3.4 \(\int (a+b \cot ^2(c+d x))^3 \, dx\)

Optimal. Leaf size=78 \[ -\frac{b \left (3 a^2-3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{b^2 (3 a-b) \cot ^3(c+d x)}{3 d}+x (a-b)^3-\frac{b^3 \cot ^5(c+d x)}{5 d} \]

[Out]

(a - b)^3*x - (b*(3*a^2 - 3*a*b + b^2)*Cot[c + d*x])/d - ((3*a - b)*b^2*Cot[c + d*x]^3)/(3*d) - (b^3*Cot[c + d
*x]^5)/(5*d)

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Rubi [A]  time = 0.0472866, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3661, 390, 203} \[ -\frac{b \left (3 a^2-3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{b^2 (3 a-b) \cot ^3(c+d x)}{3 d}+x (a-b)^3-\frac{b^3 \cot ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x]^2)^3,x]

[Out]

(a - b)^3*x - (b*(3*a^2 - 3*a*b + b^2)*Cot[c + d*x])/d - ((3*a - b)*b^2*Cot[c + d*x]^3)/(3*d) - (b^3*Cot[c + d
*x]^5)/(5*d)

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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])

Rubi steps

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

Mathematica [A]  time = 2.75154, size = 111, normalized size = 1.42 \[ -\frac{\cot ^5(c+d x) \left (b \left (15 \left (3 a^2-3 a b+b^2\right ) \tan ^4(c+d x)+5 b (3 a-b) \tan ^2(c+d x)+3 b^2\right )+\frac{15 (a-b)^3 \tan ^8(c+d x) \tanh ^{-1}\left (\sqrt{-\tan ^2(c+d x)}\right )}{\left (-\tan ^2(c+d x)\right )^{3/2}}\right )}{15 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cot[c + d*x]^2)^3,x]

[Out]

-(Cot[c + d*x]^5*((15*(a - b)^3*ArcTanh[Sqrt[-Tan[c + d*x]^2]]*Tan[c + d*x]^8)/(-Tan[c + d*x]^2)^(3/2) + b*(3*
b^2 + 5*(3*a - b)*b*Tan[c + d*x]^2 + 15*(3*a^2 - 3*a*b + b^2)*Tan[c + d*x]^4)))/(15*d)

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Maple [A]  time = 0.004, size = 116, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -{\frac{{b}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}- \left ( \cot \left ( dx+c \right ) \right ) ^{3}a{b}^{2}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}{b}^{3}}{3}}-3\,\cot \left ( dx+c \right ){a}^{2}b+3\,a{b}^{2}\cot \left ( dx+c \right ) -{b}^{3}\cot \left ( dx+c \right ) + \left ( -{a}^{3}+3\,{a}^{2}b-3\,a{b}^{2}+{b}^{3} \right ) \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( dx+c \right ) \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cot(d*x+c)^2)^3,x)

[Out]

1/d*(-1/5*b^3*cot(d*x+c)^5-cot(d*x+c)^3*a*b^2+1/3*cot(d*x+c)^3*b^3-3*cot(d*x+c)*a^2*b+3*a*b^2*cot(d*x+c)-b^3*c
ot(d*x+c)+(-a^3+3*a^2*b-3*a*b^2+b^3)*(1/2*Pi-arccot(cot(d*x+c))))

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Maxima [A]  time = 1.46974, size = 151, normalized size = 1.94 \begin{align*} a^{3} x - \frac{3 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{2} b}{d} + \frac{{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a b^{2}}{d} - \frac{{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} b^{3}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

a^3*x - 3*(d*x + c + 1/tan(d*x + c))*a^2*b/d + (3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a*b^2/d -
1/15*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*b^3/d

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Fricas [B]  time = 1.6678, size = 583, normalized size = 7.47 \begin{align*} -\frac{{\left (45 \, a^{2} b - 60 \, a b^{2} + 23 \, b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )^{3} + 45 \, a^{2} b - 30 \, a b^{2} + 13 \, b^{3} -{\left (45 \, a^{2} b - 30 \, a b^{2} + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} -{\left (45 \, a^{2} b - 60 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right ) - 15 \,{\left ({\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x \cos \left (2 \, d x + 2 \, c\right ) +{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x\right )} \sin \left (2 \, d x + 2 \, c\right )}{15 \,{\left (d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x + 2 \, c\right ) + d\right )} \sin \left (2 \, d x + 2 \, c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/15*((45*a^2*b - 60*a*b^2 + 23*b^3)*cos(2*d*x + 2*c)^3 + 45*a^2*b - 30*a*b^2 + 13*b^3 - (45*a^2*b - 30*a*b^2
+ b^3)*cos(2*d*x + 2*c)^2 - (45*a^2*b - 60*a*b^2 + 11*b^3)*cos(2*d*x + 2*c) - 15*((a^3 - 3*a^2*b + 3*a*b^2 -
b^3)*d*x*cos(2*d*x + 2*c)^2 - 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*x*cos(2*d*x + 2*c) + (a^3 - 3*a^2*b + 3*a*b^
2 - b^3)*d*x)*sin(2*d*x + 2*c))/((d*cos(2*d*x + 2*c)^2 - 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c))

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Sympy [A]  time = 0.698127, size = 126, normalized size = 1.62 \begin{align*} \begin{cases} a^{3} x - 3 a^{2} b x - \frac{3 a^{2} b \cot{\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac{a b^{2} \cot ^{3}{\left (c + d x \right )}}{d} + \frac{3 a b^{2} \cot{\left (c + d x \right )}}{d} - b^{3} x - \frac{b^{3} \cot ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{3} \cot ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \cot{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cot ^{2}{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c)**2)**3,x)

[Out]

Piecewise((a**3*x - 3*a**2*b*x - 3*a**2*b*cot(c + d*x)/d + 3*a*b**2*x - a*b**2*cot(c + d*x)**3/d + 3*a*b**2*co
t(c + d*x)/d - b**3*x - b**3*cot(c + d*x)**5/(5*d) + b**3*cot(c + d*x)**3/(3*d) - b**3*cot(c + d*x)/d, Ne(d, 0
)), (x*(a + b*cot(c)**2)**3, True))

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Giac [B]  time = 1.24844, size = 309, normalized size = 3.96 \begin{align*} \frac{3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 60 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 720 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 900 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 330 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 480 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}{\left (d x + c\right )} - \frac{720 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 900 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 330 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 60 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 35 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, b^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/480*(3*b^3*tan(1/2*d*x + 1/2*c)^5 + 60*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 35*b^3*tan(1/2*d*x + 1/2*c)^3 + 720*a^
2*b*tan(1/2*d*x + 1/2*c) - 900*a*b^2*tan(1/2*d*x + 1/2*c) + 330*b^3*tan(1/2*d*x + 1/2*c) + 480*(a^3 - 3*a^2*b
+ 3*a*b^2 - b^3)*(d*x + c) - (720*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 900*a*b^2*tan(1/2*d*x + 1/2*c)^4 + 330*b^3*ta
n(1/2*d*x + 1/2*c)^4 + 60*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 35*b^3*tan(1/2*d*x + 1/2*c)^2 + 3*b^3)/tan(1/2*d*x +
1/2*c)^5)/d