3.805 \(\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=268 \[ \frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a b \sqrt{\cot (c+d x)}}{d} \]

[Out]

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

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Rubi [A]  time = 0.28773, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3673, 3543, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a b \sqrt{\cot (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^2,x]

[Out]

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

Rule 3673

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rule 3543

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

Rule 3528

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

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 \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx &=\int \cot ^{\frac{5}{2}}(c+d x) (b+a \cot (c+d x))^2 \, dx\\ &=-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}+\int \cot ^{\frac{5}{2}}(c+d x) \left (-a^2+b^2+2 a b \cot (c+d x)\right ) \, dx\\ &=-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}+\int \cot ^{\frac{3}{2}}(c+d x) \left (-2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right ) \, dx\\ &=\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}+\int \sqrt{\cot (c+d x)} \left (a^2-b^2-2 a b \cot (c+d x)\right ) \, dx\\ &=\frac{4 a b \sqrt{\cot (c+d x)}}{d}+\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}+\int \frac{2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{4 a b \sqrt{\cot (c+d x)}}{d}+\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 \operatorname{Subst}\left (\int \frac{-2 a b+\left (-a^2+b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{4 a b \sqrt{\cot (c+d x)}}{d}+\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{4 a b \sqrt{\cot (c+d x)}}{d}+\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \sqrt{2} d}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \sqrt{2} d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 d}\\ &=\frac{4 a b \sqrt{\cot (c+d x)}}{d}+\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}-\frac{\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}\\ &=\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{4 a b \sqrt{\cot (c+d x)}}{d}+\frac{2 \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}-\frac{\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}+\frac{\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}\\ \end{align*}

Mathematica [C]  time = 1.46909, size = 215, normalized size = 0.8 \[ -\frac{\frac{2}{3} \left (a^2-b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right )-1\right )+\frac{2}{7} a^2 \cot ^{\frac{7}{2}}(c+d x)+\frac{1}{10} a b \left (8 \cot ^{\frac{5}{2}}(c+d x)-40 \sqrt{\cot (c+d x)}-5 \sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+5 \sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )+10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^2,x]

[Out]

-(((2*a^2*Cot[c + d*x]^(7/2))/7 + (2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[
c + d*x]^2]))/3 + (a*b*(-10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqr
t[Cot[c + d*x]]] - 40*Sqrt[Cot[c + d*x]] + 8*Cot[c + d*x]^(5/2) - 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]
+ Cot[c + d*x]] + 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/10)/d)

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Maple [C]  time = 0.458, size = 7157, normalized size = 26.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x)

[Out]

result too large to display

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Maxima [A]  time = 1.68228, size = 298, normalized size = 1.11 \begin{align*} -\frac{210 \, \sqrt{2}{\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt{2}{\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt{2}{\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt{2}{\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) - \frac{1680 \, a b}{\sqrt{\tan \left (d x + c\right )}} + \frac{336 \, a b}{\tan \left (d x + c\right )^{\frac{5}{2}}} - \frac{280 \,{\left (a^{2} - b^{2}\right )}}{\tan \left (d x + c\right )^{\frac{3}{2}}} + \frac{120 \, a^{2}}{\tan \left (d x + c\right )^{\frac{7}{2}}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/420*(210*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 210*sqrt(2)*(a^
2 + 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 105*sqrt(2)*(a^2 - 2*a*b - b^2)*log(s
qrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 105*sqrt(2)*(a^2 - 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x +
c)) + 1/tan(d*x + c) + 1) - 1680*a*b/sqrt(tan(d*x + c)) + 336*a*b/tan(d*x + c)^(5/2) - 280*(a^2 - b^2)/tan(d*x
+ c)^(3/2) + 120*a^2/tan(d*x + c)^(7/2))/d

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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(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

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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(cot(d*x+c)**(9/2)*(a+b*tan(d*x+c))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c) + a)^2*cot(d*x + c)^(9/2), x)