\(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx\) [111]

Optimal result

Integrand size = 36, antiderivative size = 312 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {(43 A+20 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 (2 i A-B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d} \]

[Out]

1/4*(43*A+20*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-1/8*(A-I*B)*arctanh(1/2*(a+I*a*tan(d*x+c
))^(1/2)*2^(1/2)/a^(1/2))/a^(5/2)/d*2^(1/2)+1/60*(337*A+167*I*B)*cot(d*x+c)^2/a^2/d/(a+I*a*tan(d*x+c))^(1/2)+2
1/4*(2*I*A-B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^3/d-1/12*(85*A+41*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/
2)/a^3/d+1/5*(A+I*B)*cot(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(5/2)+1/30*(23*A+13*I*B)*cot(d*x+c)^2/a/d/(a+I*a*tan(d*
x+c))^(3/2)

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {(43 A+20 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d}+\frac {21 (-B+2 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}} \]

[In]

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]

[Out]

((43*A + (20*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(5/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*
a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + ((A + I*B)*Cot[c + d*x]^2)/(5*d*(a + I*a*Tan[c + d
*x])^(5/2)) + ((23*A + (13*I)*B)*Cot[c + d*x]^2)/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((337*A + (167*I)*B)*
Cot[c + d*x]^2)/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (21*((2*I)*A - B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*
x]])/(4*a^3*d) - ((85*A + (41*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*a^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 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)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {\int \frac {\cot ^3(c+d x) \left (a (7 A+2 i B)-\frac {9}{2} a (i A-B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^3(c+d x) \left (a^2 (44 A+19 i B)-\frac {7}{4} a^2 (23 i A-13 B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{15 a^4} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 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 (\frac {5}{2} a^3 (85 A+41 i B)-\frac {5}{8} a^3 (337 i A-167 B) \tan (c+d x)\right ) \, dx}{15 a^6} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {315}{2} a^4 (2 i A-B)-\frac {15}{4} a^4 (85 A+41 i B) \tan (c+d x)\right ) \, dx}{30 a^7} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 (2 i A-B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {15}{4} a^5 (43 A+20 i B)+\frac {315}{4} a^5 (2 i A-B) \tan (c+d x)\right ) \, dx}{30 a^8} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 (2 i A-B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d}-\frac {(43 A+20 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^4}-\frac {(i A+B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{8 a^3} \\ & = \frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 (2 i A-B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 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 )}{4 a^2 d}-\frac {(43 A+20 i B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{8 a^2 d} \\ & = -\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 (2 i A-B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d}+\frac {(43 i A-20 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^3 d} \\ & = \frac {(43 A+20 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {(23 A+13 i B) \cot ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac {(337 A+167 i B) \cot ^2(c+d x)}{60 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {21 (2 i A-B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^3 d}-\frac {(85 A+41 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.70 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {12 a^{17/2} (A+i B) \cot ^2(c+d x)-\frac {1}{2} a^8 (i+\cot (c+d x)) \tan ^2(c+d x) \left (-\frac {1}{2} \sqrt {a} \csc ^3(c+d x) (-((961 A+461 i B) \cos (c+d x))+(793 A+413 i B) \cos (3 (c+d x))+18 (-57 i A+27 B+(83 i A-43 B) \cos (2 (c+d x))) \sin (c+d x))-30 (43 A+20 i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right ) (i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}+15 \sqrt {2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right ) (i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}\right )}{60 a^{17/2} d (a+i a \tan (c+d x))^{5/2}} \]

[In]

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2),x]

[Out]

(12*a^(17/2)*(A + I*B)*Cot[c + d*x]^2 - (a^8*(I + Cot[c + d*x])*Tan[c + d*x]^2*(-1/2*(Sqrt[a]*Csc[c + d*x]^3*(
-((961*A + (461*I)*B)*Cos[c + d*x]) + (793*A + (413*I)*B)*Cos[3*(c + d*x)] + 18*((-57*I)*A + 27*B + ((83*I)*A
- 43*B)*Cos[2*(c + d*x)])*Sin[c + d*x])) - 30*(43*A + (20*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]]*(I
+ Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]] + 15*Sqrt[2]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]
*Sqrt[a])]*(I + Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]]))/2)/(60*a^(17/2)*d*(a + I*a*Tan[c + d*x])^(5/2))

