3.10 \(\int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx\)

Optimal. Leaf size=165 \[ -\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^7(c+d x) \cos (c+d x)}{8 d}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{48 d}-\frac{35 a \sin ^3(c+d x) \cos (c+d x)}{192 d}-\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128} \]

[Out]

(35*a*x)/128 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (
a*Sin[c + d*x]^3)/(3*d) - (35*a*Cos[c + d*x]*Sin[c + d*x]^3)/(192*d) - (a*Sin[c + d*x]^5)/(5*d) - (7*a*Cos[c +
d*x]*Sin[c + d*x]^5)/(48*d) - (a*Sin[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]*Sin[c + d*x]^7)/(8*d)

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Rubi [A]  time = 0.145638, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3872, 2838, 2592, 302, 206, 2635, 8} \[ -\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^7(c+d x) \cos (c+d x)}{8 d}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{48 d}-\frac{35 a \sin ^3(c+d x) \cos (c+d x)}{192 d}-\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*a*x)/128 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (
a*Sin[c + d*x]^3)/(3*d) - (35*a*Cos[c + d*x]*Sin[c + d*x]^3)/(192*d) - (a*Sin[c + d*x]^5)/(5*d) - (7*a*Cos[c +
d*x]*Sin[c + d*x]^5)/(48*d) - (a*Sin[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]*Sin[c + d*x]^7)/(8*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 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

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 206

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

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^7(c+d x) \tan (c+d x) \, dx\\ &=a \int \sin ^8(c+d x) \, dx+a \int \sin ^7(c+d x) \tan (c+d x) \, dx\\ &=-\frac{a \cos (c+d x) \sin ^7(c+d x)}{8 d}+\frac{1}{8} (7 a) \int \sin ^6(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \frac{x^8}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{48 d}-\frac{a \cos (c+d x) \sin ^7(c+d x)}{8 d}+\frac{1}{48} (35 a) \int \sin ^4(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \left (-1-x^2-x^4-x^6+\frac{1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a \sin (c+d x)}{d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{35 a \cos (c+d x) \sin ^3(c+d x)}{192 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{48 d}-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos (c+d x) \sin ^7(c+d x)}{8 d}+\frac{1}{64} (35 a) \int \sin ^2(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}-\frac{35 a \cos (c+d x) \sin (c+d x)}{128 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{35 a \cos (c+d x) \sin ^3(c+d x)}{192 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{48 d}-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos (c+d x) \sin ^7(c+d x)}{8 d}+\frac{1}{128} (35 a) \int 1 \, dx\\ &=\frac{35 a x}{128}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}-\frac{35 a \cos (c+d x) \sin (c+d x)}{128 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{35 a \cos (c+d x) \sin ^3(c+d x)}{192 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{48 d}-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos (c+d x) \sin ^7(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.330181, size = 106, normalized size = 0.64 \[ \frac{a \left (-15360 \sin ^7(c+d x)-21504 \sin ^5(c+d x)-35840 \sin ^3(c+d x)-107520 \sin (c+d x)+35 (-672 \sin (2 (c+d x))+168 \sin (4 (c+d x))-32 \sin (6 (c+d x))+3 \sin (8 (c+d x))+840 c+840 d x)+107520 \tanh ^{-1}(\sin (c+d x))\right )}{107520 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(107520*ArcTanh[Sin[c + d*x]] - 107520*Sin[c + d*x] - 35840*Sin[c + d*x]^3 - 21504*Sin[c + d*x]^5 - 15360*S
in[c + d*x]^7 + 35*(840*c + 840*d*x - 672*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] - 32*Sin[6*(c + d*x)] + 3*Si
n[8*(c + d*x)])))/(107520*d)

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Maple [A]  time = 0.094, size = 164, normalized size = 1. \begin{align*} -{\frac{a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{8\,d}}-{\frac{7\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{48\,d}}-{\frac{35\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{192\,d}}-{\frac{35\,a\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{128\,d}}+{\frac{35\,ax}{128}}+{\frac{35\,ac}{128\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{a\sin \left ( dx+c \right ) }{d}}+{\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((a+a*sec(d*x+c))*sin(d*x+c)^8,x)

[Out]

-1/8*a*cos(d*x+c)*sin(d*x+c)^7/d-7/48*a*cos(d*x+c)*sin(d*x+c)^5/d-35/192*a*cos(d*x+c)*sin(d*x+c)^3/d-35/128*a*
cos(d*x+c)*sin(d*x+c)/d+35/128*a*x+35/128/d*a*c-1/7*a*sin(d*x+c)^7/d-1/5*a*sin(d*x+c)^5/d-1/3*a*sin(d*x+c)^3/d
-a*sin(d*x+c)/d+1/d*a*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 0.990094, size = 171, normalized size = 1.04 \begin{align*} -\frac{512 \,{\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a - 35 \,{\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{107520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/107520*(512*(30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 - 105*log(sin(d*x + c) + 1) + 105*lo
g(sin(d*x + c) - 1) + 210*sin(d*x + c))*a - 35*(128*sin(2*d*x + 2*c)^3 + 840*d*x + 840*c + 3*sin(8*d*x + 8*c)
+ 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a)/d

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Fricas [A]  time = 1.9058, size = 378, normalized size = 2.29 \begin{align*} \frac{3675 \, a d x + 6720 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 6720 \, a \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (1680 \, a \cos \left (d x + c\right )^{7} + 1920 \, a \cos \left (d x + c\right )^{6} - 7000 \, a \cos \left (d x + c\right )^{5} - 8448 \, a \cos \left (d x + c\right )^{4} + 11410 \, a \cos \left (d x + c\right )^{3} + 15616 \, a \cos \left (d x + c\right )^{2} - 9765 \, a \cos \left (d x + c\right ) - 22528 \, a\right )} \sin \left (d x + c\right )}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(3675*a*d*x + 6720*a*log(sin(d*x + c) + 1) - 6720*a*log(-sin(d*x + c) + 1) + (1680*a*cos(d*x + c)^7 +
1920*a*cos(d*x + c)^6 - 7000*a*cos(d*x + c)^5 - 8448*a*cos(d*x + c)^4 + 11410*a*cos(d*x + c)^3 + 15616*a*cos(d
*x + c)^2 - 9765*a*cos(d*x + c) - 22528*a)*sin(d*x + c))/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((a+a*sec(d*x+c))*sin(d*x+c)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.51213, size = 235, normalized size = 1.42 \begin{align*} \frac{3675 \,{\left (d x + c\right )} a + 13440 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 13440 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (9765 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 83825 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 321013 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 724649 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1078359 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 508683 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 140175 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 17115 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(3675*(d*x + c)*a + 13440*a*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 13440*a*log(abs(tan(1/2*d*x + 1/2*c)
- 1)) - 2*(9765*a*tan(1/2*d*x + 1/2*c)^15 + 83825*a*tan(1/2*d*x + 1/2*c)^13 + 321013*a*tan(1/2*d*x + 1/2*c)^11
+ 724649*a*tan(1/2*d*x + 1/2*c)^9 + 1078359*a*tan(1/2*d*x + 1/2*c)^7 + 508683*a*tan(1/2*d*x + 1/2*c)^5 + 1401
75*a*tan(1/2*d*x + 1/2*c)^3 + 17115*a*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d