3.1233 \(\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=188 \[ \frac{\left (a^2-b^2\right )^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 a^4 d}-\frac{4 b \left (a^2-b^2\right ) \csc (c+d x)}{a^5 d}+\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \log (\sin (c+d x))}{a^6 d}-\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d} \]

[Out]

(-4*b*(a^2 - b^2)*Csc[c + d*x])/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3)/(3
*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2*b^2
+ 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*Sin[c + d*x]))

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Rubi [A]  time = 0.170519, antiderivative size = 188, 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} \[ \frac{\left (a^2-b^2\right )^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 a^4 d}-\frac{4 b \left (a^2-b^2\right ) \csc (c+d x)}{a^5 d}+\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \log (\sin (c+d x))}{a^6 d}-\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*(a^2 - b^2)*Csc[c + d*x])/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3)/(3
*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2*b^2
+ 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*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 ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x^5 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{b^4}{a^2 x^5}-\frac{2 b^4}{a^3 x^4}+\frac{-2 a^2 b^2+3 b^4}{a^4 x^3}+\frac{4 b^2 \left (a^2-b^2\right )}{a^5 x^2}+\frac{a^4-6 a^2 b^2+5 b^4}{a^6 x}-\frac{\left (a^2-b^2\right )^2}{a^5 (a+x)^2}+\frac{-a^4+6 a^2 b^2-5 b^4}{a^6 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{4 b \left (a^2-b^2\right ) \csc (c+d x)}{a^5 d}+\frac{\left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{\left (a^4-6 a^2 b^2+5 b^4\right ) \log (\sin (c+d x))}{a^6 d}-\frac{\left (a^4-6 a^2 b^2+5 b^4\right ) \log (a+b \sin (c+d x))}{a^6 d}+\frac{\left (a^2-b^2\right )^2}{a^5 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.13748, size = 187, normalized size = 0.99 \[ \frac{\left (a^2-b^2\right )^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 a^4 d}+\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \log (\sin (c+d x))}{a^6 d}-\frac{\left (-6 a^2 b^2+a^4+5 b^4\right ) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{4 b (a-b) (a+b) \csc (c+d x)}{a^5 d}-\frac{\csc ^4(c+d x)}{4 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(a - b)*b*(a + b)*Csc[c + d*x])/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3
)/(3*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2
*b^2 + 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*Sin[c + d*x]))

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Maple [A]  time = 0.155, size = 282, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}+6\,{\frac{{b}^{2}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}-5\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{4}}{d{a}^{6}}}+{\frac{1}{da \left ( a+b\sin \left ( dx+c \right ) \right ) }}-2\,{\frac{{b}^{2}}{d{a}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{{b}^{4}}{d{a}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{1}{4\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{b}^{2}}{2\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-6\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ){b}^{2}}{d{a}^{4}}}+5\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ){b}^{4}}{d{a}^{6}}}+{\frac{2\,b}{3\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-4\,{\frac{b}{d{a}^{3}\sin \left ( dx+c \right ) }}+4\,{\frac{{b}^{3}}{d{a}^{5}\sin \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x)

[Out]

-1/d/a^2*ln(a+b*sin(d*x+c))+6/d/a^4*b^2*ln(a+b*sin(d*x+c))-5/d/a^6*ln(a+b*sin(d*x+c))*b^4+1/d/a/(a+b*sin(d*x+c
))-2/d/a^3*b^2/(a+b*sin(d*x+c))+1/d/a^5/(a+b*sin(d*x+c))*b^4-1/4/d/a^2/sin(d*x+c)^4+1/d/a^2/sin(d*x+c)^2-3/2/d
/a^4/sin(d*x+c)^2*b^2+ln(sin(d*x+c))/a^2/d-6/d/a^4*ln(sin(d*x+c))*b^2+5/d/a^6*ln(sin(d*x+c))*b^4+2/3/d/a^3*b/s
in(d*x+c)^3-4/d/a^3*b/sin(d*x+c)+4/d*b^3/a^5/sin(d*x+c)

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Maxima [A]  time = 1.01323, size = 255, normalized size = 1.36 \begin{align*} \frac{\frac{5 \, a^{3} b \sin \left (d x + c\right ) + 12 \,{\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sin \left (d x + c\right )^{4} - 3 \, a^{4} - 6 \,{\left (6 \, a^{3} b - 5 \, a b^{3}\right )} \sin \left (d x + c\right )^{3} + 2 \,{\left (6 \, a^{4} - 5 \, a^{2} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{5} b \sin \left (d x + c\right )^{5} + a^{6} \sin \left (d x + c\right )^{4}} - \frac{12 \,{\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6}} + \frac{12 \,{\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{6}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/12*((5*a^3*b*sin(d*x + c) + 12*(a^4 - 6*a^2*b^2 + 5*b^4)*sin(d*x + c)^4 - 3*a^4 - 6*(6*a^3*b - 5*a*b^3)*sin(
d*x + c)^3 + 2*(6*a^4 - 5*a^2*b^2)*sin(d*x + c)^2)/(a^5*b*sin(d*x + c)^5 + a^6*sin(d*x + c)^4) - 12*(a^4 - 6*a
^2*b^2 + 5*b^4)*log(b*sin(d*x + c) + a)/a^6 + 12*(a^4 - 6*a^2*b^2 + 5*b^4)*log(sin(d*x + c))/a^6)/d

