\(\int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [89]

Optimal result

Integrand size = 36, antiderivative size = 261 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {3 a^{5/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \]

[Out]

-3/64*a^(5/2)*(121*A-120*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d+4*a^(5/2)*(A-I*B)*arctanh(1/2*(a+I*a
*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d+1/64*a^2*(149*I*A+152*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d+1
/96*a^2*(107*A-104*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d-1/24*a^2*(11*I*A+8*B)*cot(d*x+c)^3*(a+I*a*tan(
d*x+c))^(1/2)/d-1/4*a*A*cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)/d

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {3 a^{5/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}+\frac {a^2 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \]

[In]

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

[Out]

(-3*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(64*d) + (4*Sqrt[2]*a^(5/2)*(A -
I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*((149*I)*A + 152*B)*Cot[c + d*x]*Sqrt[a +
I*a*Tan[c + d*x]])/(64*d) + (a^2*(107*A - (104*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) - (a^2
*((11*I)*A + 8*B)*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x
])^(3/2))/(4*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 ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {1}{4} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} \left (\frac {1}{2} a (11 i A+8 B)-\frac {1}{2} a (5 A-8 i B) \tan (c+d x)\right ) \, dx \\ & = -\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {1}{12} \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{4} a^2 (107 A-104 i B)-\frac {1}{4} a^2 (85 i A+88 B) \tan (c+d x)\right ) \, dx \\ & = \frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (149 i A+152 B)+\frac {3}{8} a^3 (107 A-104 i B) \tan (c+d x)\right ) \, dx}{24 a} \\ & = \frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {9}{16} a^4 (121 A-120 i B)+\frac {3}{16} a^4 (149 i A+152 B) \tan (c+d x)\right ) \, dx}{24 a^2} \\ & = \frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {1}{128} (3 a (121 A-120 i B)) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx+\left (4 a^2 (i A+B)\right ) \int \sqrt {a+i a \tan (c+d x)} \, dx \\ & = \frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {\left (8 a^3 (A-i 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 (3 a^3 (121 A-120 i B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{128 d} \\ & = \frac {4 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}-\frac {\left (3 a^2 (121 i A+120 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 )}{64 d} \\ & = -\frac {3 a^{5/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.77 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.68 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {-9 a^{5/2} (121 A-120 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+768 \sqrt {2} a^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+a^2 \cot (c+d x) \left (447 i A+456 B+(214 A-208 i B) \cot (c+d x)+(-136 i A-64 B) \cot ^2(c+d x)-48 A \cot ^3(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}}{192 d} \]

[In]

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

[Out]

(-9*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 768*Sqrt[2]*a^(5/2)*(A - I*B)*Ar
cTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] + a^2*Cot[c + d*x]*((447*I)*A + 456*B + (214*A - (208*I)*B
)*Cot[c + d*x] + ((-136*I)*A - 64*B)*Cot[c + d*x]^2 - 48*A*Cot[c + d*x]^3)*Sqrt[a + I*a*Tan[c + d*x]])/(192*d)

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {2 a^{5} \left (-\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {5}{2}}}-\frac {\frac {\left (-\frac {19 i B}{16}+\frac {149 A}{128}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}+\left (\frac {145}{48} i a B -\frac {1127}{384} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-\frac {127}{48} i B \,a^{2}+\frac {1049}{384} A \,a^{2}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {13}{16} i B \,a^{3}-\frac {107}{128} A \,a^{3}\right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{4} \tan \left (d x +c \right )^{4}}+\frac {3 \left (-120 i B +121 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{2}}\right )}{d}\)	\(206\)
default	\(\frac {2 a^{5} \left (-\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {5}{2}}}-\frac {\frac {\left (-\frac {19 i B}{16}+\frac {149 A}{128}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}+\left (\frac {145}{48} i a B -\frac {1127}{384} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-\frac {127}{48} i B \,a^{2}+\frac {1049}{384} A \,a^{2}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {13}{16} i B \,a^{3}-\frac {107}{128} A \,a^{3}\right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{4} \tan \left (d x +c \right )^{4}}+\frac {3 \left (-120 i B +121 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{2}}\right )}{d}\)	\(206\)

[In]

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

[Out]

2/d*a^5*(-1/2*(4*I*B-4*A)/a^(5/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))-1/a^2*(((-19/1
6*I*B+149/128*A)*(a+I*a*tan(d*x+c))^(7/2)+(145/48*I*a*B-1127/384*a*A)*(a+I*a*tan(d*x+c))^(5/2)+(-127/48*I*a^2*
B+1049/384*A*a^2)*(a+I*a*tan(d*x+c))^(3/2)+(13/16*I*B*a^3-107/128*A*a^3)*(a+I*a*tan(d*x+c))^(1/2))/a^4/tan(d*x
+c)^4+3/128*(121*A-120*I*B)/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. 944 vs. \(2 (206) = 412\).