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {2 a^{3} \left (-\frac {17 i B +31 A}{8 a^{5} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {5 i B +7 A}{12 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {i B +A}{10 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{16 a^{\frac {11}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {11 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {13}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (20 i B +43 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{5}}\right )}{d}\)	\(225\)
default	\(\frac {2 a^{3} \left (-\frac {17 i B +31 A}{8 a^{5} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {5 i B +7 A}{12 a^{4} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {i B +A}{10 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{16 a^{\frac {11}{2}}}+\frac {-\frac {\left (-\frac {i B}{2}-\frac {11 A}{8}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}+\left (\frac {1}{2} i a B +\frac {13}{8} a A \right ) \sqrt {a +i a \tan \left (d x +c \right )}}{a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (20 i B +43 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{5}}\right )}{d}\)	\(225\)

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/d*a^3*(-1/8/a^5*(17*I*B+31*A)/(a+I*a*tan(d*x+c))^(1/2)-1/12/a^4*(7*A+5*I*B)/(a+I*a*tan(d*x+c))^(3/2)-1/10/a^
3*(A+I*B)/(a+I*a*tan(d*x+c))^(5/2)-1/16*(A-I*B)/a^(11/2)*2^(1/2)*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/
a^(1/2))+1/a^5*(-((-1/2*I*B-11/8*A)*(a+I*a*tan(d*x+c))^(3/2)+(1/2*I*a*B+13/8*a*A)*(a+I*a*tan(d*x+c))^(1/2))/a^
2/tan(d*x+c)^2+1/8*(43*A+20*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. 923 vs. \(2 (243) = 486\).

Time = 0.29 (sec) , antiderivative size = 923, normalized size of antiderivative = 2.96 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/240*(30*sqrt(1/2)*(a^3*d*e^(9*I*d*x + 9*I*c) - 2*a^3*d*e^(7*I*d*x + 7*I*c) + a^3*d*e^(5*I*d*x + 5*I*c))*sqrt
((A^2 - 2*I*A*B - B^2)/(a^5*d^2))*log(-4*(sqrt(2)*sqrt(1/2)*(I*a^3*d*e^(2*I*d*x + 2*I*c) + I*a^3*d)*sqrt(a/(e^
(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^5*d^2)) + (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)
/(I*A + B)) - 30*sqrt(1/2)*(a^3*d*e^(9*I*d*x + 9*I*c) - 2*a^3*d*e^(7*I*d*x + 7*I*c) + a^3*d*e^(5*I*d*x + 5*I*c
))*sqrt((A^2 - 2*I*A*B - B^2)/(a^5*d^2))*log(-4*(sqrt(2)*sqrt(1/2)*(-I*a^3*d*e^(2*I*d*x + 2*I*c) - I*a^3*d)*sq
rt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^5*d^2)) + (-I*A - B)*a*e^(I*d*x + I*c))*e^(-I*d*
x - I*c)/(I*A + B)) + 15*(a^3*d*e^(9*I*d*x + 9*I*c) - 2*a^3*d*e^(7*I*d*x + 7*I*c) + a^3*d*e^(5*I*d*x + 5*I*c))
*sqrt((1849*A^2 + 1720*I*A*B - 400*B^2)/(a^5*d^2))*log(-16*(3*(43*I*A - 20*B)*a^2*e^(2*I*d*x + 2*I*c) + (43*I*
A - 20*B)*a^2 + 2*sqrt(2)*(I*a^4*d*e^(3*I*d*x + 3*I*c) + I*a^4*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c)
+ 1))*sqrt((1849*A^2 + 1720*I*A*B - 400*B^2)/(a^5*d^2)))*e^(-2*I*d*x - 2*I*c)/(-43*I*A + 20*B)) - 15*(a^3*d*e^
(9*I*d*x + 9*I*c) - 2*a^3*d*e^(7*I*d*x + 7*I*c) + a^3*d*e^(5*I*d*x + 5*I*c))*sqrt((1849*A^2 + 1720*I*A*B - 400
*B^2)/(a^5*d^2))*log(-16*(3*(43*I*A - 20*B)*a^2*e^(2*I*d*x + 2*I*c) + (43*I*A - 20*B)*a^2 + 2*sqrt(2)*(-I*a^4*
d*e^(3*I*d*x + 3*I*c) - I*a^4*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((1849*A^2 + 1720*I*A*B
- 400*B^2)/(a^5*d^2)))*e^(-2*I*d*x - 2*I*c)/(-43*I*A + 20*B)) - 2*sqrt(2)*((773*A + 403*I*B)*e^(10*I*d*x + 10
*I*c) - 6*(97*A + 42*I*B)*e^(8*I*d*x + 8*I*c) - (931*A + 431*I*B)*e^(6*I*d*x + 6*I*c) + 3*(153*A + 83*I*B)*e^(
4*I*d*x + 4*I*c) + 2*(19*A + 14*I*B)*e^(2*I*d*x + 2*I*c) + 3*A + 3*I*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a^
3*d*e^(9*I*d*x + 9*I*c) - 2*a^3*d*e^(7*I*d*x + 7*I*c) + a^3*d*e^(5*I*d*x + 5*I*c))