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Fricas [B]  time = 2.14108, size = 1241, normalized size = 6.6 \begin{align*} \frac{21 \, a^{5} - 82 \, a^{3} b^{2} + 60 \, a b^{4} + 12 \,{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (18 \, a^{5} - 77 \, a^{3} b^{2} + 60 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 12 \,{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4} +{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5} +{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\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 ) + 12 \,{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4} +{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5} +{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\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 ) -{\left (31 \, a^{4} b - 30 \, a^{2} b^{3} - 6 \,{\left (6 \, a^{4} b - 5 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \,{\left (a^{7} d \cos \left (d x + c\right )^{4} - 2 \, a^{7} d \cos \left (d x + c\right )^{2} + a^{7} d +{\left (a^{6} b d \cos \left (d x + c\right )^{4} - 2 \, a^{6} b d \cos \left (d x + c\right )^{2} + a^{6} b 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)^5*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/12*(21*a^5 - 82*a^3*b^2 + 60*a*b^4 + 12*(a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 - 2*(18*a^5 - 77*a^3*b^2
+ 60*a*b^4)*cos(d*x + c)^2 - 12*(a^5 - 6*a^3*b^2 + 5*a*b^4 + (a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 - 2*(a
^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^2 + (a^4*b - 6*a^2*b^3 + 5*b^5 + (a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x +
c)^4 - 2*(a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 12*(a^5 - 6*a^3*b
^2 + 5*a*b^4 + (a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^2 + (a^
4*b - 6*a^2*b^3 + 5*b^5 + (a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x + c)^4 - 2*(a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x +
c)^2)*sin(d*x + c))*log(-1/2*sin(d*x + c)) - (31*a^4*b - 30*a^2*b^3 - 6*(6*a^4*b - 5*a^2*b^3)*cos(d*x + c)^2)
*sin(d*x + c))/(a^7*d*cos(d*x + c)^4 - 2*a^7*d*cos(d*x + c)^2 + a^7*d + (a^6*b*d*cos(d*x + c)^4 - 2*a^6*b*d*co
s(d*x + c)^2 + a^6*b*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)**5*csc(d*x+c)**5/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.23456, size = 375, normalized size = 1.99 \begin{align*} \frac{\frac{12 \,{\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac{12 \,{\left (a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b} + \frac{12 \,{\left (a^{4} b \sin \left (d x + c\right ) - 6 \, a^{2} b^{3} \sin \left (d x + c\right ) + 5 \, b^{5} \sin \left (d x + c\right ) + 2 \, a^{5} - 8 \, a^{3} b^{2} + 6 \, a b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{6}} - \frac{25 \, a^{4} \sin \left (d x + c\right )^{4} - 150 \, a^{2} b^{2} \sin \left (d x + c\right )^{4} + 125 \, b^{4} \sin \left (d x + c\right )^{4} + 48 \, a^{3} b \sin \left (d x + c\right )^{3} - 48 \, a b^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{4} \sin \left (d x + c\right )^{2} + 18 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 8 \, a^{3} b \sin \left (d x + c\right ) + 3 \, a^{4}}{a^{6} \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)^5*csc(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/12*(12*(a^4 - 6*a^2*b^2 + 5*b^4)*log(abs(sin(d*x + c)))/a^6 - 12*(a^4*b - 6*a^2*b^3 + 5*b^5)*log(abs(b*sin(d
*x + c) + a))/(a^6*b) + 12*(a^4*b*sin(d*x + c) - 6*a^2*b^3*sin(d*x + c) + 5*b^5*sin(d*x + c) + 2*a^5 - 8*a^3*b
^2 + 6*a*b^4)/((b*sin(d*x + c) + a)*a^6) - (25*a^4*sin(d*x + c)^4 - 150*a^2*b^2*sin(d*x + c)^4 + 125*b^4*sin(d
*x + c)^4 + 48*a^3*b*sin(d*x + c)^3 - 48*a*b^3*sin(d*x + c)^3 - 12*a^4*sin(d*x + c)^2 + 18*a^2*b^2*sin(d*x + c
)^2 - 8*a^3*b*sin(d*x + c) + 3*a^4)/(a^6*sin(d*x + c)^4))/d