(I*d-c)^(3/
2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))
________________________________________________________________________________________
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(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
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Fricas [B] time = 34.2936, size = 6575, normalized size = 17.87 \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))^3/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
(((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I*a^3*c*d^5 - 24*a^3*
d^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*d^4 - 96*I*a^3*c*d^5
+ 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt(I/((-64*I*a^6*c^3 - 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a^6*d^3)*f
^2))*log((((16*I*a^3*c^2 + 32*a^3*c*d - 16*I*a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (16*I*a^3*c^2 + 32*a^3*c*d - 16*
I*a^3*d^2)*f)*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(I/((-64*I*a^6*c^3
- 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a^6*d^3)*f^2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) + 2*c)*e^(-2*I*f*x -
2*I*e)) - ((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I*a^3*c*d^5
- 24*a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*d^4 - 96*I*
a^3*c*d^5 + 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt(I/((-64*I*a^6*c^3 - 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a
^6*d^3)*f^2))*log((((-16*I*a^3*c^2 - 32*a^3*c*d + 16*I*a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (-16*I*a^3*c^2 - 32*a^
3*c*d + 16*I*a^3*d^2)*f)*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(I/((-6
4*I*a^6*c^3 - 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a^6*d^3)*f^2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) + 2*c)*e^(
-2*I*f*x - 2*I*e)) - ((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I
*a^3*c*d^5 - 24*a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*
d^4 - 96*I*a^3*c*d^5 + 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt((4*I*c^6 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*
d^3 + 2481*I*c^2*d^4 - 3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504
*a^6*c^6*d^3 - 32256*I*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c
*d^8 + 256*a^6*d^9)*f^2))*log((2*c^4 + 14*I*c^3*d - 45*c^2*d^2 - 91*I*c*d^3 + 58*d^4 + ((16*I*a^3*c^5 - 80*a^3
*c^4*d - 160*I*a^3*c^3*d^2 + 160*a^3*c^2*d^3 + 80*I*a^3*c*d^4 - 16*a^3*d^5)*f*e^(2*I*f*x + 2*I*e) + (16*I*a^3*
c^5 - 80*a^3*c^4*d - 160*I*a^3*c^3*d^2 + 160*a^3*c^2*d^3 + 80*I*a^3*c*d^4 - 16*a^3*d^5)*f)*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((4*I*c^6 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*d^3
+ 2481*I*c^2*d^4 - 3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504*a^
6*c^6*d^3 - 32256*I*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c*d^
8 + 256*a^6*d^9)*f^2)) + (2*c^4 + 12*I*c^3*d - 33*c^2*d^2 - 58*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I
*e)/((-16*I*a^3*c^5 + 80*a^3*c^4*d + 160*I*a^3*c^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f)) +
((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I*a^3*c*d^5 - 24*a^3*d
^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*d^4 - 96*I*a^3*c*d^5
+ 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt((4*I*c^6 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*d^3 + 2481*I*c^2*d^4
- 3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504*a^6*c^6*d^3 - 32256*
I*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c*d^8 + 256*a^6*d^9)*f
^2))*log((2*c^4 + 14*I*c^3*d - 45*c^2*d^2 - 91*I*c*d^3 + 58*d^4 + ((-16*I*a^3*c^5 + 80*a^3*c^4*d + 160*I*a^3*c
^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f*e^(2*I*f*x + 2*I*e) + (-16*I*a^3*c^5 + 80*a^3*c^4*d
+ 160*I*a^3*c^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f)*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((4*I*c^6 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*d^3 + 2481*I*c^2*d^4 -
3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504*a^6*c^6*d^3 - 32256*I
*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c*d^8 + 256*a^6*d^9)*f^
2)) + (2*c^4 + 12*I*c^3*d - 33*c^2*d^2 - 58*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-16*I*a^3*c^5
+ 80*a^3*c^4*d + 160*I*a^3*c^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f)) - (-2*I*c^4 + 4*c^3*d
+ 4*c*d^3 + 2*I*d^4 + (-11*I*c^4 + 32*c^3*d + 3*I*c^2*d^2 + 146*c*d^3 - 292*I*d^4)*e^(8*I*f*x + 8*I*e) + (-29
*I*c^4 + 97*c^3*d + 67*I*c^2*d^2 + 211*c*d^3 - 210*I*d^4)*e^(6*I*f*x + 6*I*e) + (-27*I*c^4 + 90*c^3*d + 71*I*c
^2*d^2 + 90*c*d^3 + 98*I*d^4)*e^(4*I*f*x + 4*I*e) + (-11*I*c^4 + 29*c^3*d + 7*I*c^2*d^2 + 29*c*d^3 + 18*I*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)))/((96*a^3*c^6
+ 192*I*a^3*c^5*d + 96*a^3*c^4*d^2 + 384*I*a^3*c^3*d^3 - 96*a^3*c^2*d^4 + 192*I*a^3*c*d^5 - 96*a^3*d^6)*f*e^(8
*I*f*x + 8*I*e) + (96*a^3*c^6 + 384*I*a^3*c^5*d - 480*a^3*c^4*d^2 - 480*a^3*c^2*d^4 - 384*I*a^3*c*d^5 + 96*a^3
*d^6)*f*e^(6*I*f*x + 6*I*e))
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+I*a*tan(f*x+e))**3/(c+d*tan(f*x+e))**(3/2),x)
[Out]
Exception raised: AttributeError
________________________________________________________________________________________
Giac [B] time = 1.84949, size = 1073, normalized size = 2.92 \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))^3/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
[Out]
-1/2*d^4*(8*(2*I*c^3 - 12*c^2*d - 33*I*c*d^2 + 58*d^3)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*
sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 +
d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/((16*a^3*c^4*d^4*f + 64*I*a^3*c^3*d^5*f - 96*a^3*c^2*d^6*f - 64*I*a^3*c*
d^7*f + 16*a^3*d^8*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) - 4/((-I*a^3*c^5*f + 3*a
^3*c^4*d*f + 2*I*a^3*c^3*d^2*f + 2*a^3*c^2*d^3*f + 3*I*a^3*c*d^4*f - a^3*d^5*f)*sqrt(d*tan(f*x + e) + c)) - I*
arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d
^2)) - I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/((a^3*c*d^4*f - I
*a^3*d^5*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - 4*(6*(d*tan(f*x + e) + c)^(5/2)
*c^2 - 12*(d*tan(f*x + e) + c)^(3/2)*c^3 + 6*sqrt(d*tan(f*x + e) + c)*c^4 + 33*I*(d*tan(f*x + e) + c)^(5/2)*c*
d - 84*I*(d*tan(f*x + e) + c)^(3/2)*c^2*d + 51*I*sqrt(d*tan(f*x + e) + c)*c^3*d - 84*(d*tan(f*x + e) + c)^(5/2
)*d^2 + 268*(d*tan(f*x + e) + c)^(3/2)*c*d^2 - 204*sqrt(d*tan(f*x + e) + c)*c^2*d^2 + 196*I*(d*tan(f*x + e) +
c)^(3/2)*d^3 - 279*I*sqrt(d*tan(f*x + e) + c)*c*d^3 + 120*sqrt(d*tan(f*x + e) + c)*d^4)/((-96*I*a^3*c^4*d^3*f
+ 384*a^3*c^3*d^4*f + 576*I*a^3*c^2*d^5*f - 384*a^3*c*d^6*f - 96*I*a^3*d^7*f)*(-I*d*tan(f*x + e) - d)^3))