\(\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [81]

Optimal result

Integrand size = 36, antiderivative size = 213 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {a^{3/2} (23 i A+22 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {2 \sqrt {2} a^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]

[Out]

1/8*a^(3/2)*(23*I*A+22*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d-2*a^(3/2)*(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*(9*A-10*I*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d-1/12*a*(7*I
*A+6*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))^(1/2)/d

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 213, 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} \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {a^{3/2} (22 B+23 i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {2 \sqrt {2} a^{3/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a (6 B+7 i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]

[In]

Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

(a^(3/2)*((23*I)*A + 22*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (2*Sqrt[2]*a^(3/2)*(I*A + B)*A
rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a*(9*A - (10*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c
+ d*x]])/(8*d) - (a*((7*I)*A + 6*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*d) - (a*A*Cot[c + d*x]^3*Sq
rt[a + I*a*Tan[c + d*x]])/(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) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {1}{2} a (7 i A+6 B)-\frac {1}{2} a (5 A-6 i B) \tan (c+d x)\right ) \, dx \\ & = -\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^2 (9 A-10 i B)-\frac {3}{4} a^2 (7 i A+6 B) \tan (c+d x)\right ) \, dx}{6 a} \\ & = \frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (23 i A+22 B)+\frac {3}{8} a^3 (9 A-10 i B) \tan (c+d x)\right ) \, dx}{6 a^2} \\ & = \frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {1}{16} (-23 i A-22 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx+(2 a (A-i B)) \int \sqrt {a+i a \tan (c+d x)} \, dx \\ & = \frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {\left (4 a^2 (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^2 (23 i A+22 B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{16 d} \\ & = -\frac {2 \sqrt {2} a^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {(a (23 A-22 i B)) \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^{3/2} (23 i A+22 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {2 \sqrt {2} a^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a (9 A-10 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a (7 i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.41 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.74 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {-3 a^{3/2} (23 i A+22 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+48 \sqrt {2} a^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+a \cot (c+d x) \left (-27 A+30 i B+2 (7 i A+6 B) \cot (c+d x)+8 A \cot ^2(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}}{24 d} \]

[In]

Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

-1/24*(-3*a^(3/2)*((23*I)*A + 22*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 48*Sqrt[2]*a^(3/2)*(I*A + B)
*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] + a*Cot[c + d*x]*(-27*A + (30*I)*B + 2*((7*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 = 0.30 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {2 i a^{4} \left (-\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {5}{2}}}+\frac {-\frac {i \left (\left (\frac {5 i B}{8}-\frac {9 A}{16}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-i a B +\frac {5}{6} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {3}{8} i B \,a^{2}-\frac {7}{16} A \,a^{2}\right ) \sqrt {a +i a \tan \left (d x +c \right )}\right )}{a^{3} \tan \left (d x +c \right )^{3}}+\frac {\left (-22 i B +23 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a^{2}}\right )}{d}\)	\(179\)
default	\(\frac {2 i a^{4} \left (-\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {5}{2}}}+\frac {-\frac {i \left (\left (\frac {5 i B}{8}-\frac {9 A}{16}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}+\left (-i a B +\frac {5}{6} a A \right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {3}{8} i B \,a^{2}-\frac {7}{16} A \,a^{2}\right ) \sqrt {a +i a \tan \left (d x +c \right )}\right )}{a^{3} \tan \left (d x +c \right )^{3}}+\frac {\left (-22 i B +23 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a^{2}}\right )}{d}\)	\(179\)

[In]

int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2*I/d*a^4*(-1/2*(2*A-2*I*B)/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*(-I*((
5/8*I*B-9/16*A)*(a+I*a*tan(d*x+c))^(5/2)+(-I*a*B+5/6*a*A)*(a+I*a*tan(d*x+c))^(3/2)+(3/8*I*B*a^2-7/16*A*a^2)*(a
+I*a*tan(d*x+c))^(1/2))/a^3/tan(d*x+c)^3+1/16*(-22*I*B+23*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. 856 vs. \(2 (166) = 332\).

Time = 0.27 (sec) , antiderivative size = 856, normalized size of antiderivative = 4.02 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/96*(96*sqrt(2)*sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/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^2*e^(I*d*x + I*c) + sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/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)) - 96*sqrt(2)*sqrt(-(A^2
- 2*I*A*B - B^2)*a^3/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^2*e^(I*d*x + I*c) - sqrt(-(A^2 - 2*I*A*B - B^2)*a^3/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)) - 3*sqrt(-(529*A^2 - 1012*I*A*B - 484*B^2)*a^3/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*(-23*I*A - 22*B)*
a^2*e^(2*I*d*x + 2*I*c) + (-23*I*A - 22*B)*a^2 + 2*sqrt(2)*sqrt(-(529*A^2 - 1012*I*A*B - 484*B^2)*a^3/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)/(23*I*A + 22*
B)) + 3*sqrt(-(529*A^2 - 1012*I*A*B - 484*B^2)*a^3/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*(-23*I*A - 22*B)*a^2*e^(2*I*d*x + 2*I*c) + (-23*I*A - 22*B)*a^2 - 2*sqrt(
2)*sqrt(-(529*A^2 - 1012*I*A*B - 484*B^2)*a^3/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)/(23*I*A + 22*B)) - 4*sqrt(2)*(7*(-7*I*A - 6*B)*a*e^(7*I*d*x + 7*I*c)
- (11*I*A - 18*B)*a*e^(5*I*d*x + 5*I*c) - (-17*I*A - 42*B)*a*e^(3*I*d*x + 3*I*c) + 3*(-7*I*A - 6*B)*a*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]

\[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)