Sympy [F]

\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(5/2),x)

[Out]

Integral((A + B*tan(c + d*x))*cot(c + d*x)**3/(I*a*(tan(c + d*x) - I))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {a^{2} {\left (\frac {4 \, {\left (315 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} {\left (2 \, A + i \, B\right )} - 5 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} {\left (211 \, A + 104 i \, B\right )} a + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (337 \, A + 167 i \, B\right )} a^{2} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (23 \, A + 13 i \, B\right )} a^{3} + 12 \, {\left (A + i \, B\right )} a^{4}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{4} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{5} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{6}} - \frac {15 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {9}{2}}} + \frac {30 \, {\left (43 \, A + 20 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}}}\right )}}{240 \, d} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-1/240*a^2*(4*(315*(I*a*tan(d*x + c) + a)^4*(2*A + I*B) - 5*(I*a*tan(d*x + c) + a)^3*(211*A + 104*I*B)*a + (I*
a*tan(d*x + c) + a)^2*(337*A + 167*I*B)*a^2 + 2*(I*a*tan(d*x + c) + a)*(23*A + 13*I*B)*a^3 + 12*(A + I*B)*a^4)
/((I*a*tan(d*x + c) + a)^(9/2)*a^4 - 2*(I*a*tan(d*x + c) + a)^(7/2)*a^5 + (I*a*tan(d*x + c) + a)^(5/2)*a^6) -
15*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^(9/2) + 30*(43*A + 20*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a)
+ sqrt(a)))/a^(9/2))/d

Giac [F]

\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{3}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^3/(I*a*tan(d*x + c) + a)^(5/2), x)

Mupad [B] (verification not implemented)

Time = 10.01 (sec) , antiderivative size = 3048, normalized size of antiderivative = 9.77 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

