3.61 \(\int \frac{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=322 \[ -\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}} \]

[Out]

((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) - ((a^2 - 2*a*b -
b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (2*a^2)/(5*d*e*(e*Cot[c + d*x]
)^(5/2)) + (4*a*b)/(3*d*e^2*(e*Cot[c + d*x])^(3/2)) - (2*(a^2 - b^2))/(d*e^3*Sqrt[e*Cot[c + d*x]]) - ((a^2 + 2
*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) + ((a^2
+ 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.427952, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3542, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]

[Out]

((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) - ((a^2 - 2*a*b -
b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (2*a^2)/(5*d*e*(e*Cot[c + d*x]
)^(5/2)) + (4*a*b)/(3*d*e^2*(e*Cot[c + d*x])^(3/2)) - (2*(a^2 - b^2))/(d*e^3*Sqrt[e*Cot[c + d*x]]) - ((a^2 + 2
*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) + ((a^2
+ 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2))

Rule 3542

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Ta
n[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Tan[e + f*x], x], x], x] /; FreeQ
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 1168

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

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{(a+b \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{\int \frac{2 a b e-\left (a^2-b^2\right ) e \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx}{e^2}\\ &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}+\frac{\int \frac{-\left (a^2-b^2\right ) e^2-2 a b e^2 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{e^4}\\ &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}+\frac{\int \frac{-2 a b e^3+\left (a^2-b^2\right ) e^3 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{e^6}\\ &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{2 a b e^4-\left (a^2-b^2\right ) e^3 x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e^6}\\ &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e^3}+\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{d e^3}\\ &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{7/2}}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{7/2}}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 d e^3}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 d e^3}\\ &=\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{7/2}}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}\\ &=\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac{4 a b}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac{2 \left (a^2-b^2\right )}{d e^3 \sqrt{e \cot (c+d x)}}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} d e^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.35641, size = 85, normalized size = 0.26 \[ \frac{2 \left (3 \left (a^2-b^2\right ) \text{Hypergeometric2F1}\left (-\frac{5}{4},1,-\frac{1}{4},-\cot ^2(c+d x)\right )+b \left (10 a \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{4},1,\frac{1}{4},-\cot ^2(c+d x)\right )+3 b\right )\right )}{15 d e (e \cot (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]

[Out]

(2*(3*(a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + b*(3*b + 10*a*Cot[c + d*x]*Hypergeometri
c2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*e*(e*Cot[c + d*x])^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.027, size = 600, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x)

[Out]

1/2/e^4/d*a*b*(e^2)^(1/4)*2^(1/2)*ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*co
t(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+1/e^4/d*a*b*(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-1/e^4/d*a*b*(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c)
)^(1/2)+1)-1/4/e^3/d/(e^2)^(1/4)*2^(1/2)*ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)
)/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))*a^2+1/4/e^3/d/(e^2)^(1/4)*2^(1/2)*ln((e
*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/
2)*2^(1/2)+(e^2)^(1/2)))*b^2-1/2/e^3/d/(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*
a^2+1/2/e^3/d/(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*b^2+1/2/e^3/d/(e^2)^(1/4)
*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*a^2-1/2/e^3/d/(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)
/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*b^2+2/5*a^2/d/e/(e*cot(d*x+c))^(5/2)-2/e^3/d/(e*cot(d*x+c))^(1/2)*a^2+2/e
^3/d/(e*cot(d*x+c))^(1/2)*b^2+4/3*a*b/d/e^2/(e*cot(d*x+c))^(3/2)

________________________________________________________________________________________

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((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [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((a+b*cot(d*x+c))**2/(e*cot(d*x+c))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((b*cot(d*x + c) + a)^2/(e*cot(d*x + c))^(7/2), x)