3.12.30 \(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \, dx\) [1130]
Optimal. Leaf size=368 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 (c-i d)^{3/2} f}+\frac {\left (2 i c^3-12 c^2 d-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{9/2} f}+\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}} \]
[Out]
-1/8*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^3/(c-I*d)^(3/2)/f+1/16*(2*I*c^3-12*c^2*d-33*I*c*d^2+58*
d^3)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^3/(c+I*d)^(9/2)/f+1/16*d*(2*c^3+9*I*c^2*d-17*c*d^2+60*I*d
^3)/a^3/(c-I*d)/(c+I*d)^4/f/(c+d*tan(f*x+e))^(1/2)-1/6/(I*c-d)/f/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3+1
/24*(3*I*c-10*d)/a/(c+I*d)^2/f/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2+1/48*(6*c^2+27*I*c*d-56*d^2)/(I*c-d
)^3/f/(c+d*tan(f*x+e))^(1/2)/(a^3+I*a^3*tan(f*x+e))
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Rubi [A]
time = 0.86, antiderivative size = 368, normalized size of antiderivative = 1.00, 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 = {3640, 3677,
3610, 3620, 3618, 65, 214} \begin {gather*} \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 {d \left (2 c^3+9 i c^2 d-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 {\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 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)}} \end {gather*}
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]
((-1/8*I)*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 -
(33*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 65
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 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 3610
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 3618
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620
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 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]
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 {\text {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 ) \text {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 {\text {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 ) \text {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*}
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Mathematica [A]
time = 7.87, size = 468, normalized size = 1.27 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (-\frac {2 \left (\sqrt {-c+i d} \left (-2 i c^4+10 c^3 d+21 i c^2 d^2-25 c d^3+58 i d^4\right ) \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )+2 i (-c-i d)^{9/2} \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right ) (\cos (3 e)+i \sin (3 e))}{(-c-i d)^{9/2} (-c+i d)^{3/2}}+\frac {\cos (e+f x) (i \cos (3 f x)+\sin (3 f x)) \left (\left (27 c^4+90 i c^3 d-71 c^2 d^2+90 i c d^3-98 d^4\right ) \cos (e+f x)+\left (13 c^4+36 i c^3 d-3 c^2 d^2+150 i c d^3+290 d^4\right ) \cos (3 (e+f x))+i \left (\left (9 c^4+40 i c^3 d-57 c^2 d^2+40 i c d^3-66 d^4\right ) \sin (e+f x)+\left (9 c^4+28 i c^3 d-3 c^2 d^2+142 i c d^3+294 d^4\right ) \sin (3 (e+f x))\right )\right ) \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))}\right )}{32 f (a+i a \tan (e+f x))^3} \end {gather*}
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)
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Maple [A]
time = 0.38, size = 596, normalized size = 1.62
