3.36 \(\int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=163 \[ \frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{7 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \cot ^3(c+d x)}{d}-\frac{5 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

(2*a^2*ArcTanh[Sin[c + d*x]])/d - (5*a^2*Cot[c + d*x])/d - (3*a^2*Cot[c + d*x]^3)/d - (7*a^2*Cot[c + d*x]^5)/(
5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d
*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/d

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Rubi [A]  time = 0.242792, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3872, 2873, 3767, 2621, 302, 207, 2620, 270} \[ \frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{7 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \cot ^3(c+d x)}{d}-\frac{5 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]

[Out]

(2*a^2*ArcTanh[Sin[c + d*x]])/d - (5*a^2*Cot[c + d*x])/d - (3*a^2*Cot[c + d*x]^3)/d - (7*a^2*Cot[c + d*x]^5)/(
5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d
*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/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 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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]

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 2620

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [B]  time = 1.23118, size = 428, normalized size = 2.63 \[ \frac{a^2 \cos (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^2 \left (-32 \csc (2 c) (-7264 \sin (c-d x)+14208 \sin (c+d x)-19536 \sin (2 (c+d x))+7104 \sin (3 (c+d x))+7104 \sin (4 (c+d x))-7104 \sin (5 (c+d x))+1776 \sin (6 (c+d x))+17288 \sin (2 c+d x)+20384 \sin (3 c+d x)-23771 \sin (c+2 d x)+7104 \sin (2 (c+2 d x))-23771 \sin (3 c+2 d x)-8960 \sin (4 c+2 d x)+19984 \sin (c+3 d x)+8644 \sin (2 c+3 d x)+8644 \sin (4 c+3 d x)-6160 \sin (5 c+3 d x)+8644 \sin (3 c+4 d x)+8644 \sin (5 c+4 d x)+6720 \sin (6 c+4 d x)-12144 \sin (3 c+5 d x)-8644 \sin (4 c+5 d x)-8644 \sin (6 c+5 d x)-1680 \sin (7 c+5 d x)+3456 \sin (4 c+6 d x)+2161 \sin (5 c+6 d x)+2161 \sin (7 c+6 d x)-9856 \sin (2 c)+17288 \sin (d x)-29056 \sin (2 d x)) \csc ^3(c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )-6881280 \cos (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+6881280 \cos (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{13762560 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]

[Out]

(a^2*Cos[c + d*x]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(-6881280*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2]] + 6881280*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 32*Csc[2*c]*Csc[(c + d*x)/2]^4*C
sc[c + d*x]^3*(-9856*Sin[2*c] + 17288*Sin[d*x] - 29056*Sin[2*d*x] - 7264*Sin[c - d*x] + 14208*Sin[c + d*x] - 1
9536*Sin[2*(c + d*x)] + 7104*Sin[3*(c + d*x)] + 7104*Sin[4*(c + d*x)] - 7104*Sin[5*(c + d*x)] + 1776*Sin[6*(c
+ d*x)] + 17288*Sin[2*c + d*x] + 20384*Sin[3*c + d*x] - 23771*Sin[c + 2*d*x] + 7104*Sin[2*(c + 2*d*x)] - 23771
*Sin[3*c + 2*d*x] - 8960*Sin[4*c + 2*d*x] + 19984*Sin[c + 3*d*x] + 8644*Sin[2*c + 3*d*x] + 8644*Sin[4*c + 3*d*
x] - 6160*Sin[5*c + 3*d*x] + 8644*Sin[3*c + 4*d*x] + 8644*Sin[5*c + 4*d*x] + 6720*Sin[6*c + 4*d*x] - 12144*Sin
[3*c + 5*d*x] - 8644*Sin[4*c + 5*d*x] - 8644*Sin[6*c + 5*d*x] - 1680*Sin[7*c + 5*d*x] + 3456*Sin[4*c + 6*d*x]
+ 2161*Sin[5*c + 6*d*x] + 2161*Sin[7*c + 6*d*x])))/(13762560*d)

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Maple [A]  time = 0.086, size = 264, normalized size = 1.6 \begin{align*} -{\frac{144\,{a}^{2}\cot \left ( dx+c \right ) }{35\,d}}-{\frac{{a}^{2}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,{a}^{2}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,{a}^{2}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{35\,d}}-{\frac{2\,{a}^{2}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{2\,{a}^{2}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2\,{a}^{2}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{{a}^{2}}{d\sin \left ( dx+c \right ) }}+2\,{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) }}-{\frac{8\,{a}^{2}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) }}-{\frac{16\,{a}^{2}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) }}+{\frac{64\,{a}^{2}}{35\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }} \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))^2,x)

