3.1319 \(\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=212 \[ \frac{\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 d}-\frac{b \left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 d}-\frac{\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 d}+\frac{b \left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 d}+\frac{b^2 \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^7 d}-\frac{b^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^6(c+d x)}{6 a d} \]

[Out]

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

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Rubi [A]  time = 0.240817, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ \frac{\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 d}-\frac{b \left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 d}-\frac{\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 d}+\frac{b \left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 d}+\frac{b^2 \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^7 d}-\frac{b^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^6(c+d x)}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2837

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 12

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

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

Mathematica [A]  time = 2.86696, size = 165, normalized size = 0.78 \[ \frac{15 a^4 \left (2 a^2-b^2\right ) \csc ^4(c+d x)+20 a^3 b \left (b^2-2 a^2\right ) \csc ^3(c+d x)-30 a^2 \left (a^2-b^2\right )^2 \csc ^2(c+d x)+60 a b \left (a^2-b^2\right )^2 \csc (c+d x)+60 \left (b^3-a^2 b\right )^2 (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))+12 a^5 b \csc ^5(c+d x)-10 a^6 \csc ^6(c+d x)}{60 a^7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a*b*(a^2 - b^2)^2*Csc[c + d*x] - 30*a^2*(a^2 - b^2)^2*Csc[c + d*x]^2 + 20*a^3*b*(-2*a^2 + b^2)*Csc[c + d*x
]^3 + 15*a^4*(2*a^2 - b^2)*Csc[c + d*x]^4 + 12*a^5*b*Csc[c + d*x]^5 - 10*a^6*Csc[c + d*x]^6 + 60*(-(a^2*b) + b
^3)^2*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(60*a^7*d)

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Maple [A]  time = 0.105, size = 330, normalized size = 1.6 \begin{align*} -{\frac{{b}^{2}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}+2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){b}^{4}}{d{a}^{5}}}-{\frac{{b}^{6}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{7}}}-{\frac{1}{6\,da \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{b}^{2}}{4\,{a}^{3}d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{2}}{{a}^{3}d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{4}}{2\,d{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{2\,b}{3\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{b}^{3}}{3\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-2\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ){b}^{4}}{d{a}^{5}}}+{\frac{{b}^{6}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{7}}}+{\frac{b}{5\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{b}{d{a}^{2}\sin \left ( dx+c \right ) }}-2\,{\frac{{b}^{3}}{d{a}^{4}\sin \left ( dx+c \right ) }}+{\frac{{b}^{5}}{d{a}^{6}\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)^7/(a+b*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.990868, size = 278, normalized size = 1.31 \begin{align*} -\frac{\frac{60 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7}} - \frac{60 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{7}} - \frac{12 \, a^{4} b \sin \left (d x + c\right ) + 60 \,{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{5} - 10 \, a^{5} - 30 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{4} - 20 \,{\left (2 \, a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )^{3} + 15 \,{\left (2 \, a^{5} - a^{3} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{6} \sin \left (d x + c\right )^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(60*(a^4*b^2 - 2*a^2*b^4 + b^6)*log(b*sin(d*x + c) + a)/a^7 - 60*(a^4*b^2 - 2*a^2*b^4 + b^6)*log(sin(d*x
+ c))/a^7 - (12*a^4*b*sin(d*x + c) + 60*(a^4*b - 2*a^2*b^3 + b^5)*sin(d*x + c)^5 - 10*a^5 - 30*(a^5 - 2*a^3*b
^2 + a*b^4)*sin(d*x + c)^4 - 20*(2*a^4*b - a^2*b^3)*sin(d*x + c)^3 + 15*(2*a^5 - a^3*b^2)*sin(d*x + c)^2)/(a^6
*sin(d*x + c)^6))/d