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {2 d^{4} \left (-\frac {i}{\left (i c +d \right ) \left (i c -d \right ) \left (i d +c \right )^{3} \sqrt {c +d \tan \left (f x +e \right )}}+\frac {\left (-i c^{4}+6 i c^{2} d^{2}-i d^{4}+4 c^{3} d -4 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{4} d^{4}}-\frac {i \left (\frac {\frac {d \left (2 i c^{6}-50 i c^{4} d^{2}-24 i c^{2} d^{4}+28 i d^{6}-15 c^{5} d +52 c^{3} d^{3}+67 c \,d^{5}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{7}-109 i c^{5} d^{2}+53 i c^{3} d^{4}+165 i c \,d^{6}-27 c^{6} d +177 c^{4} d^{3}+155 c^{2} d^{5}-49 d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{8}-102 i c^{6} d^{2}+190 i c^{4} d^{4}+254 i c^{2} d^{6}-40 i d^{8}-21 c^{7} d +225 c^{5} d^{3}+73 c^{3} d^{5}-173 c \,d^{7}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (16 i c^{6} d -120 i c^{4} d^{3}-78 i c^{2} d^{5}+58 i d^{7}+2 c^{7}-57 c^{5} d^{2}+90 c^{3} d^{4}+149 c \,d^{6}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 \left (i d -c \right ) \left (i d +c \right )^{4} d^{4}}\right )}{f \,a^{3}}\) |
\(596\) |
| default |
\(\frac {2 d^{4} \left (-\frac {i}{\left (i c +d \right ) \left (i c -d \right ) \left (i d +c \right )^{3} \sqrt {c +d \tan \left (f x +e \right )}}+\frac {\left (-i c^{4}+6 i c^{2} d^{2}-i d^{4}+4 c^{3} d -4 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{4} d^{4}}-\frac {i \left (\frac {\frac {d \left (2 i c^{6}-50 i c^{4} d^{2}-24 i c^{2} d^{4}+28 i d^{6}-15 c^{5} d +52 c^{3} d^{3}+67 c \,d^{5}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{7}-109 i c^{5} d^{2}+53 i c^{3} d^{4}+165 i c \,d^{6}-27 c^{6} d +177 c^{4} d^{3}+155 c^{2} d^{5}-49 d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{8}-102 i c^{6} d^{2}+190 i c^{4} d^{4}+254 i c^{2} d^{6}-40 i d^{8}-21 c^{7} d +225 c^{5} d^{3}+73 c^{3} d^{5}-173 c \,d^{7}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (16 i c^{6} d -120 i c^{4} d^{3}-78 i c^{2} d^{5}+58 i d^{7}+2 c^{7}-57 c^{5} d^{2}+90 c^{3} d^{4}+149 c \,d^{6}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 \left (i d -c \right ) \left (i d +c \right )^{4} d^{4}}\right )}{f \,a^{3}}\) |
\(596\) |
| | |
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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,method=_RETURNVERBOSE)
[Out]
2/f/a^3*d^4*(-I/(I*c+d)/(I*c-d)/(c+I*d)^3/(c+d*tan(f*x+e))^(1/2)+1/16*(-I*c^4+6*I*c^2*d^2-I*d^4+4*c^3*d-4*c*d^
3)/(I*d-c)^(3/2)/(c+I*d)^4/d^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))-1/16*I/(I*d-c)/(c+I*d)^4/d^4*((1/2
*d*(2*I*c^6-50*I*c^4*d^2-24*I*c^2*d^4+28*I*d^6-15*c^5*d+52*c^3*d^3+67*c*d^5)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+
d*tan(f*x+e))^(5/2)-2/3*d*(-27*c^6*d+177*c^4*d^3+155*c^2*d^5-49*d^7+3*I*c^7-109*I*c^5*d^2+53*I*c^3*d^4+165*I*c
*d^6)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)+1/2*d*(2*I*c^8-102*I*c^6*d^2+190*I*c^4*d^4+254*I*c^
2*d^6-40*I*d^8-21*c^7*d+225*c^5*d^3+73*c^3*d^5-173*c*d^7)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)
)/(-d*tan(f*x+e)+I*d)^3-1/2*(-57*c^5*d^2+90*c^3*d^4+149*c*d^6+16*I*c^6*d-120*I*c^4*d^3-78*I*c^2*d^5+58*I*d^7+2
*c^7)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))))
________________________________________________________________________________________
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/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2674 vs. \(2 (307) = 614\).
time = 4.60, size = 2674, normalized size = 7.27 \begin {gather*} \text {Too large to display} \end {gather*}
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]
-1/192*(48*((a^3*c^6 + 2*I*a^3*c^5*d + a^3*c^4*d^2 + 4*I*a^3*c^3*d^3 - a^3*c^2*d^4 + 2*I*a^3*c*d^5 - a^3*d^6)*
f*e^(8*I*f*x + 8*I*e) + (a^3*c^6 + 4*I*a^3*c^5*d - 5*a^3*c^4*d^2 - 5*a^3*c^2*d^4 - 4*I*a^3*c*d^5 + a^3*d^6)*f*
e^(6*I*f*x + 6*I*e))*sqrt(-1/64*I/((I*a^6*c^3 + 3*a^6*c^2*d - 3*I*a^6*c*d^2 - a^6*d^3)*f^2))*log(-2*(8*((I*a^3
*c^2 + 2*a^3*c*d - I*a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (I*a^3*c^2 + 2*a^3*c*d - 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(-1/64*I/((I*a^6*c^3 + 3*a^6*c^2*d - 3*I*a^6*c*d^
2 - a^6*d^3)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) - 48*((a^3*c^6 + 2*I*a^3*c^5*d +
a^3*c^4*d^2 + 4*I*a^3*c^3*d^3 - a^3*c^2*d^4 + 2*I*a^3*c*d^5 - a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (a^3*c^6 + 4*I
*a^3*c^5*d - 5*a^3*c^4*d^2 - 5*a^3*c^2*d^4 - 4*I*a^3*c*d^5 + a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt(-1/64*I/((I*
