3.687 \(\int \frac{\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=97 \[ -\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{2 \csc (c+d x)}{a d}+\frac{2 \log (\sin (c+d x))}{a d} \]

[Out]

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

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Rubi [A]  time = 0.120062, antiderivative size = 97, 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} \[ -\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{2 \csc (c+d x)}{a d}+\frac{2 \log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) + (2*Log[Sin[c + d*x]])/(a*d) + Sin[c
+ d*x]/(a*d) - Sin[c + d*x]^2/(2*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 ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a-x)^3 (a+x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a+\frac{a^5}{x^4}-\frac{a^4}{x^3}-\frac{2 a^3}{x^2}+\frac{2 a^2}{x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{2 \csc (c+d x)}{a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{2 \log (\sin (c+d x))}{a d}+\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}\\ \end{align*}

Mathematica [A]  time = 0.17654, size = 66, normalized size = 0.68 \[ \frac{-3 \sin ^2(c+d x)+6 \sin (c+d x)-2 \csc ^3(c+d x)+3 \csc ^2(c+d x)+12 \csc (c+d x)+12 \log (\sin (c+d x))}{6 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(12*Csc[c + d*x] + 3*Csc[c + d*x]^2 - 2*Csc[c + d*x]^3 + 12*Log[Sin[c + d*x]] + 6*Sin[c + d*x] - 3*Sin[c + d*x
]^2)/(6*a*d)

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Maple [A]  time = 0.129, size = 94, normalized size = 1. \begin{align*} -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,da}}+{\frac{\sin \left ( dx+c \right ) }{da}}+2\,{\frac{1}{da\sin \left ( dx+c \right ) }}+2\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\,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)^4/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.02851, size = 99, normalized size = 1.02 \begin{align*} -\frac{\frac{3 \,{\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} - \frac{12 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac{12 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/6*(3*(sin(d*x + c)^2 - 2*sin(d*x + c))/a - 12*log(sin(d*x + c))/a - (12*sin(d*x + c)^2 + 3*sin(d*x + c) - 2
)/(a*sin(d*x + c)^3))/d

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Fricas [A]  time = 1.16546, size = 289, normalized size = 2.98 \begin{align*} -\frac{12 \, \cos \left (d x + c\right )^{4} - 24 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 48 \, \cos \left (d x + c\right )^{2} - 3 \,{\left (2 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 32}{12 \,{\left (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)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/12*(12*cos(d*x + c)^4 - 24*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c))*sin(d*x + c) - 48*cos(d*x + c)^2 - 3*
(2*cos(d*x + c)^4 - 3*cos(d*x + c)^2 - 1)*sin(d*x + c) + 32)/((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)**7*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.29019, size = 117, normalized size = 1.21 \begin{align*} \frac{\frac{12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{3 \,{\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac{22 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/6*(12*log(abs(sin(d*x + c)))/a - 3*(a*sin(d*x + c)^2 - 2*a*sin(d*x + c))/a^2 - (22*sin(d*x + c)^3 - 12*sin(d
*x + c)^2 - 3*sin(d*x + c) + 2)/(a*sin(d*x + c)^3))/d