\(\int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx\) [17]

Optimal result

Integrand size = 19, antiderivative size = 131 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \]

[Out]

a*arctanh(sin(d*x+c))/d-a*cot(d*x+c)/d-a*cot(d*x+c)^3/d-3/5*a*cot(d*x+c)^5/d-1/7*a*cot(d*x+c)^7/d-a*csc(d*x+c)
/d-1/3*a*csc(d*x+c)^3/d-1/5*a*csc(d*x+c)^5/d-1/7*a*csc(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3957, 2917, 2701, 308, 213, 3852} \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^3(c+d x)}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d} \]

[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 213

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

Rule 308

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 2701

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 2917

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 3852

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]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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]

Rubi steps \begin{align*} \text {integral}& = -\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 \text {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \text {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 \text {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 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = \frac {a \text {arctanh}(\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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {16 a \cot (c+d x)}{35 d}-\frac {8 a \cot (c+d x) \csc ^2(c+d x)}{35 d}-\frac {6 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac {a \cot (c+d x) \csc ^6(c+d x)}{7 d}-\frac {a \csc ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\sin ^2(c+d x)\right )}{7 d} \]

[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)

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {a \left (-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {16}{35}-\frac {\csc \left (d x +c \right )^{6}}{7}-\frac {6 \csc \left (d x +c \right )^{4}}{35}-\frac {8 \csc \left (d x +c \right )^{2}}{35}\right ) \cot \left (d x +c \right )}{d}\)	\(103\)
default	\(\frac {a \left (-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {16}{35}-\frac {\csc \left (d x +c \right )^{6}}{7}-\frac {6 \csc \left (d x +c \right )^{4}}{35}-\frac {8 \csc \left (d x +c \right )^{2}}{35}\right ) \cot \left (d x +c \right )}{d}\)	\(103\)
parallelrisch	\(-\frac {\left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\frac {56 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {203 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+448 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+203 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+448 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-448 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right ) a}{448 d}\)	\(121\)
norman	\(\frac {-\frac {a}{448 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{40 d}-\frac {29 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{192 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {29 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{24 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{320 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\)	\(158\)
risch	\(-\frac {2 i a \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}-210 \,{\mathrm e}^{10 i \left (d x +c \right )}-455 \,{\mathrm e}^{9 i \left (d x +c \right )}+1120 \,{\mathrm e}^{8 i \left (d x +c \right )}+686 \,{\mathrm e}^{7 i \left (d x +c \right )}-2492 \,{\mathrm e}^{6 i \left (d x +c \right )}-274 \,{\mathrm e}^{5 i \left (d x +c \right )}+1360 \,{\mathrm e}^{4 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}-402 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+48\right )}{105 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\)	\(195\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (121) = 242\).

Time = 0.27 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.15 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {96 \, a \cos \left (d x + c\right )^{6} + 114 \, a \cos \left (d x + c\right )^{5} - 450 \, a \cos \left (d x + c\right )^{4} - 250 \, a \cos \left (d x + c\right )^{3} + 670 \, a \cos \left (d x + c\right )^{2} - 105 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 142 \, a \cos \left (d x + c\right ) - 352 \, a}{210 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]

[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))

Sympy [F(-1)]

Timed out. \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} + 15\right )}}{\sin \left (d x + c\right )^{7}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {6 \, {\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{210 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6720 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3045 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1015 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{6720 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 14.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {29\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (64\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a}{7}\right )}{64\,d} \]

[In]

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

[Out]

(2*a*atanh(tan(c/2 + (d*x)/2)))/d - (29*a*tan(c/2 + (d*x)/2))/(64*d) - (a*tan(c/2 + (d*x)/2)^3)/(24*d) - (a*ta
n(c/2 + (d*x)/2)^5)/(320*d) - (cot(c/2 + (d*x)/2)^7*(a/7 + (8*a*tan(c/2 + (d*x)/2)^2)/5 + (29*a*tan(c/2 + (d*x
)/2)^4)/3 + 64*a*tan(c/2 + (d*x)/2)^6))/(64*d)