\(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx\) [96]

Optimal result

Integrand size = 36, antiderivative size = 219 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {(11 A+4 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(7 i A-8 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d} \]

[Out]

1/4*(11*A+4*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)-1/2*(A-I*B)*arctanh(1/2*(a+I*a*tan(d*x+c)
)^(1/2)*2^(1/2)/a^(1/2))/d*2^(1/2)/a^(1/2)+(A+I*B)*cot(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(1/2)+1/4*(7*I*A-8*B)*cot
(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a/d-1/2*(3*A+2*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/a/d

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 219, 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 = {3677, 3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {(11 A+4 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d}+\frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(-8 B+7 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d} \]

[In]

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

[Out]

((11*A + (4*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*a
*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((A + I*B)*Cot[c + d*x]^2)/(d*Sqrt[a + I*a*Tan[c + d*
x]]) + (((7*I)*A - 8*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a*d) - ((3*A + (2*I)*B)*Cot[c + d*x]^2*Sqr
t[a + I*a*Tan[c + d*x]])/(2*a*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 3677

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

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+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (a (3 A+2 i B)-\frac {5}{2} a (i A-B) \tan (c+d x)\right ) \, dx}{a^2} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{2} a^2 (7 i A-8 B)-\frac {3}{2} a^2 (3 A+2 i B) \tan (c+d x)\right ) \, dx}{2 a^3} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(7 i A-8 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{4} a^3 (11 A+4 i B)+\frac {1}{4} a^3 (7 i A-8 B) \tan (c+d x)\right ) \, dx}{2 a^4} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(7 i A-8 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d}-\frac {(11 A+4 i B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{8 a^2}-\frac {(i A+B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{2 a} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(7 i A-8 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d}-\frac {(A-i B) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {(11 A+4 i B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{8 d} \\ & = -\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(7 i A-8 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d}+\frac {(11 i A-4 B) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{4 a d} \\ & = \frac {(11 A+4 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}+\frac {(A+i B) \cot ^2(c+d x)}{d \sqrt {a+i a \tan (c+d x)}}+\frac {(7 i A-8 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a d}-\frac {(3 A+2 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.42 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\frac {(11 A+4 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 \sqrt {2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}+\frac {-7 A-8 i B+i (A+4 i B) \cot (c+d x)-2 A \cot ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}}}{4 d} \]

[In]

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

[Out]

(((11*A + (4*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] - (2*Sqrt[2]*(A - I*B)*ArcTanh[Sqrt[a
+ I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[a] + (-7*A - (8*I)*B + I*(A + (4*I)*B)*Cot[c + d*x] - 2*A*Cot[c +
d*x]^2)/Sqrt[a + I*a*Tan[c + d*x]])/(4*d)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {i B +A}{2 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {7}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {3 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {5}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (4 i B +11 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{3}}\right )}{d}\)	\(171\)
default	\(\frac {2 a^{3} \left (-\frac {i B +A}{2 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {7}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {3 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {5}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (4 i B +11 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{3}}\right )}{d}\)	\(171\)

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d*a^3*(-1/2/a^3*(A+I*B)/(a+I*a*tan(d*x+c))^(1/2)-1/4*(A-I*B)/a^(7/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^
(1/2)*2^(1/2)/a^(1/2))+1/a^3*(-((-1/2*I*B-3/8*A)*(a+I*a*tan(d*x+c))^(3/2)+(1/2*I*a*B+5/8*a*A)*(a+I*a*tan(d*x+c
))^(1/2))/a^2/tan(d*x+c)^2+1/8*(11*A+4*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. 835 vs. \(2 (172) = 344\).