[Out]

-144/35*a^2*cot(d*x+c)/d-1/7/d*a^2*cot(d*x+c)*csc(d*x+c)^6-6/35/d*a^2*cot(d*x+c)*csc(d*x+c)^4-8/35/d*a^2*cot(d
*x+c)*csc(d*x+c)^2-2/7/d*a^2/sin(d*x+c)^7-2/5/d*a^2/sin(d*x+c)^5-2/3/d*a^2/sin(d*x+c)^3-2/d*a^2/sin(d*x+c)+2/d
*a^2*ln(sec(d*x+c)+tan(d*x+c))-1/7/d*a^2/sin(d*x+c)^7/cos(d*x+c)-8/35/d*a^2/sin(d*x+c)^5/cos(d*x+c)-16/35/d*a^
2/sin(d*x+c)^3/cos(d*x+c)+64/35/d*a^2/sin(d*x+c)/cos(d*x+c)

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Maxima [A]  time = 1.03769, size = 236, normalized size = 1.45 \begin{align*} -\frac{a^{2}{\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 )} + 3 \, a^{2}{\left (\frac{140 \, \tan \left (d x + c\right )^{6} + 70 \, \tan \left (d x + c\right )^{4} + 28 \, \tan \left (d x + c\right )^{2} + 5}{\tan \left (d x + c\right )^{7}} - 35 \, \tan \left (d x + c\right )\right )} + \frac{3 \,{\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^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, 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))^2,x, algorithm="maxima")

[Out]

-1/105*(a^2*(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)) + 3*a^2*((140*tan(d*x + c)^6 + 70*tan(d*x + c)^4 + 28*tan(d*x + c)^
2 + 5)/tan(d*x + c)^7 - 35*tan(d*x + c)) + 3*(35*tan(d*x + c)^6 + 35*tan(d*x + c)^4 + 21*tan(d*x + c)^2 + 5)*a
^2/tan(d*x + c)^7)/d

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Fricas [A]  time = 1.87472, size = 694, normalized size = 4.26 \begin{align*} -\frac{432 \, a^{2} \cos \left (d x + c\right )^{6} - 654 \, a^{2} \cos \left (d x + c\right )^{5} - 636 \, a^{2} \cos \left (d x + c\right )^{4} + 1226 \, a^{2} \cos \left (d x + c\right )^{3} + 74 \, a^{2} \cos \left (d x + c\right )^{2} - 562 \, a^{2} \cos \left (d x + c\right ) - 105 \,{\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \,{\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \, a^{2}}{105 \,{\left (d \cos \left (d x + c\right )^{5} - 2 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right )\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))^2,x, algorithm="fricas")

[Out]

-1/105*(432*a^2*cos(d*x + c)^6 - 654*a^2*cos(d*x + c)^5 - 636*a^2*cos(d*x + c)^4 + 1226*a^2*cos(d*x + c)^3 + 7
4*a^2*cos(d*x + c)^2 - 562*a^2*cos(d*x + c) - 105*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x +
c)^2 - a^2*cos(d*x + c))*log(sin(d*x + c) + 1)*sin(d*x + c) + 105*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4
+ 2*a^2*cos(d*x + c)^2 - a^2*cos(d*x + c))*log(-sin(d*x + c) + 1)*sin(d*x + c) + 105*a^2)/((d*cos(d*x + c)^5 -
2*d*cos(d*x + c)^4 + 2*d*cos(d*x + c)^2 - d*cos(d*x + c))*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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.42212, size = 227, normalized size = 1.39 \begin{align*} \frac{35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6720 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 6720 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 945 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{6720 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} - \frac{10710 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 189 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{3360 \, 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))^2,x, algorithm="giac")

[Out]

1/3360*(35*a^2*tan(1/2*d*x + 1/2*c)^3 + 6720*a^2*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 6720*a^2*log(abs(tan(1/2
*d*x + 1/2*c) - 1)) + 945*a^2*tan(1/2*d*x + 1/2*c) - 6720*a^2*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1
) - (10710*a^2*tan(1/2*d*x + 1/2*c)^6 + 1330*a^2*tan(1/2*d*x + 1/2*c)^4 + 189*a^2*tan(1/2*d*x + 1/2*c)^2 + 15*
a^2)/tan(1/2*d*x + 1/2*c)^7)/d