3.51 \(\int \cot ^4(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=69 \[ -\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (4*a^3*Cot[c + d*x]^3)/(3*d) + (3*a^3*Csc[c + d*x])/d - (4*a^3*Csc[c + d*x]^3)/
(3*d)

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Rubi [A]  time = 0.132014, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3886, 3473, 8, 2606, 2607, 30} \[ -\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{4 a^3 \csc ^3(c+d x)}{3 d}+\frac{3 a^3 \csc (c+d x)}{d}+a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]

[Out]

a^3*x + (a^3*Cot[c + d*x])/d - (4*a^3*Cot[c + d*x]^3)/(3*d) + (3*a^3*Csc[c + d*x])/d - (4*a^3*Csc[c + d*x]^3)/
(3*d)

Rule 3886

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.233152, size = 112, normalized size = 1.62 \[ \frac{a^3 \csc \left (\frac{c}{2}\right ) \csc ^3\left (\frac{1}{2} (c+d x)\right ) \left (-18 \sin \left (c+\frac{d x}{2}\right )+14 \sin \left (c+\frac{3 d x}{2}\right )-9 d x \cos \left (c+\frac{d x}{2}\right )-3 d x \cos \left (c+\frac{3 d x}{2}\right )+3 d x \cos \left (2 c+\frac{3 d x}{2}\right )-24 \sin \left (\frac{d x}{2}\right )+9 d x \cos \left (\frac{d x}{2}\right )\right )}{24 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*Csc[c/2]*Csc[(c + d*x)/2]^3*(9*d*x*Cos[(d*x)/2] - 9*d*x*Cos[c + (d*x)/2] - 3*d*x*Cos[c + (3*d*x)/2] + 3*d
*x*Cos[2*c + (3*d*x)/2] - 24*Sin[(d*x)/2] - 18*Sin[c + (d*x)/2] + 14*Sin[c + (3*d*x)/2]))/(24*d)

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Maple [A]  time = 0.065, size = 125, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}+\cot \left ( dx+c \right ) +dx+c \right ) +3\,{a}^{3} \left ( -1/3\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+1/3\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{\sin \left ( dx+c \right ) }}+1/3\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{3}}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+a*sec(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+3*a^3*(-1/3/sin(d*x+c)^3*cos(d*x+c)^4+1/3/sin(d*x+c)*cos(d*x+c)^
4+1/3*(2+cos(d*x+c)^2)*sin(d*x+c))-a^3/sin(d*x+c)^3*cos(d*x+c)^3-1/3*a^3/sin(d*x+c)^3)

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Maxima [A]  time = 1.79234, size = 122, normalized size = 1.77 \begin{align*} \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^{3} + \frac{3 \,{\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{3}}{\sin \left (d x + c\right )^{3}} - \frac{a^{3}}{\sin \left (d x + c\right )^{3}} - \frac{3 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/3*((3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 + 3*(3*sin(d*x + c)^2 - 1)*a^3/sin(d*x + c)^3 -
a^3/sin(d*x + c)^3 - 3*a^3/tan(d*x + c)^3)/d

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Fricas [A]  time = 1.11453, size = 193, normalized size = 2.8 \begin{align*} \frac{7 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \cos \left (d x + c\right ) - 5 \, a^{3} + 3 \,{\left (a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3*(7*a^3*cos(d*x + c)^2 + 2*a^3*cos(d*x + c) - 5*a^3 + 3*(a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/((d*
cos(d*x + c) - d)*sin(d*x + c))

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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)**4*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.50818, size = 68, normalized size = 0.99 \begin{align*} \frac{3 \,{\left (d x + c\right )} a^{3} + \frac{6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/3*(3*(d*x + c)*a^3 + (6*a^3*tan(1/2*d*x + 1/2*c)^2 - a^3)/tan(1/2*d*x + 1/2*c)^3)/d