3.269 \(\int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx\)

Optimal. Leaf size=84 \[ -\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x \]

[Out]

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

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Rubi [A]  time = 0.0811312, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}-a x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(a + b*Sec[c + d*x]),x]

[Out]

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

Rule 3882

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[((e*Cot[c
+ d*x])^(m + 1)*(a + b*Csc[c + d*x]))/(d*e*(m + 1)), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)*(
a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 8

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

Rubi steps

\begin{align*} \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{1}{5} \int \cot ^4(c+d x) (-5 a-4 b \sec (c+d x)) \, dx\\ &=-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}+\frac{1}{15} \int \cot ^2(c+d x) (15 a+8 b \sec (c+d x)) \, dx\\ &=-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}+\frac{1}{15} \int -15 a \, dx\\ &=-a x-\frac{\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac{\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac{\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d}\\ \end{align*}

Mathematica [C]  time = 0.0367839, size = 79, normalized size = 0.94 \[ -\frac{a \cot ^5(c+d x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(c+d x)\right )}{5 d}-\frac{b \csc ^5(c+d x)}{5 d}+\frac{2 b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(a + b*Sec[c + d*x]),x]

[Out]

-((b*Csc[c + d*x])/d) + (2*b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^5*Hypergeometr
ic2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d)

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Maple [A]  time = 0.045, size = 129, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}-\cot \left ( dx+c \right ) -dx-c \right ) +b \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{15\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{5\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+b*sec(d*x+c)),x)

[Out]

1/d*(a*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+b*(-1/5/sin(d*x+c)^5*cos(d*x+c)^6+1/15/sin(d*x+c)
^3*cos(d*x+c)^6-1/5/sin(d*x+c)*cos(d*x+c)^6-1/5*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))

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Maxima [A]  time = 1.46946, size = 107, normalized size = 1.27 \begin{align*} -\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 )} a + \frac{{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} b}{\sin \left (d x + c\right )^{5}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

-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)*a + (15*sin(d*x + c)^4 - 10
*sin(d*x + c)^2 + 3)*b/sin(d*x + c)^5)/d

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Fricas [A]  time = 0.806891, size = 343, normalized size = 4.08 \begin{align*} -\frac{23 \, a \cos \left (d x + c\right )^{5} + 15 \, b \cos \left (d x + c\right )^{4} - 35 \, a \cos \left (d x + c\right )^{3} - 20 \, b \cos \left (d x + c\right )^{2} + 15 \, a \cos \left (d x + c\right ) + 15 \,{\left (a d x \cos \left (d x + c\right )^{4} - 2 \, a d x \cos \left (d x + c\right )^{2} + a d x\right )} \sin \left (d x + c\right ) + 8 \, b}{15 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + 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)^6*(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/15*(23*a*cos(d*x + c)^5 + 15*b*cos(d*x + c)^4 - 35*a*cos(d*x + c)^3 - 20*b*cos(d*x + c)^2 + 15*a*cos(d*x +
c) + 15*(a*d*x*cos(d*x + c)^4 - 2*a*d*x*cos(d*x + c)^2 + a*d*x)*sin(d*x + c) + 8*b)/((d*cos(d*x + c)^4 - 2*d*c
os(d*x + c)^2 + 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)**6*(a+b*sec(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.28717, size = 230, normalized size = 2.74 \begin{align*} \frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 25 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 480 \,{\left (d x + c\right )} a + 330 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 150 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{330 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 150 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 25 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a + 3 \, b}{\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(cot(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

1/480*(3*a*tan(1/2*d*x + 1/2*c)^5 - 3*b*tan(1/2*d*x + 1/2*c)^5 - 35*a*tan(1/2*d*x + 1/2*c)^3 + 25*b*tan(1/2*d*
x + 1/2*c)^3 - 480*(d*x + c)*a + 330*a*tan(1/2*d*x + 1/2*c) - 150*b*tan(1/2*d*x + 1/2*c) - (330*a*tan(1/2*d*x
+ 1/2*c)^4 + 150*b*tan(1/2*d*x + 1/2*c)^4 - 35*a*tan(1/2*d*x + 1/2*c)^2 - 25*b*tan(1/2*d*x + 1/2*c)^2 + 3*a +
3*b)/tan(1/2*d*x + 1/2*c)^5)/d