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

Optimal. Leaf size=97 \[ \frac{1}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]

[Out]

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

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Rubi [A]  time = 0.064456, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 44} \[ \frac{1}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2707

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

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

Mathematica [A]  time = 0.338893, size = 62, normalized size = 0.64 \[ \frac{\frac{6 \sin ^2(c+d x)+15 \sin (c+d x)+11}{(\sin (c+d x)+1)^3}+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{6 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Log[Sin[c + d*x]] - 6*Log[1 + Sin[c + d*x]] + (11 + 15*Sin[c + d*x] + 6*Sin[c + d*x]^2)/(1 + Sin[c + d*x])^
3)/(6*a^4*d)

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Maple [A]  time = 0.046, size = 86, normalized size = 0.9 \begin{align*}{\frac{1}{3\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\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)/(a+a*sin(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.11793, size = 128, normalized size = 1.32 \begin{align*} \frac{\frac{6 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} - \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac{6 \, \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)/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/6*((6*sin(d*x + c)^2 + 15*sin(d*x + c) + 11)/(a^4*sin(d*x + c)^3 + 3*a^4*sin(d*x + c)^2 + 3*a^4*sin(d*x + c)
+ a^4) - 6*log(sin(d*x + c) + 1)/a^4 + 6*log(sin(d*x + c))/a^4)/d

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Fricas [A]  time = 1.48474, size = 405, normalized size = 4.18 \begin{align*} \frac{6 \, \cos \left (d x + c\right )^{2} + 6 \,{\left (3 \, \cos \left (d x + c\right )^{2} +{\left (\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 ) - 6 \,{\left (3 \, \cos \left (d x + c\right )^{2} +{\left (\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 \, \sin \left (d x + c\right ) - 17}{6 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (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)/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(6*cos(d*x + c)^2 + 6*(3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 4)*sin(d*x + c) - 4)*log(1/2*sin(d*x + c)) - 6
*(3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 4)*sin(d*x + c) - 4)*log(sin(d*x + c) + 1) - 15*sin(d*x + c) - 17)/(3*a
^4*d*cos(d*x + c)^2 - 4*a^4*d + (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{\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)/(a+a*sin(d*x+c))**4,x)

[Out]

Integral(cos(c + d*x)*csc(c + d*x)/(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 = 1.223, size = 93, normalized size = 0.96 \begin{align*} -\frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{6 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) + 11}{a^{4}{\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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