3.197 \(\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=221 \[ \frac{-6 a^2 b^2+a^4+5 b^4}{a^6 d (a+b \sin (c+d x))}+\frac{\left (a^2-b^2\right )^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac{\left (a^2-3 b^2\right ) \csc ^2(c+d x)}{a^5 d}-\frac{2 b \left (3 a^2-5 b^2\right ) \csc (c+d x)}{a^6 d}+\frac{\left (-12 a^2 b^2+a^4+15 b^4\right ) \log (\sin (c+d x))}{a^7 d}-\frac{\left (-12 a^2 b^2+a^4+15 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^3(c+d x)}{a^4 d}-\frac{\csc ^4(c+d x)}{4 a^3 d} \]

[Out]

(-2*b*(3*a^2 - 5*b^2)*Csc[c + d*x])/(a^6*d) + ((a^2 - 3*b^2)*Csc[c + d*x]^2)/(a^5*d) + (b*Csc[c + d*x]^3)/(a^4
*d) - Csc[c + d*x]^4/(4*a^3*d) + ((a^4 - 12*a^2*b^2 + 15*b^4)*Log[Sin[c + d*x]])/(a^7*d) - ((a^4 - 12*a^2*b^2
+ 15*b^4)*Log[a + b*Sin[c + d*x]])/(a^7*d) + (a^2 - b^2)^2/(2*a^5*d*(a + b*Sin[c + d*x])^2) + (a^4 - 6*a^2*b^2
+ 5*b^4)/(a^6*d*(a + b*Sin[c + d*x]))

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Rubi [A]  time = 0.211009, antiderivative size = 221, 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} \[ \frac{-6 a^2 b^2+a^4+5 b^4}{a^6 d (a+b \sin (c+d x))}+\frac{\left (a^2-b^2\right )^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac{\left (a^2-3 b^2\right ) \csc ^2(c+d x)}{a^5 d}-\frac{2 b \left (3 a^2-5 b^2\right ) \csc (c+d x)}{a^6 d}+\frac{\left (-12 a^2 b^2+a^4+15 b^4\right ) \log (\sin (c+d x))}{a^7 d}-\frac{\left (-12 a^2 b^2+a^4+15 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^3(c+d x)}{a^4 d}-\frac{\csc ^4(c+d x)}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]

[Out]

(-2*b*(3*a^2 - 5*b^2)*Csc[c + d*x])/(a^6*d) + ((a^2 - 3*b^2)*Csc[c + d*x]^2)/(a^5*d) + (b*Csc[c + d*x]^3)/(a^4
*d) - Csc[c + d*x]^4/(4*a^3*d) + ((a^4 - 12*a^2*b^2 + 15*b^4)*Log[Sin[c + d*x]])/(a^7*d) - ((a^4 - 12*a^2*b^2
+ 15*b^4)*Log[a + b*Sin[c + d*x]])/(a^7*d) + (a^2 - b^2)^2/(2*a^5*d*(a + b*Sin[c + d*x])^2) + (a^4 - 6*a^2*b^2
+ 5*b^4)/(a^6*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))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x^5 (a+x)^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{b^4}{a^3 x^5}-\frac{3 b^4}{a^4 x^4}+\frac{2 b^2 \left (-a^2+3 b^2\right )}{a^5 x^3}+\frac{2 \left (3 a^2 b^2-5 b^4\right )}{a^6 x^2}+\frac{a^4-12 a^2 b^2+15 b^4}{a^7 x}-\frac{\left (a^2-b^2\right )^2}{a^5 (a+x)^3}+\frac{-a^4+6 a^2 b^2-5 b^4}{a^6 (a+x)^2}+\frac{-a^4+12 a^2 b^2-15 b^4}{a^7 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{2 b \left (3 a^2-5 b^2\right ) \csc (c+d x)}{a^6 d}+\frac{\left (a^2-3 b^2\right ) \csc ^2(c+d x)}{a^5 d}+\frac{b \csc ^3(c+d x)}{a^4 d}-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\left (a^4-12 a^2 b^2+15 b^4\right ) \log (\sin (c+d x))}{a^7 d}-\frac{\left (a^4-12 a^2 b^2+15 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac{\left (a^2-b^2\right )^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac{a^4-6 a^2 b^2+5 b^4}{a^6 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 5.32652, size = 195, normalized size = 0.88 \[ \frac{\frac{4 a \left (-6 a^2 b^2+a^4+5 b^4\right )}{a+b \sin (c+d x)}+\frac{2 \left (a^3-a b^2\right )^2}{(a+b \sin (c+d x))^2}+4 a^2 \left (a^2-3 b^2\right ) \csc ^2(c+d x)-8 a b \left (3 a^2-5 b^2\right ) \csc (c+d x)+4 \left (-12 a^2 b^2+a^4+15 b^4\right ) \log (\sin (c+d x))-4 \left (-12 a^2 b^2+a^4+15 b^4\right ) \log (a+b \sin (c+d x))+4 a^3 b \csc ^3(c+d x)-a^4 \csc ^4(c+d x)}{4 a^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]

