\(\int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [394]

Optimal result

Integrand size = 21, antiderivative size = 361 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \]

[Out]

-83/64*arctanh((1+tan(f*x+e))^(1/2))/f-1/2*ln(1+2^(1/2)-(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f
/(1+2^(1/2))^(1/2)+1/2*ln(1+2^(1/2)+(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)-a
rctan(((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)/f+arctan(((2+2*2^(1
/2))^(1/2)+2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)/f+83/64*cot(f*x+e)*(1+tan(f*x+e))^(
1/2)/f+15/32*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2)/f-3/8*cot(f*x+e)^3*(1+tan(f*x+e))^(1/2)/f-1/4*cot(f*x+e)^4*(1+t
an(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3648, 3731, 3730, 3734, 12, 3566, 722, 1108, 648, 632, 210, 642, 3715, 65, 213} \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}-\frac {83 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}-\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{8 f}+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{32 f}+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]

[In]

Int[Cot[e + f*x]^5*(1 + Tan[e + f*x])^(3/2),x]

[Out]

-((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (
Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (83*A
rcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e
+ f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*
x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + (83*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(64*f) + (15*Cot[e + f*x]^2*Sqrt[1 + T
an[e + f*x]])/(32*f) - (3*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(8*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]
])/(4*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 210

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

Rule 213

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 3648

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

Rule 3715

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

Rule 3730

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

Rule 3731

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

Rule 3734

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {1}{4} \int \frac {\cot ^4(e+f x) \left (-\frac {9}{2}+\frac {7}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = -\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{12} \int \frac {\cot ^3(e+f x) \left (-\frac {45}{4}-24 \tan (e+f x)-\frac {45}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {1}{24} \int \frac {\cot ^2(e+f x) \left (\frac {249}{8}-\frac {135}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{24} \int \frac {\cot (e+f x) \left (\frac {249}{16}+48 \tan (e+f x)+\frac {249}{16} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{24} \int \frac {48}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {83}{128} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+2 \int \frac {1}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {83 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{128 f} \\ & = \frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {83 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {4 \text {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f} \\ & = -\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f} \\ & = -\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {2} f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{\sqrt {2} f}-\frac {\text {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f} \\ & = -\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}-\frac {\sqrt {2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f} \\ & = -\frac {\arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {-1+\sqrt {2}} f}+\frac {\arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {-1+\sqrt {2}} f}-\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.63 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.47 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {-83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+64 (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+64 (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+83 \cot (e+f x) \sqrt {1+\tan (e+f x)}+30 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}-24 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}-16 \cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{64 f} \]

[In]

Integrate[Cot[e + f*x]^5*(1 + Tan[e + f*x])^(3/2),x]

[Out]

(-83*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 64*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 64*(1 +
I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 83*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 30*Cot[e + f*x
]^2*Sqrt[1 + Tan[e + f*x]] - 24*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 16*Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x
]])/(64*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4798\) vs. \(2(285)=570\).

Time = 88.37 (sec) , antiderivative size = 4799, normalized size of antiderivative = 13.29

method	result	size
default	\(\text {Expression too large to display}\)	\(4799\)

[In]

