3.1104 \(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx\)

Optimal. Leaf size=136 \[ -\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{b \cot ^5(c+d x)}{5 d} \]

[Out]

(-3*a*ArcTanh[Cos[c + d*x]])/(128*d) - (b*Cot[c + d*x]^5)/(5*d) - (b*Cot[c + d*x]^7)/(7*d) - (3*a*Cot[c + d*x]
*Csc[c + d*x])/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a*
Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)

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Rubi [A]  time = 0.18414, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2838, 2611, 3768, 3770, 2607, 14} \[ -\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{b \cot ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x]),x]

[Out]

(-3*a*ArcTanh[Cos[c + d*x]])/(128*d) - (b*Cot[c + d*x]^5)/(5*d) - (b*Cot[c + d*x]^7)/(7*d) - (3*a*Cot[c + d*x]
*Csc[c + d*x])/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a*
Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+b \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{1}{8} (3 a) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{16} a \int \csc ^5(c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{b \cot ^5(c+d x)}{5 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{64} (3 a) \int \csc ^3(c+d x) \, dx\\ &=-\frac{b \cot ^5(c+d x)}{5 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{128} (3 a) \int \csc (c+d x) \, dx\\ &=-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{b \cot ^5(c+d x)}{5 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end{align*}

Mathematica [B]  time = 0.0794382, size = 279, normalized size = 2.05 \[ -\frac{a \csc ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}+\frac{a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{3 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{a \sec ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}-\frac{a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{512 d}-\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{3 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{3 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}-\frac{3 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}-\frac{2 b \cot (c+d x)}{35 d}-\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac{8 b \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{35 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x]),x]

[Out]

(-2*b*Cot[c + d*x])/(35*d) - (3*a*Csc[(c + d*x)/2]^2)/(512*d) + (a*Csc[(c + d*x)/2]^4)/(1024*d) + (a*Csc[(c +
d*x)/2]^6)/(512*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) - (b*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) + (8*b*Cot[c + d
*x]*Csc[c + d*x]^4)/(35*d) - (b*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (3*a*Log[Cos[(c + d*x)/2]])/(128*d) + (3*
a*Log[Sin[(c + d*x)/2]])/(128*d) + (3*a*Sec[(c + d*x)/2]^2)/(512*d) - (a*Sec[(c + d*x)/2]^4)/(1024*d) - (a*Sec
[(c + d*x)/2]^6)/(512*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)

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Maple [A]  time = 0.066, size = 182, normalized size = 1.3 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{64\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{128\,d}}+{\frac{3\,\cos \left ( dx+c \right ) a}{128\,d}}+{\frac{3\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{2\,b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c)),x)

[Out]

-1/8/d*a/sin(d*x+c)^8*cos(d*x+c)^5-1/16/d*a/sin(d*x+c)^6*cos(d*x+c)^5-1/64/d*a/sin(d*x+c)^4*cos(d*x+c)^5+1/128
/d*a/sin(d*x+c)^2*cos(d*x+c)^5+1/128*a*cos(d*x+c)^3/d+3/128*a*cos(d*x+c)/d+3/128/d*a*ln(csc(d*x+c)-cot(d*x+c))
-1/7/d*b/sin(d*x+c)^7*cos(d*x+c)^5-2/35/d*b/sin(d*x+c)^5*cos(d*x+c)^5

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Maxima [A]  time = 1.00146, size = 186, normalized size = 1.37 \begin{align*} \frac{35 \, a{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{256 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} b}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/8960*(35*a*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*cos(d*x + c))/(cos(d*x + c)^8 -
4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1
)) - 256*(7*tan(d*x + c)^2 + 5)*b/tan(d*x + c)^7)/d

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Fricas [A]  time = 1.85859, size = 656, normalized size = 4.82 \begin{align*} \frac{210 \, a \cos \left (d x + c\right )^{7} - 770 \, a \cos \left (d x + c\right )^{5} - 770 \, a \cos \left (d x + c\right )^{3} + 210 \, a \cos \left (d x + c\right ) - 105 \,{\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 105 \,{\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 256 \,{\left (2 \, b \cos \left (d x + c\right )^{7} - 7 \, b \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, 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)^4*csc(d*x+c)^9*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8960*(210*a*cos(d*x + c)^7 - 770*a*cos(d*x + c)^5 - 770*a*cos(d*x + c)^3 + 210*a*cos(d*x + c) - 105*(a*cos(d
*x + c)^8 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 + a)*log(1/2*cos(d*x + c) + 1/2) + 10
5*(a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2 + a)*log(-1/2*cos(d*x + c)
+ 1/2) + 256*(2*b*cos(d*x + c)^7 - 7*b*cos(d*x + c)^5)*sin(d*x + c))/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 +
6*d*cos(d*x + c)^4 - 4*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)**4*csc(d*x+c)**9*(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.38671, size = 271, normalized size = 1.99 \begin{align*} \frac{35 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 80 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 112 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 280 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 560 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1680 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 1680 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{4566 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1680 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 560 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 280 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 112 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 80 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 35 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{71680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/71680*(35*a*tan(1/2*d*x + 1/2*c)^8 + 80*b*tan(1/2*d*x + 1/2*c)^7 - 112*b*tan(1/2*d*x + 1/2*c)^5 - 280*a*tan(
1/2*d*x + 1/2*c)^4 - 560*b*tan(1/2*d*x + 1/2*c)^3 + 1680*a*log(abs(tan(1/2*d*x + 1/2*c))) + 1680*b*tan(1/2*d*x
+ 1/2*c) - (4566*a*tan(1/2*d*x + 1/2*c)^8 + 1680*b*tan(1/2*d*x + 1/2*c)^7 - 560*b*tan(1/2*d*x + 1/2*c)^5 - 28
0*a*tan(1/2*d*x + 1/2*c)^4 - 112*b*tan(1/2*d*x + 1/2*c)^3 + 80*b*tan(1/2*d*x + 1/2*c) + 35*a)/tan(1/2*d*x + 1/
2*c)^8)/d