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

Optimal. Leaf size=119 \[ -\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}+\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0845058, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2836, 12, 88} \[ -\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}+\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) + (3*a*Csc
[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Log[Sin[c + d*x]])/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 ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^8 (a-x)^3 (a+x)^4}{x^8} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^4}{x^8} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{a^7}{x^8}+\frac{a^6}{x^7}-\frac{3 a^5}{x^6}-\frac{3 a^4}{x^5}+\frac{3 a^3}{x^4}+\frac{3 a^2}{x^3}-\frac{a}{x^2}-\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \csc (c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.390895, size = 115, normalized size = 0.97 \[ -\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d}-\frac{a \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*(6*Cot[
c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d)

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Maple [A]  time = 0.065, size = 217, normalized size = 1.8 \begin{align*} -{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\,d\sin \left ( dx+c \right ) }}+{\frac{16\,a\sin \left ( dx+c \right ) }{35\,d}}+{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}a}{7\,d}}+{\frac{6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{35\,d}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{35\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x)

[Out]

-1/6*a*cot(d*x+c)^6/d+1/4/d*a*cot(d*x+c)^4-1/2*a*cot(d*x+c)^2/d-a*ln(sin(d*x+c))/d-1/7/d*a/sin(d*x+c)^7*cos(d*
x+c)^8+1/35/d*a/sin(d*x+c)^5*cos(d*x+c)^8-1/35/d*a/sin(d*x+c)^3*cos(d*x+c)^8+1/7/d*a/sin(d*x+c)*cos(d*x+c)^8+1
6/35*a*sin(d*x+c)/d+1/7/d*cos(d*x+c)^6*sin(d*x+c)*a+6/35/d*cos(d*x+c)^4*sin(d*x+c)*a+8/35/d*cos(d*x+c)^2*sin(d
*x+c)*a

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Maxima [A]  time = 1.0397, size = 127, normalized size = 1.07 \begin{align*} -\frac{420 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac{420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/420*(420*a*log(sin(d*x + c)) - (420*a*sin(d*x + c)^6 - 630*a*sin(d*x + c)^5 - 420*a*sin(d*x + c)^4 + 315*a*
sin(d*x + c)^3 + 252*a*sin(d*x + c)^2 - 70*a*sin(d*x + c) - 60*a)/sin(d*x + c)^7)/d

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Fricas [A]  time = 1.2071, size = 458, normalized size = 3.85 \begin{align*} \frac{420 \, a \cos \left (d x + c\right )^{6} - 840 \, a \cos \left (d x + c\right )^{4} + 672 \, a \cos \left (d x + c\right )^{2} - 420 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 35 \,{\left (18 \, a \cos \left (d x + c\right )^{4} - 27 \, a \cos \left (d x + c\right )^{2} + 11 \, a\right )} \sin \left (d x + c\right ) - 192 \, a}{420 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/420*(420*a*cos(d*x + c)^6 - 840*a*cos(d*x + c)^4 + 672*a*cos(d*x + c)^2 - 420*(a*cos(d*x + c)^6 - 3*a*cos(d*
x + c)^4 + 3*a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c))*sin(d*x + c) + 35*(18*a*cos(d*x + c)^4 - 27*a*cos(d*x
+ c)^2 + 11*a)*sin(d*x + c) - 192*a)/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*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)**7*csc(d*x+c)**8*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.33715, size = 143, normalized size = 1.2 \begin{align*} -\frac{420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac{1089 \, a \sin \left (d x + c\right )^{7} + 420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/420*(420*a*log(abs(sin(d*x + c))) - (1089*a*sin(d*x + c)^7 + 420*a*sin(d*x + c)^6 - 630*a*sin(d*x + c)^5 -
420*a*sin(d*x + c)^4 + 315*a*sin(d*x + c)^3 + 252*a*sin(d*x + c)^2 - 70*a*sin(d*x + c) - 60*a)/sin(d*x + c)^7)
/d