[Out]

(-8*a*b*(3*a^2 - 5*b^2)*Csc[c + d*x] + 4*a^2*(a^2 - 3*b^2)*Csc[c + d*x]^2 + 4*a^3*b*Csc[c + d*x]^3 - a^4*Csc[c
+ d*x]^4 + 4*(a^4 - 12*a^2*b^2 + 15*b^4)*Log[Sin[c + d*x]] - 4*(a^4 - 12*a^2*b^2 + 15*b^4)*Log[a + b*Sin[c +
d*x]] + (2*(a^3 - a*b^2)^2)/(a + b*Sin[c + d*x])^2 + (4*a*(a^4 - 6*a^2*b^2 + 5*b^4))/(a + b*Sin[c + d*x]))/(4*
a^7*d)

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Maple [A]  time = 0.114, size = 348, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}+12\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{2}}{d{a}^{5}}}-15\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{4}}{d{a}^{7}}}+{\frac{1}{{a}^{2}d \left ( a+b\sin \left ( dx+c \right ) \right ) }}-6\,{\frac{{b}^{2}}{d{a}^{4} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+5\,{\frac{{b}^{4}}{d{a}^{6} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{1}{2\,da \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2}}{{a}^{3}d \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{4}}{2\,d{a}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{4\,{a}^{3}d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{{a}^{3}d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{{b}^{2}}{d{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-12\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ){b}^{2}}{d{a}^{5}}}+15\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ){b}^{4}}{d{a}^{7}}}+{\frac{b}{d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-6\,{\frac{b}{d{a}^{4}\sin \left ( dx+c \right ) }}+10\,{\frac{{b}^{3}}{d{a}^{6}\sin \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x)

[Out]

-ln(a+b*sin(d*x+c))/a^3/d+12/d/a^5*ln(a+b*sin(d*x+c))*b^2-15/d/a^7*ln(a+b*sin(d*x+c))*b^4+1/a^2/d/(a+b*sin(d*x
+c))-6/d/a^4/(a+b*sin(d*x+c))*b^2+5/d/a^6/(a+b*sin(d*x+c))*b^4+1/2/a/d/(a+b*sin(d*x+c))^2-1/d/a^3/(a+b*sin(d*x
+c))^2*b^2+1/2/d/a^5/(a+b*sin(d*x+c))^2*b^4-1/4/d/a^3/sin(d*x+c)^4+1/d/a^3/sin(d*x+c)^2-3/d/a^5/sin(d*x+c)^2*b
^2+ln(sin(d*x+c))/a^3/d-12/d/a^5*ln(sin(d*x+c))*b^2+15/d/a^7*ln(sin(d*x+c))*b^4+1/d/a^4*b/sin(d*x+c)^3-6/d/a^4
*b/sin(d*x+c)+10/d*b^3/a^6/sin(d*x+c)

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Maxima [A]  time = 1.61205, size = 319, normalized size = 1.44 \begin{align*} \frac{\frac{2 \, a^{4} b \sin \left (d x + c\right ) + 4 \,{\left (a^{4} b - 12 \, a^{2} b^{3} + 15 \, b^{5}\right )} \sin \left (d x + c\right )^{5} - a^{5} + 6 \,{\left (a^{5} - 12 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \sin \left (d x + c\right )^{4} - 4 \,{\left (4 \, a^{4} b - 5 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )^{3} +{\left (4 \, a^{5} - 5 \, a^{3} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{6} b^{2} \sin \left (d x + c\right )^{6} + 2 \, a^{7} b \sin \left (d x + c\right )^{5} + a^{8} \sin \left (d x + c\right )^{4}} - \frac{4 \,{\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7}} + \frac{4 \,{\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{7}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/4*((2*a^4*b*sin(d*x + c) + 4*(a^4*b - 12*a^2*b^3 + 15*b^5)*sin(d*x + c)^5 - a^5 + 6*(a^5 - 12*a^3*b^2 + 15*a
*b^4)*sin(d*x + c)^4 - 4*(4*a^4*b - 5*a^2*b^3)*sin(d*x + c)^3 + (4*a^5 - 5*a^3*b^2)*sin(d*x + c)^2)/(a^6*b^2*s
in(d*x + c)^6 + 2*a^7*b*sin(d*x + c)^5 + a^8*sin(d*x + c)^4) - 4*(a^4 - 12*a^2*b^2 + 15*b^4)*log(b*sin(d*x + c
) + a)/a^7 + 4*(a^4 - 12*a^2*b^2 + 15*b^4)*log(sin(d*x + c))/a^7)/d