Time = 0.28 (sec) , antiderivative size = 835, normalized size of antiderivative = 3.81 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/16*(4*sqrt(2)*(a*d*e^(5*I*d*x + 5*I*c) - 2*a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt((A^2 - 2*I*A*
B - B^2)/(a*d^2))*log(-4*((-I*A - B)*a*e^(I*d*x + I*c) + (I*a*d*e^(2*I*d*x + 2*I*c) + I*a*d)*sqrt(a/(e^(2*I*d*
x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a*d^2)))*e^(-I*d*x - I*c)/(I*A + B)) - 4*sqrt(2)*(a*d*e^(5*I*d*x
+ 5*I*c) - 2*a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt((A^2 - 2*I*A*B - B^2)/(a*d^2))*log(-4*((-I*A
- B)*a*e^(I*d*x + I*c) + (-I*a*d*e^(2*I*d*x + 2*I*c) - I*a*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*
I*A*B - B^2)/(a*d^2)))*e^(-I*d*x - I*c)/(I*A + B)) + (a*d*e^(5*I*d*x + 5*I*c) - 2*a*d*e^(3*I*d*x + 3*I*c) + a*
d*e^(I*d*x + I*c))*sqrt((121*A^2 + 88*I*A*B - 16*B^2)/(a*d^2))*log(-16*(3*(11*I*A - 4*B)*a^2*e^(2*I*d*x + 2*I*
c) + (11*I*A - 4*B)*a^2 + 2*sqrt(2)*(I*a^2*d*e^(3*I*d*x + 3*I*c) + I*a^2*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x
+ 2*I*c) + 1))*sqrt((121*A^2 + 88*I*A*B - 16*B^2)/(a*d^2)))*e^(-2*I*d*x - 2*I*c)/(-11*I*A + 4*B)) - (a*d*e^(5
*I*d*x + 5*I*c) - 2*a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt((121*A^2 + 88*I*A*B - 16*B^2)/(a*d^2))
*log(-16*(3*(11*I*A - 4*B)*a^2*e^(2*I*d*x + 2*I*c) + (11*I*A - 4*B)*a^2 + 2*sqrt(2)*(-I*a^2*d*e^(3*I*d*x + 3*I
*c) - I*a^2*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((121*A^2 + 88*I*A*B - 16*B^2)/(a*d^2)))*
e^(-2*I*d*x - 2*I*c)/(-11*I*A + 4*B)) - 4*sqrt(2)*(3*(A + 2*I*B)*e^(6*I*d*x + 6*I*c) - 2*(3*A + I*B)*e^(4*I*d*
x + 4*I*c) - (7*A + 6*I*B)*e^(2*I*d*x + 2*I*c) + 2*A + 2*I*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a*d*e^(5*I*d
*x + 5*I*c) - 2*a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))

Sympy [F]

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

[In]

integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (7 \, A + 8 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (13 \, A + 12 i \, B\right )} a + 4 \, {\left (A + i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4}} - \frac {2 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}} + \frac {{\left (11 \, A + 4 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )}}{8 \, d} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-1/8*a^2*(2*((I*a*tan(d*x + c) + a)^2*(7*A + 8*I*B) - (I*a*tan(d*x + c) + a)*(13*A + 12*I*B)*a + 4*(A + I*B)*a
^2)/((I*a*tan(d*x + c) + a)^(5/2)*a^2 - 2*(I*a*tan(d*x + c) + a)^(3/2)*a^3 + sqrt(I*a*tan(d*x + c) + a)*a^4) -
2*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^(5/2) + (11*A + 4*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + s
qrt(a)))/a^(5/2))/d

Giac [F]

