3.510 \(\int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=97 \[ -\frac{a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^{10}(c+d x)}{10 d}+\frac{2 a \csc ^9(c+d x)}{9 d}+\frac{a \csc ^8(c+d x)}{4 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d} \]

[Out]

-(a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) -
(a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)

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Rubi [A]  time = 0.0818183, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^{10}(c+d x)}{10 d}+\frac{2 a \csc ^9(c+d x)}{9 d}+\frac{a \csc ^8(c+d x)}{4 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) -
(a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)

Rule 2836

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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^{12} (a-x)^2 (a+x)^3}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{a^7 \operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^3}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^7 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{12}}+\frac{a^4}{x^{11}}-\frac{2 a^3}{x^{10}}-\frac{2 a^2}{x^9}+\frac{a}{x^8}+\frac{1}{x^7}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^8(c+d x)}{4 d}+\frac{2 a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^{11}(c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 0.184271, size = 88, normalized size = 0.91 \[ -\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{2 a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \left (6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right )}{60 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^7)/(7*d) + (2*a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^11)/(11*d) - (a*(10*Csc[c + d*x]^6 -
15*Csc[c + d*x]^8 + 6*Csc[c + d*x]^10))/(60*d)

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Maple [B]  time = 0.063, size = 202, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{20\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{60\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{11\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{99\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{231\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{231\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{693\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{231\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{231} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^12*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/10/sin(d*x+c)^10*cos(d*x+c)^6-1/20/sin(d*x+c)^8*cos(d*x+c)^6-1/60/sin(d*x+c)^6*cos(d*x+c)^6)+a*(-1/
11/sin(d*x+c)^11*cos(d*x+c)^6-5/99/sin(d*x+c)^9*cos(d*x+c)^6-5/231/sin(d*x+c)^7*cos(d*x+c)^6-1/231/sin(d*x+c)^
5*cos(d*x+c)^6+1/693/sin(d*x+c)^3*cos(d*x+c)^6-1/231/sin(d*x+c)*cos(d*x+c)^6-1/231*(8/3+cos(d*x+c)^4+4/3*cos(d
*x+c)^2)*sin(d*x+c)))

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Maxima [A]  time = 1.15121, size = 95, normalized size = 0.98 \begin{align*} -\frac{2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/13860*(2310*a*sin(d*x + c)^5 + 1980*a*sin(d*x + c)^4 - 3465*a*sin(d*x + c)^3 - 3080*a*sin(d*x + c)^2 + 1386
*a*sin(d*x + c) + 1260*a)/(d*sin(d*x + c)^11)

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Fricas [A]  time = 1.12054, size = 344, normalized size = 3.55 \begin{align*} \frac{1980 \, a \cos \left (d x + c\right )^{4} - 880 \, a \cos \left (d x + c\right )^{2} + 231 \,{\left (10 \, a \cos \left (d x + c\right )^{4} - 5 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 160 \, a}{13860 \,{\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/13860*(1980*a*cos(d*x + c)^4 - 880*a*cos(d*x + c)^2 + 231*(10*a*cos(d*x + c)^4 - 5*a*cos(d*x + c)^2 + a)*sin
(d*x + c) + 160*a)/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*
cos(d*x + c)^2 - 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)**12*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.26654, size = 95, normalized size = 0.98 \begin{align*} -\frac{2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/13860*(2310*a*sin(d*x + c)^5 + 1980*a*sin(d*x + c)^4 - 3465*a*sin(d*x + c)^3 - 3080*a*sin(d*x + c)^2 + 1386
*a*sin(d*x + c) + 1260*a)/(d*sin(d*x + c)^11)