3.1.88 \(\int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^8} \, dx\) [88]

Optimal. Leaf size=127 \[ \frac {x}{a^8}-\frac {2 \cos ^7(c+d x)}{7 a d (a+a \sin (c+d x))^7}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {2 \cos ^3(c+d x)}{3 a^2 d \left (a^2+a^2 \sin (c+d x)\right )^3}+\frac {2 \cos (c+d x)}{d \left (a^8+a^8 \sin (c+d x)\right )} \]

[Out]

x/a^8-2/7*cos(d*x+c)^7/a/d/(a+a*sin(d*x+c))^7+2/5*cos(d*x+c)^5/a^3/d/(a+a*sin(d*x+c))^5-2/3*cos(d*x+c)^3/a^2/d
/(a^2+a^2*sin(d*x+c))^3+2*cos(d*x+c)/d/(a^8+a^8*sin(d*x+c))

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Rubi [A]
time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2759, 8} \begin {gather*} \frac {2 \cos (c+d x)}{d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {x}{a^8}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {2 \cos ^3(c+d x)}{3 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^8 - (2*Cos[c + d*x]^7)/(7*a*d*(a + a*Sin[c + d*x])^7) + (2*Cos[c + d*x]^5)/(5*a^3*d*(a + a*Sin[c + d*x])^5
) - (2*Cos[c + d*x]^3)/(3*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) + (2*Cos[c + d*x])/(d*(a^8 + a^8*Sin[c + d*x]))

Rule 8

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

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(275\) vs. \(2(127)=254\).
time = 6.06, size = 275, normalized size = 2.17 \begin {gather*} -\frac {2 \sqrt {2} \cos ^9(c+d x) \left (-1+\frac {1}{2} (1-\sin (c+d x))\right )^4 \left (\frac {\sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}}{\sqrt {2} \sqrt {1+\frac {1}{2} (-1+\sin (c+d x))}}+\frac {1-\sin (c+d x)}{2 \left (-1+\frac {1}{2} (1-\sin (c+d x))\right )}+\frac {(1-\sin (c+d x))^2}{12 \left (-1+\frac {1}{2} (1-\sin (c+d x))\right )^2}+\frac {(1-\sin (c+d x))^3}{40 \left (-1+\frac {1}{2} (1-\sin (c+d x))\right )^3}+\frac {(1-\sin (c+d x))^4}{112 \left (-1+\frac {1}{2} (1-\sin (c+d x))\right )^4}\right )}{a^8 d \left (1+\frac {1}{2} (-1+\sin (c+d x))\right )^{7/2} (1-\sin (c+d x))^5 (1+\sin (c+d x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*Cos[c + d*x]^9*(-1 + (1 - Sin[c + d*x])/2)^4*((ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin
[c + d*x]])/(Sqrt[2]*Sqrt[1 + (-1 + Sin[c + d*x])/2]) + (1 - Sin[c + d*x])/(2*(-1 + (1 - Sin[c + d*x])/2)) + (
1 - Sin[c + d*x])^2/(12*(-1 + (1 - Sin[c + d*x])/2)^2) + (1 - Sin[c + d*x])^3/(40*(-1 + (1 - Sin[c + d*x])/2)^
3) + (1 - Sin[c + d*x])^4/(112*(-1 + (1 - Sin[c + d*x])/2)^4)))/(a^8*d*(1 + (-1 + Sin[c + d*x])/2)^(7/2)*(1 -
Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))

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Maple [A]
time = 0.20, size = 110, normalized size = 0.87