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

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.58 (sec) , antiderivative size = 3037, normalized size of antiderivative = 13.87 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2*atanh((12*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((129*A^2)/(128*a*d^2) - ((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^1
0)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) - (3*B^2)/(1
6*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2)*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4
- (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2))/(B^3*a^5*d*8i - (1469*A^3*a^5*d)/2 + 9*A*d^3*((12769*
A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d
^4)^(1/2) + B*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/
d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*6i + 156*A*B^2*a^5*d - A^2*B*a^5*d*789i) - (226*A^2*a^2*d^2*(a + a*tan(c +
d*x)*1i)^(1/2)*((129*A^2)/(128*a*d^2) - ((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d
^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) - (3*B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2)
)^(1/2))/(B^3*a^2*d*8i - (1469*A^3*a^2*d)/2 + 156*A*B^2*a^2*d - A^2*B*a^2*d*789i + (9*A*d^3*((12769*A^4*a^10)/
(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2))
/a^3 + (B*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4
+ (A^3*B*a^10*5876i)/d^4)^(1/2)*6i)/a^3) + (16*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((129*A^2)/(128*a*d^2
) - ((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a
^10*5876i)/d^4)^(1/2)/(64*a^6) - (3*B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2))/(B^3*a^2*d*8i - (1469*A^3*a^
2*d)/2 + 156*A*B^2*a^2*d - A^2*B*a^2*d*789i + (9*A*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A
^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2))/a^3 + (B*d^3*((12769*A^4*a^10)/(4*d^
4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*6i)/a
^3) - (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((129*A^2)/(128*a*d^2) - ((12769*A^4*a^10)/(4*d^4) + (16*B^4*
a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) - (3*B^2)
/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2)*208i)/(B^3*a^2*d*8i - (1469*A^3*a^2*d)/2 + 156*A*B^2*a^2*d - A^2*B*a^
2*d*789i + (9*A*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i
)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2))/a^3 + (B*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*
B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*6i)/a^3))*((129*A^2)/(128*a*d^2) - ((127
69*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i
)/d^4)^(1/2)/(64*a^6) - (3*B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2) + 2*atanh((12*d^4*(a + a*tan(c + d*x)*
1i)^(1/2)*(((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (
A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) + (129*A^2)/(128*a*d^2) - (3*B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2
)*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^1
0*5876i)/d^4)^(1/2))/((1469*A^3*a^5*d)/2 - B^3*a^5*d*8i + 9*A*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^
4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2) + B*d^3*((12769*A^4*a^10)/
(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*
6i - 156*A*B^2*a^5*d + A^2*B*a^5*d*789i) + (226*A^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((12769*A^4*a^10)/(
4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)/(
64*a^6) + (129*A^2)/(128*a*d^2) - (3*B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2))/((1469*A^3*a^2*d)/2 - B^3*a
^2*d*8i - 156*A*B^2*a^2*d + A^2*B*a^2*d*789i + (9*A*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*
A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2))/a^3 + (B*d^3*((12769*A^4*a^10)/(4*d
^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*6i)/
a^3) - (16*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^
2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) + (129*A^2)/(128*a*d^2) - (3*
B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2))/((1469*A^3*a^2*d)/2 - B^3*a^2*d*8i - 156*A*B^2*a^2*d + A^2*B*a^2
*d*789i + (9*A*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)
/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2))/a^3 + (B*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B
^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*6i)/a^3) + (A*B*a^2*d^2*(a + a*tan(c + d*
x)*1i)^(1/2)*(((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4
+ (A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) + (129*A^2)/(128*a*d^2) - (3*B^2)/(16*a*d^2) + (A*B*9i)/(16*a*d^2))^(
1/2)*208i)/((1469*A^3*a^2*d)/2 - B^3*a^2*d*8i - 156*A*B^2*a^2*d + A^2*B*a^2*d*789i + (9*A*d^3*((12769*A^4*a^10
)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2
))/a^3 + (B*d^3*((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^10)/d^4 - (A*B^3*a^10*416i)/d^
4 + (A^3*B*a^10*5876i)/d^4)^(1/2)*6i)/a^3))*(((12769*A^4*a^10)/(4*d^4) + (16*B^4*a^10)/d^4 - (3156*A^2*B^2*a^1
0)/d^4 - (A*B^3*a^10*416i)/d^4 + (A^3*B*a^10*5876i)/d^4)^(1/2)/(64*a^6) + (129*A^2)/(128*a*d^2) - (3*B^2)/(16*
a*d^2) + (A*B*9i)/(16*a*d^2))^(1/2) - ((A*a^2 + B*a^2*1i)/d + ((7*A + B*8i)*(a + a*tan(c + d*x)*1i)^2)/(4*d) -
((13*A*a + B*a*12i)*(a + a*tan(c + d*x)*1i))/(4*d))/((a + a*tan(c + d*x)*1i)^(5/2) - 2*a*(a + a*tan(c + d*x)*
1i)^(3/2) + a^2*(a + a*tan(c + d*x)*1i)^(1/2))