3.213 \(\int \frac{\cot ^3(e+f x)}{\sqrt{d \cot (e+f x)}} \, dx\)

Optimal. Leaf size=214 \[ -\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} f} \]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e +
f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - (2*(d*Cot[e + f*x])^(3/2))/(3*d^2*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*
x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d
*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)

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Rubi [A]  time = 0.159854, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {16, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^3/Sqrt[d*Cot[e + f*x]],x]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e +
f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - (2*(d*Cot[e + f*x])^(3/2))/(3*d^2*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*
x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d
*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3476

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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]

Rubi steps

\begin{align*} \int \frac{\cot ^3(e+f x)}{\sqrt{d \cot (e+f x)}} \, dx &=\frac{\int (d \cot (e+f x))^{5/2} \, dx}{d^3}\\ &=-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}-\frac{\int \sqrt{d \cot (e+f x)} \, dx}{d}\\ &=-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}-\frac{\operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}+\frac{\operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}\\ &=-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} f}-\frac{2 (d \cot (e+f x))^{3/2}}{3 d^2 f}+\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}-\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} \sqrt{d} f}\\ \end{align*}

Mathematica [C]  time = 0.0611562, size = 47, normalized size = 0.22 \[ \frac{2 \cot ^2(e+f x) \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\cot ^2(e+f x)\right )-1\right )}{3 f \sqrt{d \cot (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^3/Sqrt[d*Cot[e + f*x]],x]

[Out]

(2*Cot[e + f*x]^2*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[e + f*x]^2]))/(3*f*Sqrt[d*Cot[e + f*x]])

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Maple [A]  time = 0.028, size = 175, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,{d}^{2}f} \left ( d\cot \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{\sqrt{2}}{4\,f}\ln \left ({ \left ( d\cot \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\cot \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\cot \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\cot \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{\sqrt{2}}{2\,f}\arctan \left ({\sqrt{2}\sqrt{d\cot \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{\sqrt{2}}{2\,f}\arctan \left ( -{\sqrt{2}\sqrt{d\cot \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(d*cot(f*x+e))^(3/2)/d^2/f+1/4/f/(d^2)^(1/4)*2^(1/2)*ln((d*cot(f*x+e)-(d^2)^(1/4)*(d*cot(f*x+e))^(1/2)*2^
(1/2)+(d^2)^(1/2))/(d*cot(f*x+e)+(d^2)^(1/4)*(d*cot(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+1/2/f/(d^2)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(d^2)^(1/4)*(d*cot(f*x+e))^(1/2)+1)-1/2/f/(d^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(d^2)^(1/4)*(
d*cot(f*x+e))^(1/2)+1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [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)^3/(d*cot(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Integral(cot(e + f*x)**3/sqrt(d*cot(e + f*x)), x)

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

[Out]

integrate(cot(f*x + e)^3/sqrt(d*cot(f*x + e)), x)