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

Optimal. Leaf size=162 \[ -\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left (a^2 \sin (c+d x)+a^2\right )^4}{d}-\frac{16 \left (a^4 \sin (c+d x)+a^4\right )^2}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d} \]

[Out]

(-128*a^8*Log[1 - Sin[c + d*x]])/d - (64*a^8*Sin[c + d*x])/d - (16*a^5*(a + a*Sin[c + d*x])^3)/(3*d) - (4*a^3*
(a + a*Sin[c + d*x])^5)/(5*d) - (a^2*(a + a*Sin[c + d*x])^6)/(3*d) - (a*(a + a*Sin[c + d*x])^7)/(7*d) - (2*(a^
2 + a^2*Sin[c + d*x])^4)/d - (16*(a^4 + a^4*Sin[c + d*x])^2)/d

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Rubi [A]  time = 0.0768911, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left (a^2 \sin (c+d x)+a^2\right )^4}{d}-\frac{16 \left (a^4 \sin (c+d x)+a^4\right )^2}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-128*a^8*Log[1 - Sin[c + d*x]])/d - (64*a^8*Sin[c + d*x])/d - (16*a^5*(a + a*Sin[c + d*x])^3)/(3*d) - (4*a^3*
(a + a*Sin[c + d*x])^5)/(5*d) - (a^2*(a + a*Sin[c + d*x])^6)/(3*d) - (a*(a + a*Sin[c + d*x])^7)/(7*d) - (2*(a^
2 + a^2*Sin[c + d*x])^4)/d - (16*(a^4 + a^4*Sin[c + d*x])^2)/d

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^7}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (-64 a^6+\frac{128 a^7}{a-x}-32 a^5 (a+x)-16 a^4 (a+x)^2-8 a^3 (a+x)^3-4 a^2 (a+x)^4-2 a (a+x)^5-(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a+a \sin (c+d x))^3}{3 d}-\frac{4 a^3 (a+a \sin (c+d x))^5}{5 d}-\frac{a^2 (a+a \sin (c+d x))^6}{3 d}-\frac{a (a+a \sin (c+d x))^7}{7 d}-\frac{2 \left (a^2+a^2 \sin (c+d x)\right )^4}{d}-\frac{16 \left (a^4+a^4 \sin (c+d x)\right )^2}{d}\\ \end{align*}

Mathematica [A]  time = 0.176694, size = 95, normalized size = 0.59 \[ \frac{a^8 \left (-\frac{1}{7} \sin ^7(c+d x)-\frac{4}{3} \sin ^6(c+d x)-\frac{29}{5} \sin ^5(c+d x)-16 \sin ^4(c+d x)-33 \sin ^3(c+d x)-60 \sin ^2(c+d x)-127 \sin (c+d x)-128 \log (1-\sin (c+d x))\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^8*(-128*Log[1 - Sin[c + d*x]] - 127*Sin[c + d*x] - 60*Sin[c + d*x]^2 - 33*Sin[c + d*x]^3 - 16*Sin[c + d*x]^
4 - (29*Sin[c + d*x]^5)/5 - (4*Sin[c + d*x]^6)/3 - Sin[c + d*x]^7/7))/d

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Maple [A]  time = 0.076, size = 149, normalized size = 0.9 \begin{align*} -128\,{\frac{{a}^{8}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+128\,{\frac{{a}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{4\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3\,d}}-{\frac{29\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-16\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-33\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}}-60\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-127\,{\frac{{a}^{8}\sin \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128/d*a^8*ln(cos(d*x+c))+128/d*a^8*ln(sec(d*x+c)+tan(d*x+c))-1/7/d*a^8*sin(d*x+c)^7-4/3/d*a^8*sin(d*x+c)^6-29
/5*a^8*sin(d*x+c)^5/d-16*a^8*sin(d*x+c)^4/d-33*a^8*sin(d*x+c)^3/d-60*a^8*sin(d*x+c)^2/d-127*a^8*sin(d*x+c)/d

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Maxima [A]  time = 0.950359, size = 147, normalized size = 0.91 \begin{align*} -\frac{15 \, a^{8} \sin \left (d x + c\right )^{7} + 140 \, a^{8} \sin \left (d x + c\right )^{6} + 609 \, a^{8} \sin \left (d x + c\right )^{5} + 1680 \, a^{8} \sin \left (d x + c\right )^{4} + 3465 \, a^{8} \sin \left (d x + c\right )^{3} + 6300 \, a^{8} \sin \left (d x + c\right )^{2} + 13440 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 13335 \, a^{8} \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(15*a^8*sin(d*x + c)^7 + 140*a^8*sin(d*x + c)^6 + 609*a^8*sin(d*x + c)^5 + 1680*a^8*sin(d*x + c)^4 + 34
65*a^8*sin(d*x + c)^3 + 6300*a^8*sin(d*x + c)^2 + 13440*a^8*log(sin(d*x + c) - 1) + 13335*a^8*sin(d*x + c))/d

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Fricas [A]  time = 1.83921, size = 302, normalized size = 1.86 \begin{align*} \frac{140 \, a^{8} \cos \left (d x + c\right )^{6} - 2100 \, a^{8} \cos \left (d x + c\right )^{4} + 10080 \, a^{8} \cos \left (d x + c\right )^{2} - 13440 \, a^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (5 \, a^{8} \cos \left (d x + c\right )^{6} - 218 \, a^{8} \cos \left (d x + c\right )^{4} + 1576 \, a^{8} \cos \left (d x + c\right )^{2} - 5808 \, a^{8}\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(140*a^8*cos(d*x + c)^6 - 2100*a^8*cos(d*x + c)^4 + 10080*a^8*cos(d*x + c)^2 - 13440*a^8*log(-sin(d*x +
c) + 1) + 3*(5*a^8*cos(d*x + c)^6 - 218*a^8*cos(d*x + c)^4 + 1576*a^8*cos(d*x + c)^2 - 5808*a^8)*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(sec(d*x+c)*(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.27936, size = 389, normalized size = 2.4 \begin{align*} \frac{2 \,{\left (6720 \, a^{8} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 13440 \, a^{8} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{17424 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 13335 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 134568 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 93870 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 442344 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 265209 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 780640 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 370308 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 780640 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 265209 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 442344 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 93870 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 134568 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 13335 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 17424 \, a^{8}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7}}\right )}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(6720*a^8*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 13440*a^8*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - (17424*a^8*ta
n(1/2*d*x + 1/2*c)^14 + 13335*a^8*tan(1/2*d*x + 1/2*c)^13 + 134568*a^8*tan(1/2*d*x + 1/2*c)^12 + 93870*a^8*tan
(1/2*d*x + 1/2*c)^11 + 442344*a^8*tan(1/2*d*x + 1/2*c)^10 + 265209*a^8*tan(1/2*d*x + 1/2*c)^9 + 780640*a^8*tan
(1/2*d*x + 1/2*c)^8 + 370308*a^8*tan(1/2*d*x + 1/2*c)^7 + 780640*a^8*tan(1/2*d*x + 1/2*c)^6 + 265209*a^8*tan(1
/2*d*x + 1/2*c)^5 + 442344*a^8*tan(1/2*d*x + 1/2*c)^4 + 93870*a^8*tan(1/2*d*x + 1/2*c)^3 + 134568*a^8*tan(1/2*
d*x + 1/2*c)^2 + 13335*a^8*tan(1/2*d*x + 1/2*c) + 17424*a^8)/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d