3.116 \(\int \frac{\cot ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\)

Optimal. Leaf size=180 \[ \frac{11 \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}+\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d} \]

[Out]

(11*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*S
qrt[a])]/(Sqrt[2]*Sqrt[a]*d) + Cot[c + d*x]^2/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((7*I)/4)*Cot[c + d*x]*Sqrt[a
+ I*a*Tan[c + d*x]])/(a*d) - (3*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*a*d)

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Rubi [A]  time = 0.568649, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3559, 3598, 3600, 3480, 206, 3599, 63, 208} \[ \frac{11 \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}+\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*S
qrt[a])]/(Sqrt[2]*Sqrt[a]*d) + Cot[c + d*x]^2/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((7*I)/4)*Cot[c + d*x]*Sqrt[a
+ I*a*Tan[c + d*x]])/(a*d) - (3*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*a*d)

Rule 3559

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(a*(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*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3598

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

Rule 3600

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

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 206

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

Rule 3599

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\cot ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{\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 (3 a-\frac{5}{2} i a \tan (c+d x)\right ) \, dx}{a^2}\\ &=\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{3 \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{7 i a^2}{2}-\frac{9}{2} a^2 \tan (c+d x)\right ) \, dx}{2 a^3}\\ &=\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}-\frac{3 \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{11 a^3}{4}+\frac{7}{4} i a^3 \tan (c+d x)\right ) \, dx}{2 a^4}\\ &=\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}-\frac{11 \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 \int \sqrt{a+i a \tan (c+d x)} \, dx}{2 a}\\ &=\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}-\frac{\operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{d}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}+\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}+\frac{(11 i) \operatorname{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 \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}+\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}\\ \end{align*}

Mathematica [A]  time = 2.42502, size = 170, normalized size = 0.94 \[ \frac{-8 e^{i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )+22 \sqrt{2} e^{i (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right )+\sqrt{1+e^{2 i (c+d x)}} \csc ^2(c+d x) (i \sin (2 (c+d x))+5 \cos (2 (c+d x))-9)}{8 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] + 22*Sqrt[2]*E^(I*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sq
rt[1 + E^((2*I)*(c + d*x))]] + Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c + d*x]^2*(-9 + 5*Cos[2*(c + d*x)] + I*Sin[2
*(c + d*x)]))/(8*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])

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Maple [B]  time = 0.438, size = 1382, normalized size = 7.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/d/a*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(11*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(((-2*co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))-16*I*cos(d*x+c)^3*sin(d*x+c)+11*I*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))*cos
(d*x+c)-4*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+
c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3-11*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*c
os(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c)-4*I*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2
)*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c)+4*I*2^(1/2)*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x
+c)+1))^(1/2))*cos(d*x+c)^2*sin(d*x+c)-4*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*co
s(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-11*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3-11*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(
1/2)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-11*
I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d
*x+c))*cos(d*x+c)^3-11*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x
+c)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^2+4*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(
I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)-11*(-2*cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2+16*cos(d*x+c)^4+28*I*sin(d*x+c)*c
os(d*x+c)+11*I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^
2*sin(d*x+c)+4*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x+c)-I-sin(d*x+c))/sin
(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+11*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(1/2))*cos(d*x+c)+11*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x
+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+11*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2))-24*cos(d*x+c)^2)/(-1+cos(d*x+c)^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.06581, size = 1790, normalized size = 9.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(3*e^(6*I*d*x + 6*I*c) - 6*e^(4*I*d*x + 4*I*c) - 7*e^(2*I*d
*x + 2*I*c) + 2)*e^(I*d*x + I*c) + 4*sqrt(2)*(a*d*e^(6*I*d*x + 6*I*c) - 2*a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I
*d*x + 2*I*c))*sqrt(1/(a*d^2))*log(1/4*(sqrt(2)*a*d*sqrt(1/(a*d^2))*e^(2*I*d*x + 2*I*c) + sqrt(2)*sqrt(a/(e^(2
*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 4*sqrt(2)*(a*d*e^(6*I*d*x
+ 6*I*c) - 2*a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a*d^2))*log(-1/4*(sqrt(2)*a*d*sqrt(1/
(a*d^2))*e^(2*I*d*x + 2*I*c) - sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x +
I*c))*e^(-I*d*x - I*c)) - 11*(a*d*e^(6*I*d*x + 6*I*c) - 2*a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))*s
qrt(1/(a*d^2))*log(-88/507*(2*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I
*c) + (3*a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt(1/(a*d^2)))/(e^(2*I*d*x + 2*I*c) - 1)) + 11*(a*d*e^(6*I*d*x + 6*I
*c) - 2*a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a*d^2))*log(-88/507*(2*sqrt(2)*sqrt(a/(e^(2
*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)*e^(I*d*x + I*c) - (3*a*d*e^(2*I*d*x + 2*I*c) + a*d)*sqrt(1/(a*
d^2)))/(e^(2*I*d*x + 2*I*c) - 1)))/(a*d*e^(6*I*d*x + 6*I*c) - 2*a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I
*c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{3}{\left (c + d x \right )}}{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(c + d*x)**3/sqrt(a*(I*tan(c + d*x) + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (d x + c\right )^{3}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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