2*atanh((192*a*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((3699*A^2)/(256*a^5*d^2) - ((13667809*A^4*a^2)/(16*d^4) + (6
38401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/
(4*d^4))^(1/2)/(64*a^6) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i)/(128*a^5*d^2))^(1/2)*((13667809*A^4*a^2)/(16*d
^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*63
62537i)/(4*d^4))^(1/2))/(B^3*d*62322i - 643278*A^3*d + 407502*A*B^2*d - A^2*B*d*887274i + (680*A*d^3*((1366780
9*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4)
+ (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) -
(8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)*328i)/a) -
(59152*A^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((3699*A^2)/(256*a^5*d^2) - ((13667809*A^4*a^2)/(16*d^4) + (6
38401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/
(4*d^4))^(1/2)/(64*a^6) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i)/(128*a^5*d^2))^(1/2))/(B^3*d*62322i - 643278*A
^3*d + 407502*A*B^2*d - A^2*B*d*887274i + (680*A*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4)
- (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d
^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*13750
79i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)*328i)/a) + (12784*B^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)
*((3699*A^2)/(256*a^5*d^2) - ((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/
(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)/(64*a^6) - (801*B^2)/(256*a^5*d^2
) + (A*B*1719i)/(128*a^5*d^2))^(1/2))/(B^3*d*62322i - 643278*A^3*d + 407502*A*B^2*d - A^2*B*d*887274i + (680*A
*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*137
5079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^
2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/
2)*328i)/a) - (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((3699*A^2)/(256*a^5*d^2) - ((13667809*A^4*a^2)/(16*d
^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*63
62537i)/(4*d^4))^(1/2)/(64*a^6) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i)/(128*a^5*d^2))^(1/2)*55072i)/(B^3*d*62
322i - 643278*A^3*d + 407502*A*B^2*d - A^2*B*d*887274i + (680*A*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4
*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^
(1/2))/a + (B*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (
A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)*328i)/a))*(-(4*d^2*((((3699*A^2*a)/4 - (801*
B^2*a)/4)/d^2 + (A*B*a*1719i)/(2*d^2))^2 + 128*a^6*((((115*A*B^3)/32 + (989*A^3*B)/128)*1i)/(a^4*d^4) - ((1849
*A^4)/256 + (1191*A^2*B^2)/256 + (25*B^4)/16)/(a^4*d^4)))^(1/2) - 3699*A^2*a + 801*B^2*a - A*B*a*3438i)/(256*a
^6*d^2))^(1/2) - ((A*a^2 + B*a^2*1i)/(5*d) + ((337*A + B*167i)*(a + a*tan(c + d*x)*1i)^2)/(60*d) + ((23*A*a +
B*a*13i)*(a + a*tan(c + d*x)*1i))/(30*d) + (21*(2*A + B*1i)*(a + a*tan(c + d*x)*1i)^4)/(4*a^2*d) - ((211*A + B
*104i)*(a + a*tan(c + d*x)*1i)^3)/(12*a*d))/((a + a*tan(c + d*x)*1i)^(9/2) - 2*a*(a + a*tan(c + d*x)*1i)^(7/2)
+ a^2*(a + a*tan(c + d*x)*1i)^(5/2)) + 2*atanh((192*a*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((13667809*A^4*a^2)/
(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a
^2*6362537i)/(4*d^4))^(1/2)/(64*a^6) + (3699*A^2)/(256*a^5*d^2) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i)/(128*a
^5*d^2))^(1/2)*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B
^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/(643278*A^3*d - B^3*d*62322i - 407502*A*B^2*d
+ A^2*B*d*887274i + (680*A*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2
)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d^3*((13667809*A^4*a^2)
/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*
a^2*6362537i)/(4*d^4))^(1/2)*328i)/a) + (59152*A^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((13667809*A^4*a^2)/
(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a
^2*6362537i)/(4*d^4))^(1/2)/(64*a^6) + (3699*A^2)/(256*a^5*d^2) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i)/(128*a
^5*d^2))^(1/2))/(643278*A^3*d - B^3*d*62322i - 407502*A*B^2*d + A^2*B*d*887274i + (680*A*d^3*((13667809*A^4*a^
2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*
B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585
*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)*328i)/a) - (12784*B
^2*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*
A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)/(64*a^6) + (3699*A^2
)/(256*a^5*d^2) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i)/(128*a^5*d^2))^(1/2))/(643278*A^3*d - B^3*d*62322i - 4
07502*A*B^2*d + A^2*B*d*887274i + (680*A*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (88775
85*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d^3*((136
67809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*
d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)*328i)/a) + (A*B*a^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((13667809*A
^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) +
(A^3*B*a^2*6362537i)/(4*d^4))^(1/2)/(64*a^6) + (3699*A^2)/(256*a^5*d^2) - (801*B^2)/(256*a^5*d^2) + (A*B*1719i
)/(128*a^5*d^2))^(1/2)*55072i)/(643278*A^3*d - B^3*d*62322i - 407502*A*B^2*d + A^2*B*d*887274i + (680*A*d^3*((
13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/
(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2))/a + (B*d^3*((13667809*A^4*a^2)/(16*d^4) + (638401*B^4*a^2)/(16*
d^4) - (8877585*A^2*B^2*a^2)/(8*d^4) - (A*B^3*a^2*1375079i)/(4*d^4) + (A^3*B*a^2*6362537i)/(4*d^4))^(1/2)*328i
)/a))*((4*d^2*((((3699*A^2*a)/4 - (801*B^2*a)/4)/d^2 + (A*B*a*1719i)/(2*d^2))^2 + 128*a^6*((((115*A*B^3)/32 +
(989*A^3*B)/128)*1i)/(a^4*d^4) - ((1849*A^4)/256 + (1191*A^2*B^2)/256 + (25*B^4)/16)/(a^4*d^4)))^(1/2) + 3699*
A^2*a - 801*B^2*a + A*B*a*3438i)/(256*a^6*d^2))^(1/2)