3.185 \(\int \frac{\cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=114 \[ -\frac{a^2-b^2}{a^3 d (a+b \sin (c+d x))}-\frac{\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-3 b^2\right ) \log (a+b \sin (c+d x))}{a^4 d}+\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.114796, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2721, 894} \[ -\frac{a^2-b^2}{a^3 d (a+b \sin (c+d x))}-\frac{\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{\left (a^2-3 b^2\right ) \log (a+b \sin (c+d x))}{a^4 d}+\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]

[Out]

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

Rule 2721

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)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.607729, size = 96, normalized size = 0.84 \[ -\frac{2 \left (a^2-3 b^2\right ) \log (\sin (c+d x))-2 \left (a^2-3 b^2\right ) \log (a+b \sin (c+d x))+a^2 \csc ^2(c+d x)+\frac{2 a (a-b) (a+b)}{a+b \sin (c+d x)}-4 a b \csc (c+d x)}{2 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 150, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{a}^{2}d}}-3\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{2}}{d{a}^{4}}}-{\frac{1}{da \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{{b}^{2}}{d{a}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{1}{2\,{a}^{2}d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{2}d}}+3\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ){b}^{2}}{d{a}^{4}}}+2\,{\frac{b}{d{a}^{3}\sin \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3/(a+b*sin(d*x+c))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.36351, size = 157, normalized size = 1.38 \begin{align*} \frac{\frac{3 \, a b \sin \left (d x + c\right ) - 2 \,{\left (a^{2} - 3 \, b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2}}{a^{3} b \sin \left (d x + c\right )^{3} + a^{4} \sin \left (d x + c\right )^{2}} + \frac{2 \,{\left (a^{2} - 3 \, b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4}} - \frac{2 \,{\left (a^{2} - 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.61861, size = 599, normalized size = 5.25 \begin{align*} -\frac{3 \, a^{2} b \sin \left (d x + c\right ) - 3 \, a^{3} + 6 \, a b^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2} -{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{2} b - 3 \, b^{3} -{\left (a^{2} b - 3 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (a^{3} - 3 \, a b^{2} -{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{2} b - 3 \, b^{3} -{\left (a^{2} b - 3 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right )}{2 \,{\left (a^{5} d \cos \left (d x + c\right )^{2} - a^{5} d +{\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/2*(3*a^2*b*sin(d*x + c) - 3*a^3 + 6*a*b^2 + 2*(a^3 - 3*a*b^2)*cos(d*x + c)^2 + 2*(a^3 - 3*a*b^2 - (a^3 - 3*
a*b^2)*cos(d*x + c)^2 + (a^2*b - 3*b^3 - (a^2*b - 3*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a)
- 2*(a^3 - 3*a*b^2 - (a^3 - 3*a*b^2)*cos(d*x + c)^2 + (a^2*b - 3*b^3 - (a^2*b - 3*b^3)*cos(d*x + c)^2)*sin(d*
x + c))*log(-1/2*sin(d*x + c)))/(a^5*d*cos(d*x + c)^2 - a^5*d + (a^4*b*d*cos(d*x + c)^2 - a^4*b*d)*sin(d*x + c
))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3/(a+b*sin(d*x+c))**2,x)

[Out]

Integral(cot(c + d*x)**3/(a + b*sin(c + d*x))**2, x)

________________________________________________________________________________________

Giac [A]  time = 1.97623, size = 223, normalized size = 1.96 \begin{align*} -\frac{\frac{2 \,{\left (a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{2 \,{\left (a^{2} b - 3 \, b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b} + \frac{2 \,{\left (a^{2} b \sin \left (d x + c\right ) - 3 \, b^{3} \sin \left (d x + c\right ) + 2 \, a^{3} - 4 \, a b^{2}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{4}} - \frac{3 \, a^{2} \sin \left (d x + c\right )^{2} - 9 \, b^{2} \sin \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) - a^{2}}{a^{4} \sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/2*(2*(a^2 - 3*b^2)*log(abs(sin(d*x + c)))/a^4 - 2*(a^2*b - 3*b^3)*log(abs(b*sin(d*x + c) + a))/(a^4*b) + 2*
(a^2*b*sin(d*x + c) - 3*b^3*sin(d*x + c) + 2*a^3 - 4*a*b^2)/((b*sin(d*x + c) + a)*a^4) - (3*a^2*sin(d*x + c)^2
- 9*b^2*sin(d*x + c)^2 + 4*a*b*sin(d*x + c) - a^2)/(a^4*sin(d*x + c)^2))/d