3.88 \(\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\)
Optimal. Leaf size=217 \[ \frac {a^{5/2} (46 B+45 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a^2 (2 B+3 i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]
[Out]
1/8*a^(5/2)*(45*I*A+46*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d-4*a^(5/2)*(I*A+B)*arctanh(1/2*(a+I*a*tan
(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d+1/8*a^2*(19*A-18*I*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d-1/4*a^2*
(3*I*A+2*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d-1/3*a*A*cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(3/2)/d
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Rubi [A] time = 0.80, antiderivative size = 217, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used =
{3593, 3598, 3600, 3480, 206, 3599, 63, 208} \[ \frac {a^{5/2} (46 B+45 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a^2 (2 B+3 i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
(a^(5/2)*((45*I)*A + 46*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (4*Sqrt[2]*a^(5/2)*(I*A + B)*A
rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*(19*A - (18*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan
[c + d*x]])/(8*d) - (a^2*((3*I)*A + 2*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^
3*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)
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 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])
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 3480
Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]
Rule 3593
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^2*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^
(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m -
1) + b*d*(n + 1)))*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] && GtQ[m, 1] && LtQ[n, -1]
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]
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 3600
Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(A*b + a*B)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[(B*c
- A*d)/(b*c + a*d), Int[((a + b*Tan[e + f*x])^m*(a - b*Tan[e + f*x]))/(c + d*Tan[e + f*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] && NeQ[A*b + a*B, 0]
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \left (\frac {3}{2} a (3 i A+2 B)-\frac {3}{2} a (A-2 i B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {1}{6} \int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^2 (19 A-18 i B)-\frac {3}{4} a^2 (13 i A+14 B) \tan (c+d x)\right ) \, dx\\ &=\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (45 i A+46 B)+\frac {3}{8} a^3 (19 A-18 i B) \tan (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\left (4 a^2 (A-i B)\right ) \int \sqrt {a+i a \tan (c+d x)} \, dx-\frac {1}{16} (a (45 i A+46 B)) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {\left (8 a^3 (i A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {\left (a^3 (45 i A+46 B)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=-\frac {4 \sqrt {2} a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {\left (a^2 (45 A-46 i B)\right ) \operatorname {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{8 d}\\ &=\frac {a^{5/2} (45 i A+46 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [B] time = 9.16, size = 634, normalized size = 2.92 \[ \frac {\cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (\csc (c) \left (\frac {1}{12} \cos (2 c)-\frac {1}{12} i \sin (2 c)\right ) \csc ^2(c+d x) (-13 i A \sin (c)-4 A \cos (c)-6 B \sin (c))+\csc (c) \left (\frac {1}{24} \cos (2 c)-\frac {1}{24} i \sin (2 c)\right ) \csc (c+d x) (-65 A \sin (d x)+54 i B \sin (d x))+\csc (c) \left (\frac {1}{24} \cos (2 c)-\frac {1}{24} i \sin (2 c)\right ) (26 i A \sin (c)+65 A \cos (c)+12 B \sin (c)-54 i B \cos (c))+A \csc (c) \left (\frac {1}{3} \cos (2 c)-\frac {1}{3} i \sin (2 c)\right ) \sin (d x) \csc ^3(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))}-\frac {i e^{-2 i c} \sqrt {e^{i d x}} (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (\sqrt {2} (45 A-46 i B) \left (\log \left (\left (-1+e^{i (c+d x)}\right )^2\right )-\log \left (\left (1+e^{i (c+d x)}\right )^2\right )+\log \left (-2 e^{i (c+d x)} \left (1+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )+3 e^{2 i (c+d x)}+2 \sqrt {2} \sqrt {1+e^{2 i (c+d x)}}+3\right )-\log \left (2 e^{i (c+d x)} \left (1+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )+3 e^{2 i (c+d x)}+2 \sqrt {2} \sqrt {1+e^{2 i (c+d x)}}+3\right )\right )+256 (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{32 \sqrt {2} d \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
((-1/32*I)*Sqrt[E^(I*d*x)]*(256*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(45*A - (46*I)*B)*(Log[(-1 + E^(I
*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(
c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) +
2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))*(a
+ I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(Sqrt[2]*d*E^((2*I)*c)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c +
d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Si
n[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*(65*A*Cos[c] - (54*I)*B*Cos[c] + (26*I)*A*Sin[c] + 12*B*Sin[c])*(Cos[2*
c]/24 - (I/24)*Sin[2*c]) + Csc[c]*Csc[c + d*x]^2*(-4*A*Cos[c] - (13*I)*A*Sin[c] - 6*B*Sin[c])*(Cos[2*c]/12 - (
I/12)*Sin[2*c]) + A*Csc[c]*Csc[c + d*x]^3*(Cos[2*c]/3 - (I/3)*Sin[2*c])*Sin[d*x] + Csc[c]*Csc[c + d*x]*(Cos[2*
c]/24 - (I/24)*Sin[2*c])*(-65*A*Sin[d*x] + (54*I)*B*Sin[d*x]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x
]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))
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fricas [B] time = 0.73, size = 868, normalized size = 4.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/96*(3*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3
