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

Optimal. Leaf size=115 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0530714, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 88} \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2707

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 - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

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

Mathematica [A]  time = 0.383324, size = 111, normalized size = 0.97 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Csc[c + d*x])/d + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^5)/(5*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) - (a*Sin[c + d*x])/d

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Maple [A]  time = 0.069, size = 195, normalized size = 1.7 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{d\sin \left ( dx+c \right ) }}-{\frac{16\,a\sin \left ( dx+c \right ) }{5\,d}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}a}{d}}-{\frac{6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{5\,d}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{5\,d}}-{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/d*a/sin(d*x+c)^5*cos(d*x+c)^8+1/5/d*a/sin(d*x+c)^3*cos(d*x+c)^8-1/d*a/sin(d*x+c)*cos(d*x+c)^8-16/5*a*sin(
d*x+c)/d-1/d*cos(d*x+c)^6*sin(d*x+c)*a-6/5/d*cos(d*x+c)^4*sin(d*x+c)*a-8/5/d*cos(d*x+c)^2*sin(d*x+c)*a-1/6*a*c
ot(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

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Maxima [A]  time = 1.03066, size = 123, normalized size = 1.07 \begin{align*} -\frac{60 \, a \log \left (\sin \left (d x + c\right )\right ) + 60 \, a \sin \left (d x + c\right ) + \frac{180 \, a \sin \left (d x + c\right )^{5} + 90 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 45 \, a \sin \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) + 10 \, a}{\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)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(60*a*log(sin(d*x + c)) + 60*a*sin(d*x + c) + (180*a*sin(d*x + c)^5 + 90*a*sin(d*x + c)^4 - 60*a*sin(d*x
+ c)^3 - 45*a*sin(d*x + c)^2 + 12*a*sin(d*x + c) + 10*a)/sin(d*x + c)^6)/d

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Fricas [A]  time = 1.16157, size = 412, normalized size = 3.58 \begin{align*} \frac{90 \, a \cos \left (d x + c\right )^{4} - 135 \, a \cos \left (d x + c\right )^{2} - 60 \,{\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 ) - 12 \,{\left (5 \, a \cos \left (d x + c\right )^{6} - 30 \, a \cos \left (d x + c\right )^{4} + 40 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) + 55 \, a}{60 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(90*a*cos(d*x + c)^4 - 135*a*cos(d*x + c)^2 - 60*(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)) - 12*(5*a*cos(d*x + c)^6 - 30*a*cos(d*x + c)^4 + 40*a*cos(d*x + c)^2 - 16*a)*si
n(d*x + c) + 55*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - 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)**7*csc(d*x+c)**7*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.30576, size = 140, normalized size = 1.22 \begin{align*} -\frac{60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a \sin \left (d x + c\right ) - \frac{147 \, a \sin \left (d x + c\right )^{6} - 180 \, a \sin \left (d x + c\right )^{5} - 90 \, a \sin \left (d x + c\right )^{4} + 60 \, a \sin \left (d x + c\right )^{3} + 45 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right ) - 10 \, a}{\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)^7*csc(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/60*(60*a*log(abs(sin(d*x + c))) + 60*a*sin(d*x + c) - (147*a*sin(d*x + c)^6 - 180*a*sin(d*x + c)^5 - 90*a*s
in(d*x + c)^4 + 60*a*sin(d*x + c)^3 + 45*a*sin(d*x + c)^2 - 12*a*sin(d*x + c) - 10*a)/sin(d*x + c)^6)/d