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

Optimal. Leaf size=361 \[ -\frac {\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}-\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {83 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{8 f}+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{32 f}+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]

[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]  time = 0.68, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3567, 3650, 3649, 3653, 12, 3485, 708, 1094, 634, 618, 204, 628, 3634, 63, 207} \[ -\frac {\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}-\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}} f}-\frac {83 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{8 f}+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{32 f}+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]

Antiderivative was successfully verified.

[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 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 204

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

Rule 207

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

Rule 618

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 628

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 634

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 708

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 1094

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 3485

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 3567

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 3634

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 3649

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*T
an[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 3650

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 3653

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*} \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx &=-\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 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {83 \tanh ^{-1}\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 \operatorname {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}\\ &=-\frac {83 \tanh ^{-1}\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 {\operatorname {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 {\operatorname {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 \tanh ^{-1}\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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 \tanh ^{-1}\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} \operatorname {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} \operatorname {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 {\tan ^{-1}\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 {\tan ^{-1}\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 \tanh ^{-1}\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*}

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Mathematica [C]  time = 1.45, size = 169, normalized size = 0.47 \[ \frac {-83 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )+64 (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {\tan (e+f x)+1}}{\sqrt {1-i}}\right )+64 (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {\tan (e+f x)+1}}{\sqrt {1+i}}\right )-16 \sqrt {\tan (e+f x)+1} \cot ^4(e+f x)-24 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)+30 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)+83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.58, size = 1130, normalized size = 3.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(16*8^(1/4)*(2*f*cos(f*x + e)^4 - 4*f*cos(f*x + e)^2 - sqrt(2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos(f*x + e)^
2 + f^3)*sqrt(f^(-4)) + 2*f)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(2*(2*sqrt(2)*f^2*sqrt(f^(
-4))*cos(f*x + e) + 8^(1/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x
+ e))*(f^(-4))^(1/4)*cos(f*x + e) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) - 16*8^(1/4)*(2*f*cos(f*x +
e)^4 - 4*f*cos(f*x + e)^2 - sqrt(2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos(f*x + e)^2 + f^3)*sqrt(f^(-4)) + 2*f)*sqr
t(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(2*(2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) - 8^(1/4)*sqrt
(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4)*cos(f*x + e
) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) - 83*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*log(sqrt((cos(
f*x + e) + sin(f*x + e))/cos(f*x + e)) + 1) + 83*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*log(sqrt((cos(f*x + e
) + sin(f*x + e))/cos(f*x + e)) - 1) - 2*(46*cos(f*x + e)^4 - 30*cos(f*x + e)^2 + (107*cos(f*x + e)^3 - 83*cos
(f*x + e))*sin(f*x + e))*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 64*8^(1/4)*sqrt(2)*(f^5*cos(f*x +
e)^4 - 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*arctan(-1/8*8^(3/4)*sqr
t(2)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4)
+ 1/8*8^(3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 8^(1/4)
*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4)*cos(f*
x + e) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - f^2*sqrt(f^(-4)) - sqrt(2))/f^4 - 64*
8^(1/4)*sqrt(2)*(f^5*cos(f*x + e)^4 - 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4)
)^(1/4)*arctan(-1/8*8^(3/4)*sqrt(2)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((cos(f*x + e) + sin(f*x + e)
)/cos(f*x + e))*(f^(-4))^(3/4) + 1/8*8^(3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((2*sqrt(2)*f^2*sqrt
(f^(-4))*cos(f*x + e) - 8^(1/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*sqrt((cos(f*x + e) + sin(f*x + e))/cos(
f*x + e))*(f^(-4))^(1/4)*cos(f*x + e) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + f^2*sq
rt(f^(-4)) + sqrt(2))/f^4)/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.41, size = 14663, normalized size = 40.62 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [B]  time = 3.98, size = 193, normalized size = 0.53 \[ \frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,83{}\mathrm {i}}{64\,f}-\frac {\frac {45\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{64}-\frac {165\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{64}+\frac {219\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{64}-\frac {83\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{64}}{f-4\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+6\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^4}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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