*d*e^(2*I*d*x + 2*I*c) - d)*log(((2160*I*A + 2208*B)*a^3*e^(2*I*d*x + 2*I*c) + (720*I*A + 736*B)*a^3 + 32*sqrt
(2)*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*a^5/d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2
*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((45*I*A + 46*B)*a)) - 3*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*
a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(((2160*I*A + 2208
*B)*a^3*e^(2*I*d*x + 2*I*c) + (720*I*A + 736*B)*a^3 - 32*sqrt(2)*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*a^5/
d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((45*
I*A + 46*B)*a)) - 24*sqrt(-(128*A^2 - 256*I*A*B - 128*B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x +
4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(((16*I*A + 16*B)*a^3*e^(I*d*x + I*c) + sqrt(2)*sqrt(-(128*A^2 - 256*
I*A*B - 128*B^2)*a^5/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((4*
I*A + 4*B)*a^2)) + 24*sqrt(-(128*A^2 - 256*I*A*B - 128*B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x +
4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(((16*I*A + 16*B)*a^3*e^(I*d*x + I*c) - sqrt(2)*sqrt(-(128*A^2 - 256
*I*A*B - 128*B^2)*a^5/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((4
*I*A + 4*B)*a^2)) + 4*sqrt(2)*((91*I*A + 66*B)*a^2*e^(7*I*d*x + 7*I*c) + (-7*I*A - 42*B)*a^2*e^(5*I*d*x + 5*I*
c) + (-59*I*A - 66*B)*a^2*e^(3*I*d*x + 3*I*c) + (39*I*A + 42*B)*a^2*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*
c) + 1)))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^4, x)
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maple [B] time = 3.46, size = 2506, normalized size = 11.55 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x)
[Out]
-1/48/d*(192*A*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*2^(1/2))*2^(1/2)+132*B*cos(d*x+c)^2+192*I*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arct
an(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)-182*I*A*cos(d*x+c)^4-130*I*A*cos(d*x+c)^3+166*I*A
*cos(d*x+c)^2+114*I*A*cos(d*x+c)+182*A*cos(d*x+c)^3*sin(d*x+c)-114*A*cos(d*x+c)*sin(d*x+c)-192*A*cos(d*x+c)*si
n(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)
-132*B*cos(d*x+c)^4+108*B*cos(d*x+c)+52*A*cos(d*x+c)^2*sin(d*x+c)-108*B*cos(d*x+c)^3-135*A*sin(d*x+c)*(-2*cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-138*B*sin(d*x+c)*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-132*I*B*co
s(d*x+c)^3*sin(d*x+c)-24*I*B*sin(d*x+c)*cos(d*x+c)^2+108*I*B*cos(d*x+c)*sin(d*x+c)+192*I*A*cos(d*x+c)^3*sin(d*
x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+
c)*2^(1/2))*2^(1/2)-192*I*B*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(-2*cos(d*
x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)+192*I*A*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/
2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)-135*A*cos(d*x+c)*si
n(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-192*A*sin(d*x+c)*
(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)-138*B*co
s(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)
+cos(d*x+c)-1)/sin(d*x+c))-192*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)-135*I*A*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(
1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+138*I*B*sin(d*x+c)*(-2*co
s(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-192*I*B*cos(d*x+c)^2*sin(d*x+c)*
(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)-192*I*A*
cos(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*si
n(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)+192*I*B*cos(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan
(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)-192*B*cos(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)+135*I*A*
cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*
x+c)+cos(d*x+c)-1)/sin(d*x+c))-138*I*B*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(
-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+135*I*A*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-
(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-138*I*B*cos(d*x+c)^2*sin(d*x+c)*(-
2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-192*I*A*sin(d*x+c)*(-2*cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^
(1/2)+192*B*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x
+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)+192*A*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)+192*B*cos(d*x+c)^2*sin(d*x+c)*(-2*cos
(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*
2^(1/2)-135*I*A*cos(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+138*I*B*cos(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2
)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+135*A*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+138*B*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c
)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+135*A*cos(d*x+c)^2*s
in(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+138*B*cos(d*x+c)
^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d
*x+c)-1)/sin(d*x+c)))*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/(-1+cos(d*x+c))/(I*sin(d*x+c)+cos(d*x+c)-
1)/(1+cos(d*x+c))^2*a^2
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maxima [A] time = 0.89, size = 249, normalized size = 1.15 \[ \frac {i \, {\left (\frac {96 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} - \frac {3 \, {\left (45 \, A - 46 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (19 \, A - 18 i \, B\right )} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (11 \, A - 12 i \, B\right )} a + 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (13 \, A - 14 i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} - a^{3}}\right )} a^{3}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/48*I*(96*sqrt(2)*(A - I*B)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*t