a^6*c^3 + 3*a^6*c^2*d - 3*I*a^6*c*d^2 - a^6*d^3)*f^2))*log(-2*(8*((-I*a^3*c^2 - 2*a^3*c*d + I*a^3*d^2)*f*e^(2*
I*f*x + 2*I*e) + (-I*a^3*c^2 - 2*a^3*c*d + 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(-1/64*I/((I*a^6*c^3 + 3*a^6*c^2*d - 3*I*a^6*c*d^2 - a^6*d^3)*f^2)) - (c - I*d)*e^(2*
I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) - 3*((a^3*c^6 + 2*I*a^3*c^5*d + a^3*c^4*d^2 + 4*I*a^3*c^3*d^3 - a^3*
c^2*d^4 + 2*I*a^3*c*d^5 - a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (a^3*c^6 + 4*I*a^3*c^5*d - 5*a^3*c^4*d^2 - 5*a^3*c^
2*d^4 - 4*I*a^3*c*d^5 + 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)/((I*a^6*c^9 - 9*a^6*c^8*d - 36*I*a^6*c^7*d^2 + 84*a^6*c^6*d^3 +
126*I*a^6*c^5*d^4 - 126*a^6*c^4*d^5 - 84*I*a^6*c^3*d^6 + 36*a^6*c^2*d^7 + 9*I*a^6*c*d^8 - a^6*d^9)*f^2))*log(
-1/16*(2*c^4 + 14*I*c^3*d - 45*c^2*d^2 - 91*I*c*d^3 + 58*d^4 - ((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*e^(2*I*f*x + 2*I*e) + (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)*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
)/((I*a^6*c^9 - 9*a^6*c^8*d - 36*I*a^6*c^7*d^2 + 84*a^6*c^6*d^3 + 126*I*a^6*c^5*d^4 - 126*a^6*c^4*d^5 - 84*I*a
^6*c^3*d^6 + 36*a^6*c^2*d^7 + 9*I*a^6*c*d^8 - 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)/((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)) + 3*((a^3*c^6 + 2*I*a^3*c^5*d + a^3*c^4*d^2 + 4*I*a^3*c^3*d^3 - a^3*c^2*d^4 + 2*I*a^3
*c*d^5 - a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (a^3*c^6 + 4*I*a^3*c^5*d - 5*a^3*c^4*d^2 - 5*a^3*c^2*d^4 - 4*I*a^3*c
*d^5 + 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)/((I*a^6*c^9 - 9*a^6*c^8*d - 36*I*a^6*c^7*d^2 + 84*a^6*c^6*d^3 + 126*I*a^6*c^5*d^
4 - 126*a^6*c^4*d^5 - 84*I*a^6*c^3*d^6 + 36*a^6*c^2*d^7 + 9*I*a^6*c*d^8 - a^6*d^9)*f^2))*log(-1/16*(2*c^4 + 14
*I*c^3*d - 45*c^2*d^2 - 91*I*c*d^3 + 58*d^4 - ((-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*e^(2*I*f*x + 2*I*e) + (-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)*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)/((I*a^6*c^9 -
9*a^6*c^8*d - 36*I*a^6*c^7*d^2 + 84*a^6*c^6*d^3 + 126*I*a^6*c^5*d^4 - 126*a^6*c^4*d^5 - 84*I*a^6*c^3*d^6 + 36
*a^6*c^2*d^7 + 9*I*a^6*c*d^8 - 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)/((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*(-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)))/((a^3*c^6 + 2*I*a^3*c^5*d + a^3*c^4*d^2 + 4*I*a^3*c^3*d^3 - a^3*c^2*d^4 + 2*I*a^3*c*d^5 -
a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (a^3*c^6 + 4*I*a^3*c^5*d - 5*a^3*c^4*d^2 - 5*a^3*c^2*d^4 - 4*I*a^3*c*d^5 + a^
3*d^6)*f*e^(6*I*f*x + 6*I*e))
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {i \int \frac {1}{c \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - 3 i c \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 3 c \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + i c \sqrt {c + d \tan {\left (e + f x \right )}} + d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )} - 3 i d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - 3 d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} + i d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}\, dx}{a^{3}} \end {gather*}
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]
I*Integral(1/(c*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3 - 3*I*c*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2 - 3*
c*sqrt(c + d*tan(e + f*x))*tan(e + f*x) + I*c*sqrt(c + d*tan(e + f*x)) + d*sqrt(c + d*tan(e + f*x))*tan(e + f*
x)**4 - 3*I*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3 - 3*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2 + I*d*sq
rt(c + d*tan(e + f*x))*tan(e + f*x)), x)/a**3
________________________________________________________________________________________
Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 785 vs. \(2 (307) = 614\).
