\(\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [88]

Optimal result

Integrand size = 36, antiderivative size = 217 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {a^{5/2} (45 i A+46 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]

[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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3674, 3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {a^{5/2} (46 B+45 i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a^2 (2 B+3 i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]

[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 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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 3561

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 3674

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] && E
qQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3679

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 3680

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 3681

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*} \text {integral}& = -\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 ) \text {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 ) \text {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) \text {arctanh}\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 ) \text {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) \text {arctanh}\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) \text {arctanh}\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*}

Mathematica [A] (verified)

Time = 5.54 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.74 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {-3 a^{5/2} (45 i A+46 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+96 \sqrt {2} a^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+a^2 \cot (c+d x) \left (-57 A+54 i B+2 (13 i A+6 B) \cot (c+d x)+8 A \cot ^2(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}}{24 d} \]

[In]

Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

-1/24*(-3*a^(5/2)*((45*I)*A + 46*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 96*Sqrt[2]*a^(5/2)*(I*A + B)
*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] + a^2*Cot[c + d*x]*(-57*A + (54*I)*B + 2*((13*I)*A + 6*
B)*Cot[c + d*x] + 8*A*Cot[c + d*x]^2)*Sqrt[a + I*a*Tan[c + d*x]])/d

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {2 i a^{4} \left (-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {3}{2}}}+\frac {-\frac {i \left (\left (\frac {9 i B}{8}-\frac {19 A}{16}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-2 i a B +\frac {11}{6} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {7}{8} i B \,a^{2}-\frac {13}{16} A \,a^{2}\right ) \sqrt {a +i a \tan \left (d x +c \right )}\right )}{a^{3} \tan \left (d x +c \right )^{3}}+\frac {\left (-46 i B +45 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a}\right )}{d}\)	\(179\)
default	\(\frac {2 i a^{4} \left (-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {3}{2}}}+\frac {-\frac {i \left (\left (\frac {9 i B}{8}-\frac {19 A}{16}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-2 i a B +\frac {11}{6} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {7}{8} i B \,a^{2}-\frac {13}{16} A \,a^{2}\right ) \sqrt {a +i a \tan \left (d x +c \right )}\right )}{a^{3} \tan \left (d x +c \right )^{3}}+\frac {\left (-46 i B +45 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a}\right )}{d}\)	\(179\)

[In]

int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2*I/d*a^4*(-1/2*(-4*I*B+4*A)/a^(3/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+1/a*(-I*((9
/8*I*B-19/16*A)*(a+I*a*tan(d*x+c))^(5/2)+(-2*I*a*B+11/6*a*A)*(a+I*a*tan(d*x+c))^(3/2)+(7/8*I*B*a^2-13/16*A*a^2
)*(a+I*a*tan(d*x+c))^(1/2))/a^3/tan(d*x+c)^3+1/16*(-46*I*B+45*A)/a^(1/2)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1
/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (170) = 340\).

Time = 0.28 (sec) , antiderivative size = 868, normalized size of antiderivative = 4.00 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[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*(192*sqrt(2)*sqrt(-(A^2 - 2*I*A*B - 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(4*((-I*A - B)*a^3*e^(I*d*x + I*c) + sqrt(-(A^2 - 2*I*A*B - 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)/((-I*A - B)*a^2)) - 192*sqrt(2)*sqrt(-
(A^2 - 2*I*A*B - 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(4*((-I*A - B)*a^3*e^(I*d*x + I*c) - sqrt(-(A^2 - 2*I*A*B - 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)/((-I*A - B)*a^2)) - 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(-16*(3*(-45*I*A
- 46*B)*a^3*e^(2*I*d*x + 2*I*c) + (-45*I*A - 46*B)*a^3 + 2*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)/((4
5*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(-16*(3*(-45*I*A - 46*B)*a^3*e^(2*I*d*x + 2*I*c) + (-45*I*A - 46*
B)*a^3 - 2*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)) + 4*sqrt(2)*((91*I*A + 66*B)*a
^2*e^(7*I*d*x + 7*I*c) - 7*(I*A + 6*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) - 3*
(-13*I*A - 14*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)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {i \, {\left (\frac {96 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} - \frac {3 \, {\left (45 \, A - 46 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (19 \, A - 18 i \, B\right )} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (11 \, A - 12 i \, B\right )} a + 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (13 \, A - 14 i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} - a^{3}}\right )} a^{3}}{48 \, d} \]

[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

Giac [F]

\[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 3048, normalized size of antiderivative = 14.05 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[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)