3.256 \(\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=131 \[ \frac{6}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{4 \csc (c+d x)}{a^4 d}+\frac{10 \log (\sin (c+d x))}{a^4 d}-\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]

[Out]

(4*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^4*d) + (10*Log[Sin[c + d*x]])/(a^4*d) - (10*Log[1 + Sin[c + d*x
]])/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 6/(d*(a^4 + a^4*Sin[c +
d*x]))

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Rubi [A]  time = 0.117688, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 44} \[ \frac{6}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{4 \csc (c+d x)}{a^4 d}+\frac{10 \log (\sin (c+d x))}{a^4 d}-\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]

[Out]

(4*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^4*d) + (10*Log[Sin[c + d*x]])/(a^4*d) - (10*Log[1 + Sin[c + d*x
]])/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 6/(d*(a^4 + a^4*Sin[c +
d*x]))

Rule 2833

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

Rule 12

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^3}{x^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^4 x^3}-\frac{4}{a^5 x^2}+\frac{10}{a^6 x}-\frac{1}{a^3 (a+x)^4}-\frac{3}{a^4 (a+x)^3}-\frac{6}{a^5 (a+x)^2}-\frac{10}{a^6 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{4 \csc (c+d x)}{a^4 d}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{10 \log (\sin (c+d x))}{a^4 d}-\frac{10 \log (1+\sin (c+d x))}{a^4 d}+\frac{1}{3 a d (a+a \sin (c+d x))^3}+\frac{3}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{6}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 3.69481, size = 85, normalized size = 0.65 \[ \frac{\frac{36}{\sin (c+d x)+1}+\frac{9}{(\sin (c+d x)+1)^2}+\frac{2}{(\sin (c+d x)+1)^3}-3 \csc ^2(c+d x)+24 \csc (c+d x)+60 \log (\sin (c+d x))-60 \log (\sin (c+d x)+1)}{6 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]

[Out]

(24*Csc[c + d*x] - 3*Csc[c + d*x]^2 + 60*Log[Sin[c + d*x]] - 60*Log[1 + Sin[c + d*x]] + 2/(1 + Sin[c + d*x])^3
+ 9/(1 + Sin[c + d*x])^2 + 36/(1 + Sin[c + d*x]))/(6*a^4*d)

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Maple [A]  time = 0.055, size = 120, normalized size = 0.9 \begin{align*}{\frac{1}{3\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{3}{2\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+6\,{\frac{1}{d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-10\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{1}{2\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{1}{d{a}^{4}\sin \left ( dx+c \right ) }}+10\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^4,x)

[Out]

1/3/d/a^4/(1+sin(d*x+c))^3+3/2/d/a^4/(1+sin(d*x+c))^2+6/d/a^4/(1+sin(d*x+c))-10*ln(1+sin(d*x+c))/a^4/d-1/2/d/a
^4/sin(d*x+c)^2+4/d/a^4/sin(d*x+c)+10*ln(sin(d*x+c))/a^4/d

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Maxima [A]  time = 1.08489, size = 170, normalized size = 1.3 \begin{align*} \frac{\frac{60 \, \sin \left (d x + c\right )^{4} + 150 \, \sin \left (d x + c\right )^{3} + 110 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 3}{a^{4} \sin \left (d x + c\right )^{5} + 3 \, a^{4} \sin \left (d x + c\right )^{4} + 3 \, a^{4} \sin \left (d x + c\right )^{3} + a^{4} \sin \left (d x + c\right )^{2}} - \frac{60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac{60 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/6*((60*sin(d*x + c)^4 + 150*sin(d*x + c)^3 + 110*sin(d*x + c)^2 + 15*sin(d*x + c) - 3)/(a^4*sin(d*x + c)^5 +
3*a^4*sin(d*x + c)^4 + 3*a^4*sin(d*x + c)^3 + a^4*sin(d*x + c)^2) - 60*log(sin(d*x + c) + 1)/a^4 + 60*log(sin
(d*x + c))/a^4)/d

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Fricas [A]  time = 1.54763, size = 643, normalized size = 4.91 \begin{align*} \frac{60 \, \cos \left (d x + c\right )^{4} - 230 \, \cos \left (d x + c\right )^{2} + 60 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) + 4\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 60 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) + 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (10 \, \cos \left (d x + c\right )^{2} - 11\right )} \sin \left (d x + c\right ) + 167}{6 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{4} - 7 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{4} - 5 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(60*cos(d*x + c)^4 - 230*cos(d*x + c)^2 + 60*(3*cos(d*x + c)^4 - 7*cos(d*x + c)^2 + (cos(d*x + c)^4 - 5*co
s(d*x + c)^2 + 4)*sin(d*x + c) + 4)*log(1/2*sin(d*x + c)) - 60*(3*cos(d*x + c)^4 - 7*cos(d*x + c)^2 + (cos(d*x
+ c)^4 - 5*cos(d*x + c)^2 + 4)*sin(d*x + c) + 4)*log(sin(d*x + c) + 1) - 15*(10*cos(d*x + c)^2 - 11)*sin(d*x
+ c) + 167)/(3*a^4*d*cos(d*x + c)^4 - 7*a^4*d*cos(d*x + c)^2 + 4*a^4*d + (a^4*d*cos(d*x + c)^4 - 5*a^4*d*cos(d
*x + c)^2 + 4*a^4*d)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin{\left (c + d x \right )} + 1}\, dx}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**3/(a+a*sin(d*x+c))**4,x)

[Out]

Integral(cos(c + d*x)*csc(c + d*x)**3/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x
) + 1), x)/a**4

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Giac [A]  time = 1.26951, size = 131, normalized size = 1. \begin{align*} -\frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{60 \, \sin \left (d x + c\right )^{4} + 150 \, \sin \left (d x + c\right )^{3} + 110 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 3}{a^{4}{\left (\sin \left (d x + c\right ) + 1\right )}^{3} \sin \left (d x + c\right )^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(60*log(abs(sin(d*x + c) + 1))/a^4 - 60*log(abs(sin(d*x + c)))/a^4 - (60*sin(d*x + c)^4 + 150*sin(d*x + c
)^3 + 110*sin(d*x + c)^2 + 15*sin(d*x + c) - 3)/(a^4*(sin(d*x + c) + 1)^3*sin(d*x + c)^2))/d