time = 1.40, size = 785, normalized size = 2.13 \begin {gather*} \frac {2 \, d^{4}}{{\left (-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\right )} \sqrt {d \tan \left (f x + e\right ) + c}} - \frac {{\left (2 i \, c^{3} - 12 \, c^{2} d - 33 i \, c d^{2} + 58 \, d^{3}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{8 \, {\left (a^{3} c^{4} f + 4 i \, a^{3} c^{3} d f - 6 \, a^{3} c^{2} d^{2} f - 4 i \, a^{3} c d^{3} f + a^{3} d^{4} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {i \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{4 \, {\left (a^{3} c f - i \, a^{3} d f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (6 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} d - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} d + 6 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} d + 33 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} - 84 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 51 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} - 84 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{3} + 268 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} - 204 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} + 196 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} - 279 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4} + 120 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{5}\right )}}{-96 \, {\left (i \, a^{3} c^{4} f - 4 \, a^{3} c^{3} d f - 6 i \, a^{3} c^{2} d^{2} f + 4 \, a^{3} c d^{3} f + i \, a^{3} d^{4} f\right )} {\left (-i \, d \tan \left (f x + e\right ) - d\right )}^{3}} \end {gather*}
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]
2*d^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)) - 1/8*(2*I*c^3 - 12*c^2*d - 33*I*c*d^2 + 58*d^3)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c -
sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) + I*sqrt(-2*c + 2*sqrt(c^2 + d^2))
*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((a^3*c^4*f + 4*I*a^3*c^3*d*f - 6*a^3*c^2*d^2*f - 4*I*a^
3*c*d^3*f + a^3*d^4*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/4*I*arctan(2*(sqrt(
d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt(-
2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((a^3*c*f - I*a^3*d*f)*sqrt(-2*c
+ 2*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + 2*(6*(d*tan(f*x + e) + c)^(5/2)*c^2*d - 12*(d*tan(f*
x + e) + c)^(3/2)*c^3*d + 6*sqrt(d*tan(f*x + e) + c)*c^4*d + 33*I*(d*tan(f*x + e) + c)^(5/2)*c*d^2 - 84*I*(d*t
an(f*x + e) + c)^(3/2)*c^2*d^2 + 51*I*sqrt(d*tan(f*x + e) + c)*c^3*d^2 - 84*(d*tan(f*x + e) + c)^(5/2)*d^3 + 2
68*(d*tan(f*x + e) + c)^(3/2)*c*d^3 - 204*sqrt(d*tan(f*x + e) + c)*c^2*d^3 + 196*I*(d*tan(f*x + e) + c)^(3/2)*
d^4 - 279*I*sqrt(d*tan(f*x + e) + c)*c*d^4 + 120*sqrt(d*tan(f*x + e) + c)*d^5)/((-96*I*a^3*c^4*f + 384*a^3*c^3
*d*f + 576*I*a^3*c^2*d^2*f - 384*a^3*c*d^3*f - 96*I*a^3*d^4*f)*(-I*d*tan(f*x + e) - d)^3)
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Mupad [B]
time = 13.88, size = 2500, normalized size = 6.79 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + a*tan(e + f*x)*1i)^3*(c + d*tan(e + f*x))^(3/2)),x)
[Out]
log(6960*a^3*d^15*f - ((-(c*d^12*6300i + 3360*d^13 - 945*c^2*d^11 + c^3*d^10*1365i + 315*c^4*d^9 + c^5*d^8*693
i + 672*c^6*d^7 - c^7*d^6*288i - 72*c^8*d^5 + c^9*d^4*8i - a^6*c^10*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16
)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a
^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^12*c^8
*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089
*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 +
a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*
a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (8914
5*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d
^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a^6
*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^
16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c
^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 +
56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*4i + a^6*d^10*f^2*(4*(((29973*c^2*d
^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64
- (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d
^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*
c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)
*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a
^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4)
+ (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 3
4*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*
c^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*
d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6
- 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 +
70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*4i + 24*a^6*c*
d^9*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^
8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4
*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*
c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7
)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56
*a^12*c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))
*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^
11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a
^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6
*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339
*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^
2 + 56*a^6*c^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2)
)^2)^(1/2) + 24*a^6*c^9*d*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^
10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*
d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*
c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c
^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*
a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 +
8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^
6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 3...
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