int(cot(f*x+e)^5*(1+tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/128/f*(1+tan(f*x+e))^(1/2)/(cot(f*x+e)^2+cot(f*x+e))^(1/2)/((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1
)^2)^(1/2)*(128*cot(f*x+e)^2*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*
x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*
arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2
^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x
+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))-332*cot(f*x+e)*csc(f*x+e)^2*((cos(f*x+e)+s
in(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)-64*cot
(f*x+e)^2*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos
(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^
(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)
*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(
-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)-96*cot(f*x+e)^2*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))
*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+498*cot(f*x+e)*csc(f*x+e
)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1
/2)-384*cot(f*x+e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/
2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos
(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))
^(1/2))+896*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2
^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)
*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1
/2))^(1/2))-128*cot(f*x+e)^3*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(
1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*
((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*si
n(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f
*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))+256*cot(f*x+e)^2*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/
2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*co
s(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(
f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos
(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))+428*cot(
f*x+e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e
))^(1/2)*2^(1/2)-184*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2
)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)-256*cot(f*x+e)^2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2
^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh((
(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*
x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)+166*cot(f*x+e)^2*(1+2^(1/2))^(1/2)*(cot(f*x+e
)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)
-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2^(1/2)+640*cot(f*x+e)*csc(f*x+e)
*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(
f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2
*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)+249*
cot(f*x+e)*csc(f*x+e)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e
))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x
+e)+1)^2)^(1/2))+192*cot(f*x+e)*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos
(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/
2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^
2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(
f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)+384*cot(f*x+e)^2*csc(f*x+e)*((c
os(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+
e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin
(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))+256*cot(f*x+e)^3
*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(
f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2
*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)-640*
cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin
(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*
sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*
2^(1/2)-642*cot(f*x+e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*
x+e)^2+cot(f*x+e))^(1/2)+276*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/
2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)-896*cot(f*x+e)*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(
1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((c
os(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+
e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))-249*cot(f*x+e)^2*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*
x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc(f
*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))-166*cot(f*x+e)*csc(f*x+e)*(1+2^(1/2))^(1/2)
*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2
*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2^(1/2)+64*cot(f*x+e)^
2*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot
(f*x+e))^(1/2)*2^(1/2)+64*cot(f*x+e)^3*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x
+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*a
rctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^
(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+
e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)-192*cot(f*x+e)^2*(-2+2*2^(1/2))^(1/
2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/
2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+
e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2
)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^
(1/2)-4))*2^(1/2)-256*cot(f*x+e)*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*co
s(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1
/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)
^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos
(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)))*2^(1/2)/(1+2^(1/2))^(1/2)/(3*2^(1/2)-
4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.23 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {64 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 64 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 64 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} + 64 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 83 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{4} + 83 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{4} + 2 \, {\left (83 \, \tan \left (f x + e\right )^{3} + 30 \, \tan \left (f x + e\right )^{2} - 24 \, \tan \left (f x + e\right ) - 16\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{128 \, f \tan \left (f x + e\right )^{4}} \]

[In]

integrate(cot(f*x+e)^5*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/128*(64*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^
4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 - 64*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(-
sqrt(2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 -
64*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*log(sqrt(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f
^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 + 64*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*log(-sqrt(2)*(f
^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 - 83*log(sqrt
(tan(f*x + e) + 1) + 1)*tan(f*x + e)^4 + 83*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^4 + 2*(83*tan(f*x + e
)^3 + 30*tan(f*x + e)^2 - 24*tan(f*x + e) - 16)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^4)

Sympy [F]

\[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{5}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**5*(1+tan(f*x+e))**(3/2),x)

[Out]

Integral((tan(e + f*x) + 1)**(3/2)*cot(e + f*x)**5, x)

Maxima [F(-1)]

Timed out. \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^5*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^5*(1+tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 4.96 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.53 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,83{}\mathrm {i}}{64\,f}-\frac {\frac {45\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{64}-\frac {165\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{64}+\frac {219\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{64}-\frac {83\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{64}}{f-4\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+6\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^4}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]

[In]

int(cot(e + f*x)^5*(tan(e + f*x) + 1)^(3/2),x)

[Out]

(atan((tan(e + f*x) + 1)^(1/2)*1i)*83i)/(64*f) - ((45*(tan(e + f*x) + 1)^(1/2))/64 - (165*(tan(e + f*x) + 1)^(
3/2))/64 + (219*(tan(e + f*x) + 1)^(5/2))/64 - (83*(tan(e + f*x) + 1)^(7/2))/64)/(f - 4*f*(tan(e + f*x) + 1) +
6*f*(tan(e + f*x) + 1)^2 - 4*f*(tan(e + f*x) + 1)^3 + f*(tan(e + f*x) + 1)^4) + atan(f*((- 1/2 - 1i/2)/f^2)^(
1/2)*(tan(e + f*x) + 1)^(1/2))*((- 1/2 - 1i/2)/f^2)^(1/2)*2i - atan(f*((- 1/2 + 1i/2)/f^2)^(1/2)*(tan(e + f*x)
+ 1)^(1/2))*((- 1/2 + 1i/2)/f^2)^(1/2)*2i