method	result	size
risch	\(\frac {x}{a^{8}}+\frac {48 i {\mathrm e}^{5 i \left (d x +c \right )}+16 \,{\mathrm e}^{6 i \left (d x +c \right )}-\frac {352 i {\mathrm e}^{3 i \left (d x +c \right )}}{3}-\frac {352 \,{\mathrm e}^{4 i \left (d x +c \right )}}{3}+\frac {464 \,{\mathrm e}^{2 i \left (d x +c \right )}}{5}+\frac {464 i {\mathrm e}^{i \left (d x +c \right )}}{15}-\frac {704}{105}}{d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7}}\)	\(100\)
derivativedivides	\(\frac {-\frac {256}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {896}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {160}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{8}}\)	\(110\)
default	\(\frac {-\frac {256}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {896}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {160}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{8}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^8*(-128/7/(tan(1/2*d*x+1/2*c)+1)^7+64/(tan(1/2*d*x+1/2*c)+1)^6-448/5/(tan(1/2*d*x+1/2*c)+1)^5+64/(tan(1/
2*d*x+1/2*c)+1)^4-80/3/(tan(1/2*d*x+1/2*c)+1)^3+8/(tan(1/2*d*x+1/2*c)+1)^2+arctan(tan(1/2*d*x+1/2*c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (121) = 242\).
time = 0.53, size = 295, normalized size = 2.32 \begin {gather*} \frac {2 \, {\left (\frac {8 \, {\left (\frac {133 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {294 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {490 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {175 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 19\right )}}{a^{8} + \frac {7 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {35 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {21 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {7 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{8}}\right )}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(8*(133*sin(d*x + c)/(cos(d*x + c) + 1) + 294*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 490*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 + 175*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 105*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 19)/
(a^8 + 7*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 21*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 35*a^8*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 + 35*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 21*a^8*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 7*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) + 105*arctan(sin(d
*x + c)/(cos(d*x + c) + 1))/a^8)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (121) = 242\).
time = 0.36, size = 244, normalized size = 1.92 \begin {gather*} \frac {{\left (105 \, d x - 352\right )} \cos \left (d x + c\right )^{4} - {\left (315 \, d x + 568\right )} \cos \left (d x + c\right )^{3} - 24 \, {\left (35 \, d x - 31\right )} \cos \left (d x + c\right )^{2} + 840 \, d x + 60 \, {\left (7 \, d x + 12\right )} \cos \left (d x + c\right ) - {\left ({\left (105 \, d x + 352\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (35 \, d x - 18\right )} \cos \left (d x + c\right )^{2} - 840 \, d x - 60 \, {\left (7 \, d x + 16\right )} \cos \left (d x + c\right ) - 240\right )} \sin \left (d x + c\right ) - 240}{105 \, {\left (a^{8} d \cos \left (d x + c\right )^{4} - 3 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \, a^{8} d \cos \left (d x + c\right )^{2} + 4 \, a^{8} d \cos \left (d x + c\right ) + 8 \, a^{8} d - {\left (a^{8} d \cos \left (d x + c\right )^{3} + 4 \, a^{8} d \cos \left (d x + c\right )^{2} - 4 \, a^{8} d \cos \left (d x + c\right ) - 8 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*((105*d*x - 352)*cos(d*x + c)^4 - (315*d*x + 568)*cos(d*x + c)^3 - 24*(35*d*x - 31)*cos(d*x + c)^2 + 840
*d*x + 60*(7*d*x + 12)*cos(d*x + c) - ((105*d*x + 352)*cos(d*x + c)^3 + 12*(35*d*x - 18)*cos(d*x + c)^2 - 840*
d*x - 60*(7*d*x + 16)*cos(d*x + c) - 240)*sin(d*x + c) - 240)/(a^8*d*cos(d*x + c)^4 - 3*a^8*d*cos(d*x + c)^3 -
8*a^8*d*cos(d*x + c)^2 + 4*a^8*d*cos(d*x + c) + 8*a^8*d - (a^8*d*cos(d*x + c)^3 + 4*a^8*d*cos(d*x + c)^2 - 4*
a^8*d*cos(d*x + c) - 8*a^8*d)*sin(d*x + c))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 7.13, size = 99, normalized size = 0.78 \begin {gather*} \frac {\frac {105 \, {\left (d x + c\right )}}{a^{8}} + \frac {16 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 175 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 490 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 294 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 133 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*(d*x + c)/a^8 + 16*(105*tan(1/2*d*x + 1/2*c)^5 + 175*tan(1/2*d*x + 1/2*c)^4 + 490*tan(1/2*d*x + 1/2
*c)^3 + 294*tan(1/2*d*x + 1/2*c)^2 + 133*tan(1/2*d*x + 1/2*c) + 19)/(a^8*(tan(1/2*d*x + 1/2*c) + 1)^7))/d

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Mupad [B]
time = 7.77, size = 91, normalized size = 0.72 \begin {gather*} \frac {x}{a^8}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {304\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {304}{105}}{a^8\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^8 + ((304*tan(c/2 + (d*x)/2))/15 + (224*tan(c/2 + (d*x)/2)^2)/5 + (224*tan(c/2 + (d*x)/2)^3)/3 + (80*tan(c
/2 + (d*x)/2)^4)/3 + 16*tan(c/2 + (d*x)/2)^5 + 304/105)/(a^8*d*(tan(c/2 + (d*x)/2) + 1)^7)

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