3.688 \(\int \frac{\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=94 \[ -\frac{\sin (c+d x)}{a d}-\frac{\csc ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{a d}-\frac{2 \csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]

[Out]

(-2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d) + Log[Sin[c +
d*x]]/(a*d) - Sin[c + d*x]/(a*d)

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Rubi [A]  time = 0.119448, antiderivative size = 94, 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, 88} \[ -\frac{\sin (c+d x)}{a d}-\frac{\csc ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{a d}-\frac{2 \csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^2*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

(-2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d) + Log[Sin[c +
d*x]]/(a*d) - Sin[c + d*x]/(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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.315159, size = 66, normalized size = 0.7 \[ -\frac{12 \sin (c+d x)+3 \csc ^4(c+d x)-4 \csc ^3(c+d x)-12 \csc ^2(c+d x)+24 \csc (c+d x)-12 \log (\sin (c+d x))}{12 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

-(24*Csc[c + d*x] - 12*Csc[c + d*x]^2 - 4*Csc[c + d*x]^3 + 3*Csc[c + d*x]^4 - 12*Log[Sin[c + d*x]] + 12*Sin[c
+ d*x])/(12*a*d)

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Maple [A]  time = 0.133, size = 93, normalized size = 1. \begin{align*} -{\frac{\sin \left ( dx+c \right ) }{da}}-2\,{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{1}{4\,da \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}+{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.00428, size = 97, normalized size = 1.03 \begin{align*} \frac{\frac{12 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac{12 \, \sin \left (d x + c\right )}{a} - \frac{24 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{a \sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(12*log(sin(d*x + c))/a - 12*sin(d*x + c)/a - (24*sin(d*x + c)^3 - 12*sin(d*x + c)^2 - 4*sin(d*x + c) + 3
)/(a*sin(d*x + c)^4))/d

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Fricas [A]  time = 1.15473, size = 281, normalized size = 2.99 \begin{align*} -\frac{12 \, \cos \left (d x + c\right )^{2} - 12 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 9}{12 \,{\left (a d \cos \left (d x + c\right )^{4} - 2 \, 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(cos(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/12*(12*cos(d*x + c)^2 - 12*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*sin(d*x + c)) + 4*(3*cos(d*x + c
)^4 - 12*cos(d*x + c)^2 + 8)*sin(d*x + c) - 9)/(a*d*cos(d*x + c)^4 - 2*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(cos(d*x+c)**7*csc(d*x+c)**5/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.32078, size = 112, normalized size = 1.19 \begin{align*} \frac{\frac{12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{12 \, \sin \left (d x + c\right )}{a} - \frac{25 \, \sin \left (d x + c\right )^{4} + 24 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{a \sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/12*(12*log(abs(sin(d*x + c)))/a - 12*sin(d*x + c)/a - (25*sin(d*x + c)^4 + 24*sin(d*x + c)^3 - 12*sin(d*x +
c)^2 - 4*sin(d*x + c) + 3)/(a*sin(d*x + c)^4))/d