an(d*x + c) + a)))/sqrt(a) - 3*(45*A - 46*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x +
c) + a) + sqrt(a)))/sqrt(a) + 2*(3*(I*a*tan(d*x + c) + a)^(5/2)*(19*A - 18*I*B) - 8*(I*a*tan(d*x + c) + a)^(3/
2)*(11*A - 12*I*B)*a + 3*sqrt(I*a*tan(d*x + c) + a)*(13*A - 14*I*B)*a^2)/((I*a*tan(d*x + c) + a)^3 - 3*(I*a*ta
n(d*x + c) + a)^2*a + 3*(I*a*tan(d*x + c) + a)*a^2 - a^3))*a^3/d
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mupad [B] time = 8.47, size = 3048, normalized size = 14.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)
[Out]
2*atanh((23*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128*d^2) - (4073*A^2*a^5)/(512*d^2) - (
(529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A
^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2))/(4*((A^3*a^11*d*1771i)/32 + (663*B
^3*a^11*d)/4 - (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*
a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^
22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (
A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16)) - (6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128*d^
2) - (4073*A^2*a^5)/(512*d^2) - ((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4)
+ (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2)*((
529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^
3*B*a^22*253i)/(8*d^4))^(1/2))/((A^3*a^14*d*1771i)/32 + (663*B^3*a^14*d)/4 + (A*B^2*a^14*d*2167i)/8 - (797*A^2
*B*a^14*d)/16 - (A*a^3*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A
*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*a^3*d^3*((529*A^4*a^22)/(64*d^4) + (2
89*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/
2))/2) + (17*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128*d^2) - (4073*A^2*a^5)/(512*d^2) -
((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (
A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2))/((A^3*a^11*d*1771i)/32 + (663*B^3
*a^11*d)/4 - (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^
22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22
)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*
B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16) + (A*B*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128
*d^2) - (4073*A^2*a^5)/(512*d^2) - ((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d
^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2)
*11i)/((A^3*a^11*d*1771i)/32 + (663*B^3*a^11*d)/4 - (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) +
(149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*d^3*(
(529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A
^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16))*((1041*B^2*a^5)/(128*d^2)
- (4073*A^2*a^5)/(512*d^2) - ((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) +
(A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2) + 2*
atanh((6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a
^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) - (4073*A^2*a^5)/(512*d^2
) + (1041*B^2*a^5)/(128*d^2) + (A*B*a^5*2059i)/(128*d^2))^(1/2)*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d
^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/((A^3*a^14*d*
1771i)/32 + (663*B^3*a^14*d)/4 + (A*B^2*a^14*d*2167i)/8 - (797*A^2*B*a^14*d)/16 + (A*a^3*d^3*((529*A^4*a^22)/(
64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/
(8*d^4))^(1/2)*13i)/4 + (7*B*a^3*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8
*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2) + (23*A^2*a^8*d^2*(a + a*tan(c + d*x)
*1i)^(1/2)*(((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)
/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) - (4073*A^2*a^5)/(512*d^2) + (1041*B^2*a^5)/(128*d^2) + (
A*B*a^5*2059i)/(128*d^2))^(1/2))/(4*((A^3*a^11*d*1771i)/32 + (663*B^3*a^11*d)/4 + (A*d^3*((529*A^4*a^22)/(64*d
^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d
^4))^(1/2)*13i)/4 + (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) +
(A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/
16)) + (17*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149
*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) - (4073*A^2*a^5
)/(512*d^2) + (1041*B^2*a^5)/(128*d^2) + (A*B*a^5*2059i)/(128*d^2))^(1/2))/((A^3*a^11*d*1771i)/32 + (663*B^3*a
^11*d)/4 + (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22
*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 + (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/
(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*B^
2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16) + (A*B*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((529*A^4*a^22)/(64*d
^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d
^4))^(1/2)/(64*a^6) - (4073*A^2*a^5)/(512*d^2) + (1041*B^2*a^5)/(128*d^2) + (A*B*a^5*2059i)/(128*d^2))^(1/2)*1
1i)/((A^3*a^11*d*1771i)/32 + (663*B^3*a^11*d)/4 + (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (
149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 + (7*B*d^3*((5
29*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3
*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16))*(((529*A^4*a^22)/(64*d^4) +
(289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^
(1/2)/(64*a^6) - (4073*A^2*a^5)/(512*d^2) + (1041*B^2*a^5)/(128*d^2) + (A*B*a^5*2059i)/(128*d^2))^(1/2) - (((1
3*A*a^5 - B*a^5*14i)*(a + a*tan(c + d*x)*1i)^(1/2)*1i)/(8*d) - ((11*A*a^4 - B*a^4*12i)*(a + a*tan(c + d*x)*1i)
^(3/2)*1i)/(3*d) + ((19*A*a^3 - B*a^3*18i)*(a + a*tan(c + d*x)*1i)^(5/2)*1i)/(8*d))/(3*a*(a + a*tan(c + d*x)*1
i)^2 - 3*a^2*(a + a*tan(c + d*x)*1i) - (a + a*tan(c + d*x)*1i)^3 + a^3)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
[Out]
Timed out
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