3.405 \(\int \frac{\cot ^5(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx\)
Optimal. Leaf size=269 \[ \frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{\left (1-\sqrt{2}\right ) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left (5 \sqrt{2}-7\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f}-\frac{115 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{64 f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{\left (1+\sqrt{2}\right ) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left (7+5 \sqrt{2}\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}+\frac{7 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac{13 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}-\frac{13 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{64 f} \]
[Out]
(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan
[e + f*x]])])/(2*f) - (115*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2]
+ (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (13*Cot[e + f*x]*Sqr
t[1 + Tan[e + f*x]])/(64*f) + (13*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(96*f) + (7*Cot[e + f*x]^3*Sqrt[1 + T
an[e + f*x]])/(24*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(4*f)
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Rubi [A] time = 0.599141, antiderivative size = 269, normalized size of antiderivative
= 1., number of steps used = 14, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\)
= 0.524, Rules used = {3569, 3649, 3650, 3654, 12, 3536, 3535, 203, 207, 3634, 63}
\[ \frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{\left (1-\sqrt{2}\right ) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left (5 \sqrt{2}-7\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f}-\frac{115 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{64 f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{\left (1+\sqrt{2}\right ) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left (7+5 \sqrt{2}\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}+\frac{7 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac{13 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}-\frac{13 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{64 f} \]
Antiderivative was successfully verified.
[In]
Int[Cot[e + f*x]^5/Sqrt[1 + Tan[e + f*x]],x]
[Out]
(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan
[e + f*x]])])/(2*f) - (115*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2]
+ (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (13*Cot[e + f*x]*Sqr
t[1 + Tan[e + f*x]])/(64*f) + (13*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(96*f) + (7*Cot[e + f*x]^3*Sqrt[1 + T
an[e + f*x]])/(24*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(4*f)
Rule 3569
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))
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 3654
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + 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, 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]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 3536
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> With[{q =
Rt[a^2 + b^2, 2]}, Dist[1/(2*q), Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*
x]], x], x] - Dist[1/(2*q), Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*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] && NeQ[2
*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])
Rule 3535
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(-2*
d^2)/f, Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*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] && EqQ[2
*a*c*d - b*(c^2 - d^2), 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 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 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]
Rubi steps
\begin{align*} \int \frac{\cot ^5(e+f x)}{\sqrt{1+\tan (e+f x)}} \, 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{7}{2}+4 \tan (e+f x)+\frac{7}{2} \tan ^2(e+f x)\right )}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 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{13}{4}+\frac{35}{4} \tan ^2(e+f x)\right )}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 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{39}{8}-24 \tan (e+f x)-\frac{39}{8} \tan ^2(e+f x)\right )}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=-\frac{13 \cot (e+f x) \sqrt{1+\tan (e+f x)}}{64 f}+\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 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{345}{16}-\frac{39}{16} \tan ^2(e+f x)\right )}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=-\frac{13 \cot (e+f x) \sqrt{1+\tan (e+f x)}}{64 f}+\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 f}-\frac{\cot ^4(e+f x) \sqrt{1+\tan (e+f x)}}{4 f}+\frac{1}{24} \int -\frac{24 \tan (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx+\frac{115}{128} \int \frac{\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=-\frac{13 \cot (e+f x) \sqrt{1+\tan (e+f x)}}{64 f}+\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 f}-\frac{\cot ^4(e+f x) \sqrt{1+\tan (e+f x)}}{4 f}+\frac{115 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\tan (e+f x)\right )}{128 f}-\int \frac{\tan (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=-\frac{13 \cot (e+f x) \sqrt{1+\tan (e+f x)}}{64 f}+\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 f}-\frac{\cot ^4(e+f x) \sqrt{1+\tan (e+f x)}}{4 f}+\frac{\int \frac{1+\left (-1-\sqrt{2}\right ) \tan (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx}{2 \sqrt{2}}-\frac{\int \frac{1+\left (-1+\sqrt{2}\right ) \tan (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx}{2 \sqrt{2}}+\frac{115 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{64 f}\\ &=-\frac{115 \tanh ^{-1}\left (\sqrt{1+\tan (e+f x)}\right )}{64 f}-\frac{13 \cot (e+f x) \sqrt{1+\tan (e+f x)}}{64 f}+\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 f}-\frac{\cot ^4(e+f x) \sqrt{1+\tan (e+f x)}}{4 f}-\frac{\left (4-3 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (-1+\sqrt{2}\right )-4 \left (-1+\sqrt{2}\right )^2+x^2} \, dx,x,\frac{1-2 \left (-1+\sqrt{2}\right )-\left (-1+\sqrt{2}\right ) \tan (e+f x)}{\sqrt{1+\tan (e+f x)}}\right )}{2 f}-\frac{\left (4+3 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (-1-\sqrt{2}\right )-4 \left (-1-\sqrt{2}\right )^2+x^2} \, dx,x,\frac{1-2 \left (-1-\sqrt{2}\right )-\left (-1-\sqrt{2}\right ) \tan (e+f x)}{\sqrt{1+\tan (e+f x)}}\right )}{2 f}\\ &=\frac{\sqrt{-1+\sqrt{2}} \tan ^{-1}\left (\frac{3-2 \sqrt{2}+\left (1-\sqrt{2}\right ) \tan (e+f x)}{\sqrt{2 \left (-7+5 \sqrt{2}\right )} \sqrt{1+\tan (e+f x)}}\right )}{2 f}-\frac{115 \tanh ^{-1}\left (\sqrt{1+\tan (e+f x)}\right )}{64 f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{3+2 \sqrt{2}+\left (1+\sqrt{2}\right ) \tan (e+f x)}{\sqrt{2 \left (7+5 \sqrt{2}\right )} \sqrt{1+\tan (e+f x)}}\right )}{2 f}-\frac{13 \cot (e+f x) \sqrt{1+\tan (e+f x)}}{64 f}+\frac{13 \cot ^2(e+f x) \sqrt{1+\tan (e+f x)}}{96 f}+\frac{7 \cot ^3(e+f x) \sqrt{1+\tan (e+f x)}}{24 f}-\frac{\cot ^4(e+f x) \sqrt{1+\tan (e+f x)}}{4 f}\\ \end{align*}
Mathematica [C] time = 1.3819, size = 169, normalized size = 0.63 \[ \frac{-345 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )+\frac{192 \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )}{\sqrt{1-i}}+\frac{192 \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )}{\sqrt{1+i}}-48 \sqrt{\tan (e+f x)+1} \cot ^4(e+f x)+56 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)+26 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)-39 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{192 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[e + f*x]^5/Sqrt[1 + Tan[e + f*x]],x]
[Out]
(-345*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (192*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (192*A
rcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] - 39*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 26*Cot[e +
f*x]^2*Sqrt[1 + Tan[e + f*x]] + 56*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 48*Cot[e + f*x]^4*Sqrt[1 + Tan[e +
f*x]])/(192*f)
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Maple [C] time = 0.576, size = 13599, normalized size = 50.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)^5/(1+tan(f*x+e))^(1/2),x)
[Out]
result too large to display
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^5/(1+tan(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [B] time = 2.27252, size = 3681, normalized size = 13.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^5/(1+tan(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
1/384*(96*(1/2)^(1/4)*(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + sqrt(1/2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos(f*x +
e)^2 + f^3)*sqrt(f^(-4)) + f)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log((2*sqrt(1/2)*f^2*sqr
t(f^(-4))*cos(f*x + e) + (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt
(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + s
in(f*x + e))/cos(f*x + e)) - 96*(1/2)^(1/4)*(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + sqrt(1/2)*(f^3*cos(f*x +
e)^4 - 2*f^3*cos(f*x + e)^2 + f^3)*sqrt(f^(-4)) + f)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*lo
g((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e) - (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(
f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1
/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 345*(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) + 345*(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*(74*cos(f*x + e)^4 - 26*cos(f*x + e)^2 - (95*cos(f*x + e)^3 - 39*cos(
f*x + e))*sin(f*x + e))*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 384*(1/2)^(3/4)*(f^5*cos(f*x + e)^4
- 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*arctan(2*(1/2)^(3/4)*(f^
5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos
(f*x + e) + (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqr
t(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/
cos(f*x + e))*(f^(-4))^(3/4) - 2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(
-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - f^2*sqrt(f^(-4)) - 2*sqrt(1/2))/f^
4 + 384*(1/2)^(3/4)*(f^5*cos(f*x + e)^4 - 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*
(f^(-4))^(1/4)*arctan(2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)
*sqrt((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e) - (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*
cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4)
)^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - 2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1
/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/
4) + f^2*sqrt(f^(-4)) + 2*sqrt(1/2))/f^4)/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{5}{\left (e + f x \right )}}{\sqrt{\tan{\left (e + f x \right )} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)**5/(1+tan(f*x+e))**(1/2),x)
[Out]
Integral(cot(e + f*x)**5/sqrt(tan(e + f*x) + 1), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )^{5}}{\sqrt{\tan \left (f x + e\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^5/(1+tan(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(cot(f*x + e)^5/sqrt(tan(f*x + e) + 1), x)