\(\int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx\) [733]

Optimal result

Integrand size = 26, antiderivative size = 86 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {8 (-1)^{3/4} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^3}{3 d \sqrt {\cot (c+d x)}}-\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]

[Out]

8*(-1)^(3/4)*a^3*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-2/3*(I*a^3+a^3*cot(d*x+c))/d/cot(d*x+c)^(3/2)-16/3*a^3
/d/cot(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3754, 3634, 3672, 3614, 214} \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {8 (-1)^{3/4} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {16 a^3}{3 d \sqrt {\cot (c+d x)}} \]

[In]

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

[Out]

(8*(-1)^(3/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (16*a^3)/(3*d*Sqrt[Cot[c + d*x]]) - (2*(I*a^3 +
a^3*Cot[c + d*x]))/(3*d*Cot[c + d*x]^(3/2))

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 3614

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2*(c^2/f), S
ubst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]

Rule 3634

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*((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 - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(
m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && Lt
Q[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3672

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

Rule 3754

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x))^3}{\cot ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int \frac {(i a+a \cot (c+d x)) \left (-4 i a^2-2 a^2 \cot (c+d x)\right )}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {16 a^3}{3 d \sqrt {\cot (c+d x)}}-\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int \frac {-6 i a^3-6 a^3 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {16 a^3}{3 d \sqrt {\cot (c+d x)}}-\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (48 a^6\right ) \text {Subst}\left (\int \frac {1}{6 i a^3-6 a^3 x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {8 (-1)^{3/4} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^3}{3 d \sqrt {\cot (c+d x)}}-\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=-\frac {2 i a^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-12 (-1)^{3/4} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)} (-9 i+\tan (c+d x))\right )}{3 d} \]

[In]

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

[Out]

(((-2*I)/3)*a^3*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-12*(-1)^(3/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] +
Sqrt[Tan[c + d*x]]*(-9*I + Tan[c + d*x])))/d

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (71 ) = 142\).

Time = 1.43 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {2 i}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {6}{\sqrt {\cot \left (d x +c \right )}}-i \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )-\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )\right )}{d}\)	\(201\)
default	\(\frac {a^{3} \left (-\frac {2 i}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {6}{\sqrt {\cot \left (d x +c \right )}}-i \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )-\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )\right )}{d}\)	\(201\)

[In]

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

[Out]

1/d*a^3*(-2/3*I/cot(d*x+c)^(3/2)-6/cot(d*x+c)^(1/2)-I*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(1+c
ot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)
))-2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(
1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(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. 341 vs. \(2 (70) = 140\).

Time = 0.26 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.97 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {3 \, \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 3 \, \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 16 \, {\left (-5 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

1/12*(3*sqrt(-64*I*a^6/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(1/4*(8*I*a^3*e^(2*I*d*x
+ 2*I*c) + sqrt(-64*I*a^6/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x +
2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/a^3) - 3*sqrt(-64*I*a^6/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*
c) + d)*log(1/4*(8*I*a^3*e^(2*I*d*x + 2*I*c) + sqrt(-64*I*a^6/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt((I*e^
(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/a^3) - 16*(-5*I*a^3*e^(4*I*d*x + 4*I*c
) + I*a^3*e^(2*I*d*x + 2*I*c) + 4*I*a^3)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(d*e^(4*
I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

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

[In]

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

[Out]

-I*a**3*(Integral(I*sqrt(cot(c + d*x)), x) + Integral(-3*tan(c + d*x)*sqrt(cot(c + d*x)), x) + Integral(tan(c
+ d*x)**3*sqrt(cot(c + d*x)), x) + Integral(-3*I*tan(c + d*x)**2*sqrt(cot(c + d*x)), x))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).

Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=-\frac {3 \, {\left (\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - 2 \, {\left (-i \, a^{3} - \frac {9 \, a^{3}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{3 \, d} \]

[In]

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

[Out]

-1/3*(3*((2*I + 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + (2*I + 2)*sqrt(2)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + (I - 1)*sqrt(2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c)
+ 1) - (I - 1)*sqrt(2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))*a^3 - 2*(-I*a^3 - 9*a^3/tan(d*x
+ c))*tan(d*x + c)^(3/2))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\int \sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]

[In]

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

[Out]

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