[Out]

Integral((I*a*(tan(c + d*x) - I))**(3/2)*(A + B*tan(c + d*x))*cot(c + d*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.19 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {i \, a^{3} {\left (\frac {48 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {3 \, {\left (23 \, A - 22 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (9 \, A - 10 i \, B\right )} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (5 \, A - 6 i \, B\right )} a + 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (7 \, A - 6 i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{2} + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{3} - a^{4}}\right )}}{48 \, d} \]

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/48*I*a^3*(48*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) - 3*(23*A - 22*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)^(5/2)*(9*A - 10*I*B) - 8*(I*a*tan(d*x + c) + a)^
(3/2)*(5*A - 6*I*B)*a + 3*sqrt(I*a*tan(d*x + c) + a)*(7*A - 6*I*B)*a^2)/((I*a*tan(d*x + c) + a)^3*a - 3*(I*a*t
an(d*x + c) + a)^2*a^2 + 3*(I*a*tan(d*x + c) + a)*a^3 - a^4))/d

Giac [F]

\[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(3/2)*cot(d*x + c)^4, x)

Mupad [B] (verification not implemented)

Time = 9.30 (sec) , antiderivative size = 3084, normalized size of antiderivative = 14.48 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

2*atanh((6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a
^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51
i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2)*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4)
+ (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/((A^3*a^11*d*663i)/
32 + (133*B^3*a^11*d)/4 + (A*B^2*a^11*d*387i)/8 + (89*A^2*B*a^11*d)/16 + (A*a^2*d^3*((289*A^4*a^18)/(64*d^4) +
(49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/
2)*7i)/4 + (3*B*a^2*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3
*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/2) + (17*A^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((249
*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B
^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d
^2))^(1/2))/(4*((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 + (A*d^3*
((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^
3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) + (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*
B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*a))) + (7*B^2*a^6*d^2*(a +
a*tan(c + d*x)*1i)^(1/2)*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a^18)/(64*d^4) + (49*
B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(6
4*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2))/((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (8
9*A^2*B*a^8*d)/16 + (A*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*
B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) + (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^
4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*
a)) + (A*B*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A
^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^1
8*51i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2)*3i)/((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4
+ (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 + (A*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*
A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) + (3*B*d^3*((289*
A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^
18*51i)/(8*d^4))^(1/2))/(2*a)))*((249*B^2*a^3)/(128*d^2) - (1041*A^2*a^3)/(512*d^2) - ((289*A^4*a^18)/(64*d^4)
+ (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(
1/2)/(64*a^6) + (A*B*a^3*509i)/(128*d^2))^(1/2) + 2*atanh((17*A^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((289
*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a
^18*51i)/(8*d^4))^(1/2)/(64*a^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) + (A*B*a^3*509i)/(128*d^
2))^(1/2))/(4*((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 - (A*d^3*(
(289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3
*B*a^18*51i)/(8*d^4))^(1/2)*7i)/(4*a) - (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B
^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*a))) - (6*d^4*(a + a*tan(c +
d*x)*1i)^(1/2)*(((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*2
1i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(64*a^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) +
(A*B*a^3*509i)/(128*d^2))^(1/2)*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4)
+ (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/((A^3*a^11*d*663i)/32 + (133*B^3*a^11*d)/4 + (A*
B^2*a^11*d*387i)/8 + (89*A^2*B*a^11*d)/16 - (A*a^2*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101
*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/4 - (3*B*a^2*d^3*((289
*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a
^18*51i)/(8*d^4))^(1/2))/2) + (7*B^2*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((289*A^4*a^18)/(64*d^4) + (49*B^4
*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)/(64*a
^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) + (A*B*a^3*509i)/(128*d^2))^(1/2))/((A^3*a^8*d*663i)/
32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 - (A*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^
4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2)*7i)/
(4*a) - (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*2
1i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*a)) + (A*B*a^6*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((289*A^4*
a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*5
1i)/(8*d^4))^(1/2)/(64*a^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) + (A*B*a^3*509i)/(128*d^2))^(
1/2)*3i)/((A^3*a^8*d*663i)/32 + (133*B^3*a^8*d)/4 + (A*B^2*a^8*d*387i)/8 + (89*A^2*B*a^8*d)/16 - (A*d^3*((289*
A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^
18*51i)/(8*d^4))^(1/2)*7i)/(4*a) - (3*B*d^3*((289*A^4*a^18)/(64*d^4) + (49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^
18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2))/(2*a)))*(((289*A^4*a^18)/(64*d^4) +
(49*B^4*a^18)/(4*d^4) + (101*A^2*B^2*a^18)/(8*d^4) + (A*B^3*a^18*21i)/(2*d^4) + (A^3*B*a^18*51i)/(8*d^4))^(1/2
)/(64*a^6) - (1041*A^2*a^3)/(512*d^2) + (249*B^2*a^3)/(128*d^2) + (A*B*a^3*509i)/(128*d^2))^(1/2) - (((9*A*a^2
- B*a^2*10i)*(a + a*tan(c + d*x)*1i)^(5/2)*1i)/(8*d) - ((5*A*a^3 - B*a^3*6i)*(a + a*tan(c + d*x)*1i)^(3/2)*1i
)/(3*d) + ((35*A*a^4 - B*a^4*30i)*(a + a*tan(c + d*x)*1i)^(1/2)*1i)/(40*d))/(3*a*(a + a*tan(c + d*x)*1i)^2 - 3
*a^2*(a + a*tan(c + d*x)*1i) - (a + a*tan(c + d*x)*1i)^3 + a^3)