3.507 \(\int \cot ^5(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=81 \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d} \]

[Out]

-(a*Cot[c + d*x]^6)/(6*d) - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) -
(a*Csc[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.121172, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2607, 14, 2606, 270} \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^6)/(6*d) - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) -
(a*Csc[c + d*x]^7)/(7*d)

Rule 2834

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 270

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

Rubi steps

\begin{align*} \int \cot ^5(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.155217, size = 88, normalized size = 1.09 \[ -\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \left (3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right )}{24 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*(6*Csc[c + d*x]^4 - 8*C
sc[c + d*x]^6 + 3*Csc[c + d*x]^8))/(24*d)

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Maple [B]  time = 0.066, size = 148, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{35} \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 ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^9*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/7/sin(d*x+c)^7*cos(d*x+c)^6-1/35/sin(d*x+c)^5*cos(d*x+c)^6+1/105/sin(d*x+c)^3*cos(d*x+c)^6-1/35/sin
(d*x+c)*cos(d*x+c)^6-1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+a*(-1/8/sin(d*x+c)^8*cos(d*x+c)^6-1/
24/sin(d*x+c)^6*cos(d*x+c)^6))

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Maxima [A]  time = 1.08151, size = 95, normalized size = 1.17 \begin{align*} -\frac{280 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, a \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, a \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/840*(280*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 336*a*sin(d*x + c)^3 - 280*a*sin(d*x + c)^2 + 120*a*sin(
d*x + c) + 105*a)/(d*sin(d*x + c)^8)

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Fricas [A]  time = 1.0961, size = 289, normalized size = 3.57 \begin{align*} -\frac{210 \, a \cos \left (d x + c\right )^{4} - 140 \, a \cos \left (d x + c\right )^{2} + 8 \,{\left (35 \, a \cos \left (d x + c\right )^{4} - 28 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 35 \, a}{840 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/840*(210*a*cos(d*x + c)^4 - 140*a*cos(d*x + c)^2 + 8*(35*a*cos(d*x + c)^4 - 28*a*cos(d*x + c)^2 + 8*a)*sin(
d*x + c) + 35*a)/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + 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(cos(d*x+c)**5*csc(d*x+c)**9*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.21974, size = 95, normalized size = 1.17 \begin{align*} -\frac{280 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, a \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, a \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/840*(280*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 336*a*sin(d*x + c)^3 - 280*a*sin(d*x + c)^2 + 120*a*sin(
d*x + c) + 105*a)/(d*sin(d*x + c)^8)