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

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

[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 = 0.46, 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 = {3648, 3731, 3730, 3734, 12, 3566, 722, 1108, 648, 632, 210, 642, 3715, 65, 213} \begin {gather*} -\frac {\sqrt {1+\sqrt {2}} \text {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}} \text {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 {\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} \end {gather*}

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 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*} \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 \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 \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 \text {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 {\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 \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 {\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 \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} \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 {\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] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 21.98, size = 4084, normalized size = 11.31 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[e + f*x]*(-23/32 + (107*Cot[e + f*x])/64 + (31*Csc[e + f*x]^2)/32 - (3*Cot[e + f*x]*Csc[e + f*x]^2)/8 - C
sc[e + f*x]^4/4)*(1 + Tan[e + f*x])^(3/2))/(f*(Cos[e + f*x] + Sin[e + f*x])) + (Cos[e + f*x]*(83*EllipticF[Arc
Sin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*El
lipticPi[-1 - Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]
], -3 - 2*Sqrt[2]] - (256*I)*EllipticPi[(-I)*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 +
Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] + (256*I)*EllipticPi[I*(1 + Sqrt[2]), ArcSin[(2^(1/4)*
Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*EllipticPi[1 +
Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt
[2]])*((83*Csc[e + f*x]*Sqrt[Sec[e + f*x]])/(128*Sqrt[Cos[e + f*x] + Sin[e + f*x]]) + (Csc[e + f*x]*Sqrt[Sec[e
+ f*x]]*Sin[2*(e + f*x)])/Sqrt[Cos[e + f*x] + Sin[e + f*x]])*Sqrt[-((1 + Tan[(e + f*x)/2])/((-2 + Sqrt[2])*(-
1 + Tan[(e + f*x)/2])))]*(1 + Tan[e + f*x])^(3/2))/(32*2^(1/4)*f*(Cos[e + f*x] + Sin[e + f*x])*Sqrt[(Cos[e + f
*x] + Sin[e + f*x])/(-1 + Sin[e + f*x])]*(((83*EllipticF[ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan
[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*EllipticPi[-1 - Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + T
an[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - (256*I)*EllipticPi[(-I)*(1 +
Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqr
t[2]] + (256*I)*EllipticPi[I*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2]
)])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*EllipticPi[1 + Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2
])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]])*Sqrt[Sec[e + f*x]]*(Cos[e + f*x] - Sin[e + f
*x])*Sqrt[-((1 + Tan[(e + f*x)/2])/((-2 + Sqrt[2])*(-1 + Tan[(e + f*x)/2])))])/(64*2^(1/4)*Sqrt[Cos[e + f*x] +
Sin[e + f*x]]*Sqrt[(Cos[e + f*x] + Sin[e + f*x])/(-1 + Sin[e + f*x])]) + ((83*EllipticF[ArcSin[(2^(1/4)*Sqrt[
(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*EllipticPi[-1 - Sqrt
[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]]
- (256*I)*EllipticPi[(-I)*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])]
)/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] + (256*I)*EllipticPi[I*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e +
f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*EllipticPi[1 + Sqrt[2], ArcSin[(2
^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]])*Sec[e + f*x]
^(3/2)*Sin[e + f*x]*Sqrt[Cos[e + f*x] + Sin[e + f*x]]*Sqrt[-((1 + Tan[(e + f*x)/2])/((-2 + Sqrt[2])*(-1 + Tan[
(e + f*x)/2])))])/(64*2^(1/4)*Sqrt[(Cos[e + f*x] + Sin[e + f*x])/(-1 + Sin[e + f*x])]) - ((83*EllipticF[ArcSin
[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*Ellip
ticPi[-1 - Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]],
-3 - 2*Sqrt[2]] - (256*I)*EllipticPi[(-I)*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan
[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] + (256*I)*EllipticPi[I*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqr
t[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*EllipticPi[1 + Sqr
t[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]
])*Sqrt[Sec[e + f*x]]*Sqrt[Cos[e + f*x] + Sin[e + f*x]]*((Cos[e + f*x] - Sin[e + f*x])/(-1 + Sin[e + f*x]) - (
Cos[e + f*x]*(Cos[e + f*x] + Sin[e + f*x]))/(-1 + Sin[e + f*x])^2)*Sqrt[-((1 + Tan[(e + f*x)/2])/((-2 + Sqrt[2
])*(-1 + Tan[(e + f*x)/2])))])/(64*2^(1/4)*((Cos[e + f*x] + Sin[e + f*x])/(-1 + Sin[e + f*x]))^(3/2)) + ((83*E
llipticF[ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt
[2]] - 83*EllipticPi[-1 - Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[
2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - (256*I)*EllipticPi[(-I)*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x
)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] + (256*I)*EllipticPi[I*(1 + Sqrt[2]), ArcS
in[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 83*Ell
ipticPi[1 + Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]],
-3 - 2*Sqrt[2]])*Sqrt[Sec[e + f*x]]*Sqrt[Cos[e + f*x] + Sin[e + f*x]]*(-1/2*Sec[(e + f*x)/2]^2/((-2 + Sqrt[2]
)*(-1 + Tan[(e + f*x)/2])) + (Sec[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]))/(2*(-2 + Sqrt[2])*(-1 + Tan[(e + f*x)
/2])^2)))/(64*2^(1/4)*Sqrt[(Cos[e + f*x] + Sin[...

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.77, size = 14663, normalized size = 40.62

method	result	size
default	\(\text {Expression too large to display}\)	\(14663\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1200 vs. \(2 (300) = 600\).
time = 1.26, size = 1200, normalized size = 3.32 \begin {gather*} \frac {16 \cdot 8^{\frac {1}{4}} {\left (2 \, f \cos \left (f x + e\right )^{4} - 4 \, f \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{4} - 2 \, f^{3} \cos \left (f x + e\right )^{2} + f^{3}\right )} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) - 16 \cdot 8^{\frac {1}{4}} {\left (2 \, f \cos \left (f x + e\right )^{4} - 4 \, f \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{4} - 2 \, f^{3} \cos \left (f x + e\right )^{2} + f^{3}\right )} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )}}{\cos \left (f x + e\right )}\right ) - 83 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + 1\right ) + 83 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - 1\right ) - 2 \, {\left (46 \, \cos \left (f x + e\right )^{4} - 30 \, \cos \left (f x + e\right )^{2} + {\left (107 \, \cos \left (f x + e\right )^{3} - 83 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - \frac {64 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (f^{5} \cos \left (f x + e\right )^{4} - 2 \, f^{5} \cos \left (f x + e\right )^{2} + f^{5}\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (-\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right )}{f^{4}} - \frac {64 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (f^{5} \cos \left (f x + e\right )^{4} - 2 \, f^{5} \cos \left (f x + e\right )^{2} + f^{5}\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (-\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + \frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right )}{f^{4}}}{128 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{5}{\left (e + f x \right )}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [B]
time = 3.98, size = 193, normalized size = 0.53 \begin {gather*} \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} \end {gather*}

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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