3.17 \(\int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx\)

Optimal. Leaf size=131 \[ -\frac{a \cot ^7(c+d x)}{7 d}-\frac{3 a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/d - (3*a*Cot[c + d*x]^5)/(5*d) - (a*Cot[
c + d*x]^7)/(7*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]
^7)/(7*d)

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Rubi [A]  time = 0.117004, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3872, 2838, 2621, 302, 207, 3767} \[ -\frac{a \cot ^7(c+d x)}{7 d}-\frac{3 a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^8*(a + a*Sec[c + d*x]),x]

[Out]

(a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/d - (3*a*Cot[c + d*x]^5)/(5*d) - (a*Cot[
c + d*x]^7)/(7*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]
^7)/(7*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

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 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

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

Mathematica [C]  time = 0.0481466, size = 113, normalized size = 0.86 \[ -\frac{a \csc ^7(c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},\sin ^2(c+d x)\right )}{7 d}-\frac{16 a \cot (c+d x)}{35 d}-\frac{a \cot (c+d x) \csc ^6(c+d x)}{7 d}-\frac{6 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{8 a \cot (c+d x) \csc ^2(c+d x)}{35 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^8*(a + a*Sec[c + d*x]),x]

[Out]

(-16*a*Cot[c + d*x])/(35*d) - (8*a*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) - (6*a*Cot[c + d*x]*Csc[c + d*x]^4)/(35
*d) - (a*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (a*Csc[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, Sin[c + d*x]^
2])/(7*d)

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Maple [A]  time = 0.122, size = 149, normalized size = 1.1 \begin{align*} -{\frac{16\,a\cot \left ( dx+c \right ) }{35\,d}}-{\frac{a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{35\,d}}-{\frac{a}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{a}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{a}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a}{d\sin \left ( dx+c \right ) }}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^8*(a+a*sec(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.02553, size = 157, normalized size = 1.2 \begin{align*} -\frac{a{\left (\frac{2 \,{\left (105 \, \sin \left (d x + c\right )^{6} + 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} + 15\right )}}{\sin \left (d x + c\right )^{7}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac{6 \,{\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{210 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(a*(2*(105*sin(d*x + c)^6 + 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 + 15)/sin(d*x + c)^7 - 105*log(sin(d*
x + c) + 1) + 105*log(sin(d*x + c) - 1)) + 6*(35*tan(d*x + c)^6 + 35*tan(d*x + c)^4 + 21*tan(d*x + c)^2 + 5)*a
/tan(d*x + c)^7)/d

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Fricas [B]  time = 1.81187, size = 745, normalized size = 5.69 \begin{align*} -\frac{96 \, a \cos \left (d x + c\right )^{6} + 114 \, a \cos \left (d x + c\right )^{5} - 450 \, a \cos \left (d x + c\right )^{4} - 250 \, a \cos \left (d x + c\right )^{3} + 670 \, a \cos \left (d x + c\right )^{2} - 105 \,{\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \,{\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 142 \, a \cos \left (d x + c\right ) - 352 \, a}{210 \,{\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^8*(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/210*(96*a*cos(d*x + c)^6 + 114*a*cos(d*x + c)^5 - 450*a*cos(d*x + c)^4 - 250*a*cos(d*x + c)^3 + 670*a*cos(d
*x + c)^2 - 105*(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c
) - a)*log(sin(d*x + c) + 1)*sin(d*x + c) + 105*(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*
a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(-sin(d*x + c) + 1)*sin(d*x + c) + 142*a*cos(d*x + c) - 352*a)/((d*c
os(d*x + c)^5 - d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 + d*cos(d*x + c) - 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(csc(d*x+c)**8*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 2.06734, size = 184, normalized size = 1.4 \begin{align*} -\frac{21 \, a \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 )^{3} - 6720 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 6720 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 3045 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{6720 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1015 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 168 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6720*(21*a*tan(1/2*d*x + 1/2*c)^5 + 280*a*tan(1/2*d*x + 1/2*c)^3 - 6720*a*log(abs(tan(1/2*d*x + 1/2*c) + 1)
) + 6720*a*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 3045*a*tan(1/2*d*x + 1/2*c) + (6720*a*tan(1/2*d*x + 1/2*c)^6 +
1015*a*tan(1/2*d*x + 1/2*c)^4 + 168*a*tan(1/2*d*x + 1/2*c)^2 + 15*a)/tan(1/2*d*x + 1/2*c)^7)/d