3.543 \(\int \frac{\cot ^5(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\csc ^7(c+d x)}{7 a d}+\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{\csc ^4(c+d x)}{4 a d} \]

[Out]

-Csc[c + d*x]^4/(4*a*d) + Csc[c + d*x]^5/(5*a*d) + Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d)

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Rubi [A]  time = 0.10912, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 75} \[ -\frac{\csc ^7(c+d x)}{7 a d}+\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{\csc ^4(c+d x)}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[c + d*x]^4/(4*a*d) + Csc[c + d*x]^5/(5*a*d) + Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d)

Rule 2836

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.10259, size = 48, normalized size = 0.66 \[ \frac{\csc ^4(c+d x) \left (-60 \csc ^3(c+d x)+70 \csc ^2(c+d x)+84 \csc (c+d x)-105\right )}{420 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^4*(-105 + 84*Csc[c + d*x] + 70*Csc[c + d*x]^2 - 60*Csc[c + d*x]^3))/(420*a*d)

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Maple [A]  time = 0.155, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{da} \left ( -{\frac{1}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{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+a*sin(d*x+c)),x)

[Out]

1/d/a*(-1/7/sin(d*x+c)^7+1/5/sin(d*x+c)^5-1/4/sin(d*x+c)^4+1/6/sin(d*x+c)^6)

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Maxima [A]  time = 1.03947, size = 62, normalized size = 0.85 \begin{align*} -\frac{105 \, \sin \left (d x + c\right )^{3} - 84 \, \sin \left (d x + c\right )^{2} - 70 \, \sin \left (d x + c\right ) + 60}{420 \, a 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+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/420*(105*sin(d*x + c)^3 - 84*sin(d*x + c)^2 - 70*sin(d*x + c) + 60)/(a*d*sin(d*x + c)^7)

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Fricas [A]  time = 1.07927, size = 217, normalized size = 2.97 \begin{align*} \frac{84 \, \cos \left (d x + c\right )^{2} - 35 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 24}{420 \,{\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 )} \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+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/420*(84*cos(d*x + c)^2 - 35*(3*cos(d*x + c)^2 - 1)*sin(d*x + c) - 24)/((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)*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+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.25472, size = 62, normalized size = 0.85 \begin{align*} -\frac{105 \, \sin \left (d x + c\right )^{3} - 84 \, \sin \left (d x + c\right )^{2} - 70 \, \sin \left (d x + c\right ) + 60}{420 \, a 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+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/420*(105*sin(d*x + c)^3 - 84*sin(d*x + c)^2 - 70*sin(d*x + c) + 60)/(a*d*sin(d*x + c)^7)