Time = 0.28 (sec) , antiderivative size = 944, normalized size of antiderivative = 3.62 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(1536*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d
*e^(4*I*d*x + 4*I*c) - 4*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)*(I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A -
B)*a^2)) - 1536*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c)
+ 6*d*e^(4*I*d*x + 4*I*c) - 4*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)*(-I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((
-I*A - B)*a^2)) + 9*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*
x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(16*(3*(-121*I*A - 120*B)*a^3*e^(2*I*d*
x + 2*I*c) + (-121*I*A - 120*B)*a^3 + 2*sqrt(2)*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(I*d*e^(3*
I*d*x + 3*I*c) + I*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)/((-121*I*A - 120
*B)*a)) - 9*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*
c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(16*(3*(-121*I*A - 120*B)*a^3*e^(2*I*d*x + 2*I*
c) + (-121*I*A - 120*B)*a^3 + 2*sqrt(2)*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(-I*d*e^(3*I*d*x +
3*I*c) - I*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)/((-121*I*A - 120*B)*a))
- 4*sqrt(2)*(13*(65*A - 56*I*B)*a^2*e^(9*I*d*x + 9*I*c) - 2*(215*A - 392*I*B)*a^2*e^(7*I*d*x + 7*I*c) - 4*(35
*A - 104*I*B)*a^2*e^(5*I*d*x + 5*I*c) + 2*(407*A - 392*I*B)*a^2*e^(3*I*d*x + 3*I*c) - 3*(107*A - 104*I*B)*a^2*
e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(
4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F(-1)]

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

[In]

integrate(cot(d*x+c)**5*(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 = 0.31 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.12 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {a^{4} {\left (\frac {768 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {9 \, {\left (121 \, A - 120 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} {\left (149 \, A - 152 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (1127 \, A - 1160 i \, B\right )} a + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (1049 \, A - 1016 i \, B\right )} a^{2} - 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (107 \, A - 104 i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} + 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + a^{5}}\right )}}{384 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/384*a^4*(768*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*tan(d*x + c) + a)))/a^(3/2) - 9*(121*A - 120*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan
(d*x + c) + a) + sqrt(a)))/a^(3/2) + 2*(3*(I*a*tan(d*x + c) + a)^(7/2)*(149*A - 152*I*B) - (I*a*tan(d*x + c) +
a)^(5/2)*(1127*A - 1160*I*B)*a + (I*a*tan(d*x + c) + a)^(3/2)*(1049*A - 1016*I*B)*a^2 - 3*sqrt(I*a*tan(d*x +
c) + a)*(107*A - 104*I*B)*a^3)/((I*a*tan(d*x + c) + a)^4*a - 4*(I*a*tan(d*x + c) + a)^3*a^2 + 6*(I*a*tan(d*x +
c) + a)^2*a^3 - 4*(I*a*tan(d*x + c) + a)*a^4 + a^5))/d

Giac [F]

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

[In]

integrate(cot(d*x+c)^5*(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)^5, x)

Mupad [B] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 3094, normalized size of antiderivative = 11.85 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^5*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

(((107*A*a^6 - B*a^6*104i)*(a + a*tan(c + d*x)*1i)^(1/2))/(64*d) - ((149*A*a^3 - B*a^3*152i)*(a + a*tan(c + d*
x)*1i)^(7/2))/(64*d) - ((1049*A*a^5 - B*a^5*1016i)*(a + a*tan(c + d*x)*1i)^(3/2))/(192*d) + ((1127*A*a^4 - B*a
^4*1160i)*(a + a*tan(c + d*x)*1i)^(5/2))/(192*d))/((a + a*tan(c + d*x)*1i)^4 - 4*a^3*(a + a*tan(c + d*x)*1i) -
4*a*(a + a*tan(c + d*x)*1i)^3 + 6*a^2*(a + a*tan(c + d*x)*1i)^2 + a^4) - 2*atanh((384*d^4*(a + a*tan(c + d*x)
*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B
^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (40
73*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(6
4*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2
))/((431443*A^3*a^14*d)/256 - B^3*a^14*d*3542i + (21783*A*B^2*a^14*d)/4 + (A^2*B*a^14*d*6993i)/32 + 214*A*a^3*
d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*
1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) - B*a^3*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^
4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*
d^4))^(1/2)*208i) + (697*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4
*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d
^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(
1/2))/(4*((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i + 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^
22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4)
)^(1/2) - B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) +
(A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^
11*d*6993i)/32)) + (368*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*
a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^
4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1
/2))/((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i + 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/
(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1
/2) - B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B
^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d
*6993i)/32) + (A*B*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64
*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)
/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*196i)
/((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i + 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*
d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)
- B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a
^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d*699
3i)/32))*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3
*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073
*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2) - 2*atanh((697*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1
/2)*((262841*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2
*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*
a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2))/(4*((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i - 214*A*d^
3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*11
27i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) + B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22
)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^
(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d*6993i)/32)) - (384*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((262
841*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2
048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*a^5)/(512
*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*
A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2))/((431443*A^3*
a^14*d)/256 - B^3*a^14*d*3542i + (21783*A*B^2*a^14*d)/4 + (A^2*B*a^14*d*6993i)/32 - 214*A*a^3*d^3*((485809*A^4
*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4)
+ (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) + B*a^3*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4)
+ (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i
) + (368*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((262841*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*
d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*
34153i)/(8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2))/((431443*A
^3*a^11*d)/256 - B^3*a^11*d*3542i - 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (112
29*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) + B*d^3*((4
85809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/
(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d*6993i)/32) + (
A*B*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((262841*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*d^4) + (5
29*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(
8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*196i)/((431443*A^3*a
^11*d)/256 - B^3*a^11*d*3542i - 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A
^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) + B*d^3*((48580
9*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128
*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d*6993i)/32))*((2628
41*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(20
48*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*a^5)/(512*
d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)