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Fricas [B]  time = 2.42375, size = 1708, normalized size = 7.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/4*(9*a^6 - 77*a^4*b^2 + 90*a^2*b^4 + 6*(a^6 - 12*a^4*b^2 + 15*a^2*b^4)*cos(d*x + c)^4 - (16*a^6 - 149*a^4*b
^2 + 180*a^2*b^4)*cos(d*x + c)^2 + 4*((a^4*b^2 - 12*a^2*b^4 + 15*b^6)*cos(d*x + c)^6 - a^6 + 11*a^4*b^2 - 3*a^
2*b^4 - 15*b^6 - (a^6 - 9*a^4*b^2 - 21*a^2*b^4 + 45*b^6)*cos(d*x + c)^4 + (2*a^6 - 21*a^4*b^2 - 6*a^2*b^4 + 45
*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5 + (a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^4 - 2*(
a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) - 4*((a^4*b^2 - 12*a^2*b^
4 + 15*b^6)*cos(d*x + c)^6 - a^6 + 11*a^4*b^2 - 3*a^2*b^4 - 15*b^6 - (a^6 - 9*a^4*b^2 - 21*a^2*b^4 + 45*b^6)*c
os(d*x + c)^4 + (2*a^6 - 21*a^4*b^2 - 6*a^2*b^4 + 45*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5 +
(a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^2)*sin(d*x + c
))*log(-1/2*sin(d*x + c)) - 2*(5*a^5*b + 14*a^3*b^3 - 30*a*b^5 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c
)^4 - 2*(2*a^5*b + 19*a^3*b^3 - 30*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))/(a^7*b^2*d*cos(d*x + c)^6 - (a^9 + 3*a
^7*b^2)*d*cos(d*x + c)^4 + (2*a^9 + 3*a^7*b^2)*d*cos(d*x + c)^2 - (a^9 + a^7*b^2)*d - 2*(a^8*b*d*cos(d*x + c)^
4 - 2*a^8*b*d*cos(d*x + c)^2 + a^8*b*d)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{5}{\left (c + d x \right )}}{\left (a + b \sin{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5/(a+b*sin(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)**5/(a + b*sin(c + d*x))**3, x)

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Giac [A]  time = 2.10396, size = 441, normalized size = 2. \begin{align*} \frac{\frac{12 \,{\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac{12 \,{\left (a^{4} b - 12 \, a^{2} b^{3} + 15 \, b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac{6 \,{\left (3 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 36 \, a^{2} b^{4} \sin \left (d x + c\right )^{2} + 45 \, b^{6} \sin \left (d x + c\right )^{2} + 8 \, a^{5} b \sin \left (d x + c\right ) - 84 \, a^{3} b^{3} \sin \left (d x + c\right ) + 100 \, a b^{5} \sin \left (d x + c\right ) + 6 \, a^{6} - 50 \, a^{4} b^{2} + 56 \, a^{2} b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} a^{7}} - \frac{25 \, a^{4} \sin \left (d x + c\right )^{4} - 300 \, a^{2} b^{2} \sin \left (d x + c\right )^{4} + 375 \, b^{4} \sin \left (d x + c\right )^{4} + 72 \, a^{3} b \sin \left (d x + c\right )^{3} - 120 \, a b^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{4} \sin \left (d x + c\right )^{2} + 36 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{3} b \sin \left (d x + c\right ) + 3 \, a^{4}}{a^{7} \sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/12*(12*(a^4 - 12*a^2*b^2 + 15*b^4)*log(abs(sin(d*x + c)))/a^7 - 12*(a^4*b - 12*a^2*b^3 + 15*b^5)*log(abs(b*s
in(d*x + c) + a))/(a^7*b) + 6*(3*a^4*b^2*sin(d*x + c)^2 - 36*a^2*b^4*sin(d*x + c)^2 + 45*b^6*sin(d*x + c)^2 +
8*a^5*b*sin(d*x + c) - 84*a^3*b^3*sin(d*x + c) + 100*a*b^5*sin(d*x + c) + 6*a^6 - 50*a^4*b^2 + 56*a^2*b^4)/((b
*sin(d*x + c) + a)^2*a^7) - (25*a^4*sin(d*x + c)^4 - 300*a^2*b^2*sin(d*x + c)^4 + 375*b^4*sin(d*x + c)^4 + 72*
a^3*b*sin(d*x + c)^3 - 120*a*b^3*sin(d*x + c)^3 - 12*a^4*sin(d*x + c)^2 + 36*a^2*b^2*sin(d*x + c)^2 - 12*a^3*b
*sin(d*x + c) + 3*a^4)/(a^7*sin(d*x + c)^4))/d