\(\int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx\) [504]

Optimal result

Integrand size = 34, antiderivative size = 53 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {2 \sqrt [4]{-1} a (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a A \sqrt {\cot (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3662, 3673, 3614, 214} \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {2 a A \sqrt {\cot (c+d x)}}{d}-\frac {2 \sqrt [4]{-1} a (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]

[In]

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

[Out]

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

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 3662

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

Rule 3673

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

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

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.21 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {2 i a \sqrt {\cot (c+d x)} \left (-i A+\sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}\right )}{d} \]

[In]

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

[Out]

((-2*I)*a*Sqrt[Cot[c + d*x]]*((-I)*A + (-1)^(1/4)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[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. 507 vs. \(2 (44 ) = 88\).

Time = 0.36 (sec) , antiderivative size = 508, normalized size of antiderivative = 9.58

method	result	size
derivativedivides	\(\frac {a \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (2 i A \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 i A \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+i A \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 i B \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 i B \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+i B \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-2 A \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-2 A \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-A \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 B \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 B \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+B \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-8 A \right )}{4 d}\)	\(508\)
default	\(\frac {a \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (2 i A \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 i A \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+i A \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 i B \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 i B \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+i B \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-2 A \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-2 A \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-A \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 B \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+2 B \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+B \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-8 A \right )}{4 d}\)	\(508\)

[In]

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

[Out]

1/4*a/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(2*I*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(1/2)+2*I
*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(1/2)+I*A*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)
)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(1/2)+2*I*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2
^(1/2)*tan(d*x+c)^(1/2)+2*I*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(1/2)+I*B*ln(-(2^(1/2)*ta
n(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(1/2)-2*A*arctan(1+2^
(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(1/2)-2*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(1
/2)-A*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^
(1/2)+2*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(1/2)+2*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*
2^(1/2)*tan(d*x+c)^(1/2)+B*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)
)*2^(1/2)*tan(d*x+c)^(1/2)-8*A)

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. 316 vs. \(2 (43) = 86\).

Time = 0.24 (sec) , antiderivative size = 316, normalized size of antiderivative = 5.96 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {4 \, A a \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} d \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) + \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} d \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right )}{2 \, d} \]

[In]

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

[Out]

-1/2*(4*A*a*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)) - sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^2/d
^2)*d*log(-2*((A - I*B)*a*e^(2*I*d*x + 2*I*c) - (I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(-(-I*A^2 - 2*A*B + I*B^2)
*a^2/d^2)*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)/((I*A + B)*a)) + s
qrt(-(-I*A^2 - 2*A*B + I*B^2)*a^2/d^2)*d*log(-2*((A - I*B)*a*e^(2*I*d*x + 2*I*c) - (-I*d*e^(2*I*d*x + 2*I*c) +
I*d)*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^2/d^2)*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)/((I*A + B)*a)))/d

Sympy [F]

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

[In]

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

[Out]

I*a*(Integral(-I*A*cot(c + d*x)**(3/2), x) + Integral(A*tan(c + d*x)*cot(c + d*x)**(3/2), x) + Integral(B*tan(
c + d*x)**2*cot(c + d*x)**(3/2), x) + Integral(-I*B*tan(c + d*x)*cot(c + d*x)**(3/2), 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. 155 vs. \(2 (43) = 86\).

Time = 0.37 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.92 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {{\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a + \frac {8 \, A a}{\sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

[In]

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

[Out]

-1/4*((2*sqrt(2)*((I - 1)*A + (I + 1)*B)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*((I
- 1)*A + (I + 1)*B)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*(-(I + 1)*A + (I - 1)*B)*l
og(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*(-(I + 1)*A + (I - 1)*B)*log(-sqrt(2)/sqrt(tan(d
*x + c)) + 1/tan(d*x + c) + 1))*a + 8*A*a/sqrt(tan(d*x + c)))/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]

[In]

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

[Out]

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