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Fricas [B]  time = 1.67498, size = 1031, normalized size = 4.86 \begin{align*} \frac{10 \, a^{6} - 45 \, a^{4} b^{2} + 30 \, a^{2} b^{4} + 30 \,{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} - 15 \,{\left (2 \, a^{6} - 7 \, a^{4} b^{2} + 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 60 \,{\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} + 2 \, a^{2} b^{4} - b^{6} - 3 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 60 \,{\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} + 2 \, a^{2} b^{4} - b^{6} - 3 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 4 \,{\left (8 \, a^{5} b - 25 \, a^{3} b^{3} + 15 \, a b^{5} + 15 \,{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{4} - 5 \,{\left (4 \, a^{5} b - 11 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \,{\left (a^{7} d \cos \left (d x + c\right )^{6} - 3 \, a^{7} d \cos \left (d x + c\right )^{4} + 3 \, a^{7} d \cos \left (d x + c\right )^{2} - a^{7} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/60*(10*a^6 - 45*a^4*b^2 + 30*a^2*b^4 + 30*(a^6 - 2*a^4*b^2 + a^2*b^4)*cos(d*x + c)^4 - 15*(2*a^6 - 7*a^4*b^2
+ 4*a^2*b^4)*cos(d*x + c)^2 - 60*((a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*
(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^4 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*log(b*sin(d*x + c)
+ a) + 60*((a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*(a^4*b^2 - 2*a^2*b^4 + b
^6)*cos(d*x + c)^4 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*log(-1/2*sin(d*x + c)) - 4*(8*a^5*b - 25*a^
3*b^3 + 15*a*b^5 + 15*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c)^4 - 5*(4*a^5*b - 11*a^3*b^3 + 6*a*b^5)*cos(d*x
+ c)^2)*sin(d*x + c))/(a^7*d*cos(d*x + c)^6 - 3*a^7*d*cos(d*x + c)^4 + 3*a^7*d*cos(d*x + c)^2 - a^7*d)

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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)**7/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.35675, size = 406, normalized size = 1.92 \begin{align*} \frac{\frac{60 \,{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac{60 \,{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b} - \frac{147 \, a^{4} b^{2} \sin \left (d x + c\right )^{6} - 294 \, a^{2} b^{4} \sin \left (d x + c\right )^{6} + 147 \, b^{6} \sin \left (d x + c\right )^{6} - 60 \, a^{5} b \sin \left (d x + c\right )^{5} + 120 \, a^{3} b^{3} \sin \left (d x + c\right )^{5} - 60 \, a b^{5} \sin \left (d x + c\right )^{5} + 30 \, a^{6} \sin \left (d x + c\right )^{4} - 60 \, a^{4} b^{2} \sin \left (d x + c\right )^{4} + 30 \, a^{2} b^{4} \sin \left (d x + c\right )^{4} + 40 \, a^{5} b \sin \left (d x + c\right )^{3} - 20 \, a^{3} b^{3} \sin \left (d x + c\right )^{3} - 30 \, a^{6} \sin \left (d x + c\right )^{2} + 15 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{5} b \sin \left (d x + c\right ) + 10 \, a^{6}}{a^{7} \sin \left (d x + c\right )^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/60*(60*(a^4*b^2 - 2*a^2*b^4 + b^6)*log(abs(sin(d*x + c)))/a^7 - 60*(a^4*b^3 - 2*a^2*b^5 + b^7)*log(abs(b*sin
(d*x + c) + a))/(a^7*b) - (147*a^4*b^2*sin(d*x + c)^6 - 294*a^2*b^4*sin(d*x + c)^6 + 147*b^6*sin(d*x + c)^6 -
60*a^5*b*sin(d*x + c)^5 + 120*a^3*b^3*sin(d*x + c)^5 - 60*a*b^5*sin(d*x + c)^5 + 30*a^6*sin(d*x + c)^4 - 60*a^
4*b^2*sin(d*x + c)^4 + 30*a^2*b^4*sin(d*x + c)^4 + 40*a^5*b*sin(d*x + c)^3 - 20*a^3*b^3*sin(d*x + c)^3 - 30*a^
6*sin(d*x + c)^2 + 15*a^4*b^2*sin(d*x + c)^2 - 12*a^5*b*sin(d*x + c) + 10*a^6)/(a^7*sin(d*x + c)^6))/d