3.1210 \(\int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx\)

Optimal. Leaf size=65 \[ -\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{b \cot ^6(c+d x)}{6 d} \]

[Out]

-(b*Cot[c + d*x]^6)/(6*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 = 0.12248, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 30} \[ -\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{b \cot ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*Cot[c + d*x]^6)/(6*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 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]

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 ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx+b \int \cot ^5(c+d x) \csc ^2(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{b \operatorname{Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{b \cot ^6(c+d x)}{6 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{b \cot ^6(c+d x)}{6 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 = 0.026354, size = 65, normalized size = 1. \[ -\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{b \cot ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*Cot[c + d*x]^6)/(6*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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Maple [B]  time = 0.064, size = 128, normalized size = 2. \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 ) -{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^8*(a+b*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))-1/6*b/sin(d*x+c)^6*cos(d*x+c)^6)

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Maxima [A]  time = 0.988483, size = 95, normalized size = 1.46 \begin{align*} -\frac{105 \, b \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, b \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, b \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^8*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/210*(105*b*sin(d*x + c)^5 + 70*a*sin(d*x + c)^4 - 105*b*sin(d*x + c)^3 - 84*a*sin(d*x + c)^2 + 35*b*sin(d*x
+ c) + 30*a)/(d*sin(d*x + c)^7)

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Fricas [A]  time = 1.62298, size = 273, normalized size = 4.2 \begin{align*} \frac{70 \, a \cos \left (d x + c\right )^{4} - 56 \, a \cos \left (d x + c\right )^{2} + 35 \,{\left (3 \, b \cos \left (d x + c\right )^{4} - 3 \, b \cos \left (d x + c\right )^{2} + b\right )} \sin \left (d x + c\right ) + 16 \, a}{210 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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(cos(d*x+c)^5*csc(d*x+c)^8*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/210*(70*a*cos(d*x + c)^4 - 56*a*cos(d*x + c)^2 + 35*(3*b*cos(d*x + c)^4 - 3*b*cos(d*x + c)^2 + b)*sin(d*x +
c) + 16*a)/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(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(cos(d*x+c)**5*csc(d*x+c)**8*(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.17815, size = 95, normalized size = 1.46 \begin{align*} -\frac{105 \, b \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, b \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, b \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^8*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/210*(105*b*sin(d*x + c)^5 + 70*a*sin(d*x + c)^4 - 105*b*sin(d*x + c)^3 - 84*a*sin(d*x + c)^2 + 35*b*sin(d*x
+ c) + 30*a)/(d*sin(d*x + c)^7)