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

Optimal. Leaf size=253 \[ -\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}-\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}-\frac{2 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{f} \]

[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 - (2*Ar
cTanh[Sqrt[1 + Tan[e + f*x]]])/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)

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Rubi [A]  time = 0.282313, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632, Rules used = {3573, 12, 3485, 708, 1094, 634, 618, 204, 628, 3634, 63, 207} \[ -\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}-\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}-\frac{2 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]*(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 - (2*Ar
cTanh[Sqrt[1 + Tan[e + f*x]]])/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)

Rule 3573

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1
/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e +
f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*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]

Rule 12

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

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 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 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 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 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 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 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]

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])

Rubi steps

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

Mathematica [C]  time = 0.172511, size = 78, normalized size = 0.31 \[ \frac{-2 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )+(1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )+(1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2
)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/f

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Maple [C]  time = 0.44, size = 2391, normalized size = 9.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/f/(2+2^(1/2))*((cos(f*x+e)+sin(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)+1)^2*(cos(f*x+e)-1)^2*(1+sin(f*x+e))*
(16*I*2^(1/2)*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e
)+1)/cos(f*x+e))^(1/2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*EllipticPi(1/2
*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2
)))^(1/2))-16*I*2^(1/2)*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)
-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*Elli
pticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),-I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2)
)/(2+2^(1/2)))^(1/2))-16*EllipticF(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^
(1/2))/(2+2^(1/2)))^(1/2))*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x
+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*2
^(1/2)+6*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/
cos(f*x+e))^(1/2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*EllipticE(1/2*2^(1/
2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)+12*((1+2^
(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/
2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(1/2)
)*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)+3*(
2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*
2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticF(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2
)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*
x+e))/cos(f*x+e))^(1/2)-3*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^
(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticE(1/2*
2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((2+
2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)-12*EllipticF(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e
))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/
2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(
f*x+e)-1)/cos(f*x+e))^(1/2)+12*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos
(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*(-(2^(1/2)*sin(f*x+e)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/
2)*EllipticE(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1
/2))+4*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos
(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticF(1/2*2^(1/2)*((2+2^(1/2)
)*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(-1+sin(f*
x+e))/cos(f*x+e))^(1/2)-3*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^
(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticE(1/2*
2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((2+2^(1/2))
*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)-2*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x
+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(
1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-2
^(1/2))/(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)+4*(2^(1/2)*(cos(f*x+e)*2^(1
/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+
e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f
*x+e))^(1/2),-2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(
f*x+e))^(1/2))*4^(1/2)/sin(f*x+e)^4/(cos(f*x+e)+sin(f*x+e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\tan \left (f x + e\right ) + 1\right )}^{\frac{3}{2}} \cot \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.95619, size = 2460, normalized size = 9.72 \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)*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/8*(4*8^(1/4)*sqrt(2)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*(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*s
qrt(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)) + 4*8^(1/4)*sqrt(2
)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*(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*sqrt(f^(-4)) + sqrt(2)) + 8^(1/4)*(sqrt(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)) - 8^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4)) - 2*f)*sqrt(2*sqr
t(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*c
os(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) + 8*log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 1) - 8*
log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 1))/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\tan \left (f x + e\right ) + 1\right )}^{\frac{3}{2}} \cot \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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