3.51 \(\int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d} \]

[Out]

-Cot[c + d*x]^6/(6*a*d) + Csc[c + d*x]/(a*d) - (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^5/(5*a*d)

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Rubi [A]  time = 0.0933341, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2606, 194} \[ -\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x]),x]

[Out]

-Cot[c + d*x]^6/(6*a*d) + Csc[c + d*x]/(a*d) - (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^5/(5*a*d)

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -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]

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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^5(c+d x) \csc (c+d x) \, dx}{a}+\frac{\int \cot ^5(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{a d}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc (c+d x)}{a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc ^5(c+d x)}{5 a d}\\ \end{align*}

Mathematica [A]  time = 0.149153, size = 61, normalized size = 0.9 \[ \frac{\csc ^6(c+d x) (78 \sin (c+d x)-5 (7 \sin (3 (c+d x))-3 \sin (5 (c+d x))+5)-15 \cos (4 (c+d x)))}{240 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x]),x]

[Out]

(Csc[c + d*x]^6*(-15*Cos[4*(c + d*x)] + 78*Sin[c + d*x] - 5*(5 + 7*Sin[3*(c + d*x)] - 3*Sin[5*(c + d*x)])))/(2
40*a*d)

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Maple [A]  time = 0.095, size = 67, normalized size = 1. \begin{align*}{\frac{1}{da} \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{-1}+{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{2}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+a*sin(d*x+c)),x)

[Out]

1/d/a*(1/sin(d*x+c)+1/5/sin(d*x+c)^5+1/2/sin(d*x+c)^4-1/6/sin(d*x+c)^6-2/3/sin(d*x+c)^3-1/2/sin(d*x+c)^2)

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Maxima [A]  time = 1.12848, size = 89, normalized size = 1.31 \begin{align*} \frac{30 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/30*(30*sin(d*x + c)^5 - 15*sin(d*x + c)^4 - 20*sin(d*x + c)^3 + 15*sin(d*x + c)^2 + 6*sin(d*x + c) - 5)/(a*d
*sin(d*x + c)^6)

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Fricas [A]  time = 1.41345, size = 248, normalized size = 3.65 \begin{align*} \frac{15 \, \cos \left (d x + c\right )^{4} - 15 \, \cos \left (d x + c\right )^{2} - 2 \,{\left (15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 5}{30 \,{\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/30*(15*cos(d*x + c)^4 - 15*cos(d*x + c)^2 - 2*(15*cos(d*x + c)^4 - 20*cos(d*x + c)^2 + 8)*sin(d*x + c) + 5)/
(a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*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)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.4623, size = 89, normalized size = 1.31 \begin{align*} \frac{30 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 5}{30 \, a d \sin \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/30*(30*sin(d*x + c)^5 - 15*sin(d*x + c)^4 - 20*sin(d*x + c)^3 + 15*sin(d*x + c)^2 + 6*sin(d*x + c) - 5)/(a*d
*sin(d*x + c)^6)