3.44 \(\int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=61 \[ \frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3}{2 d (1-\cos (c+d x))^2}+\frac{a^3 \log (1-\cos (c+d x))}{d} \]

[Out]

-a^3/(2*d*(1 - Cos[c + d*x])^2) + (2*a^3)/(d*(1 - Cos[c + d*x])) + (a^3*Log[1 - Cos[c + d*x]])/d

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Rubi [A]  time = 0.058681, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 43} \[ \frac{2 a^3}{d (1-\cos (c+d x))}-\frac{a^3}{2 d (1-\cos (c+d x))^2}+\frac{a^3 \log (1-\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]

[Out]

-a^3/(2*d*(1 - Cos[c + d*x])^2) + (2*a^3)/(d*(1 - Cos[c + d*x])) + (a^3*Log[1 - Cos[c + d*x]])/d

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cot ^5(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^2}{(a-a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{a^3 (-1+x)^3}-\frac{2}{a^3 (-1+x)^2}-\frac{1}{a^3 (-1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^3}{2 d (1-\cos (c+d x))^2}+\frac{2 a^3}{d (1-\cos (c+d x))}+\frac{a^3 \log (1-\cos (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.170142, size = 72, normalized size = 1.18 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (\csc ^4\left (\frac{1}{2} (c+d x)\right )-8 \csc ^2\left (\frac{1}{2} (c+d x)\right )-16 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{64 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]

[Out]

-(a^3*(1 + Cos[c + d*x])^3*(-8*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 - 16*Log[Sin[(c + d*x)/2]])*Sec[(c + d*
x)/2]^6)/(64*d)

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Maple [A]  time = 0.077, size = 68, normalized size = 1.1 \begin{align*}{\frac{{a}^{3}}{d \left ( -1+\sec \left ( dx+c \right ) \right ) }}-{\frac{{a}^{3}}{2\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x)

[Out]

1/d*a^3/(-1+sec(d*x+c))-1/2/d*a^3/(-1+sec(d*x+c))^2+1/d*a^3*ln(-1+sec(d*x+c))-1/d*a^3*ln(sec(d*x+c))

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Maxima [A]  time = 1.37319, size = 80, normalized size = 1.31 \begin{align*} \frac{2 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{4 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3}}{\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.02003, size = 213, normalized size = 3.49 \begin{align*} -\frac{4 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3} - 2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [B]  time = 1.51033, size = 186, normalized size = 3.05 \begin{align*} \frac{8 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 8 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (a^{3} + \frac{6 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{12 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(8*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 8*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) + 1)) - (a^3 + 6*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 12*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2)*(cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2)/d