3.1.62 \(\int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [62]

Optimal. Leaf size=104 \[ \frac {7 x}{16 a^2}+\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )} \]

[Out]

7/16*x/a^2+7/30*cos(d*x+c)^5/a^2/d+7/16*cos(d*x+c)*sin(d*x+c)/a^2/d+7/24*cos(d*x+c)^3*sin(d*x+c)/a^2/d+1/6*cos
(d*x+c)^7/d/(a^2+a^2*sin(d*x+c))

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Rubi [A]
time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2758, 2761, 2715, 8} \begin {gather*} \frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{24 a^2 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac {7 x}{16 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]

[Out]

(7*x)/(16*a^2) + (7*Cos[c + d*x]^5)/(30*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (7*Cos[c + d*x]^3*
Sin[c + d*x])/(24*a^2*d) + Cos[c + d*x]^7/(6*d*(a^2 + a^2*Sin[c + d*x]))

Rule 8

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

Rule 2715

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] && Integ
erQ[2*n]

Rule 2758

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

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \frac {\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{6 a}\\ &=\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \cos ^4(c+d x) \, dx}{6 a^2}\\ &=\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \cos ^2(c+d x) \, dx}{8 a^2}\\ &=\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int 1 \, dx}{16 a^2}\\ &=\frac {7 x}{16 a^2}+\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.74, size = 151, normalized size = 1.45 \begin {gather*} -\frac {\cos ^9(c+d x) \left (-210 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (96+39 \sin (c+d x)-327 \sin ^2(c+d x)+202 \sin ^3(c+d x)+86 \sin ^4(c+d x)-136 \sin ^5(c+d x)+40 \sin ^6(c+d x)\right )\right )}{240 a^2 d (-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])^2,x]

[Out]

-1/240*(Cos[c + d*x]^9*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c +
d*x]]*(96 + 39*Sin[c + d*x] - 327*Sin[c + d*x]^2 + 202*Sin[c + d*x]^3 + 86*Sin[c + d*x]^4 - 136*Sin[c + d*x]^5
+ 40*Sin[c + d*x]^6)))/(a^2*d*(-1 + Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))

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Maple [A]
time = 0.16, size = 181, normalized size = 1.74

method	result	size
risch	\(\frac {7 x}{16 a^{2}}+\frac {\cos \left (d x +c \right )}{4 a^{2} d}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}+\frac {\cos \left (5 d x +5 c \right )}{40 a^{2} d}+\frac {\sin \left (4 d x +4 c \right )}{64 a^{2} d}+\frac {\cos \left (3 d x +3 c \right )}{8 a^{2} d}+\frac {17 \sin \left (2 d x +2 c \right )}{64 a^{2} d}\)	\(107\)
derivativedivides	\(\frac {\frac {2 \left (-\frac {9 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {89 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {89 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}+\frac {2}{5}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\)	\(181\)
default	\(\frac {\frac {2 \left (-\frac {9 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {89 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {89 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}+\frac {2}{5}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

2/d/a^2*((-9/16*tan(1/2*d*x+1/2*c)^11+2*tan(1/2*d*x+1/2*c)^10-89/48*tan(1/2*d*x+1/2*c)^9+2*tan(1/2*d*x+1/2*c)^
8+11/8*tan(1/2*d*x+1/2*c)^7+4*tan(1/2*d*x+1/2*c)^6-11/8*tan(1/2*d*x+1/2*c)^5+4*tan(1/2*d*x+1/2*c)^4+89/48*tan(
1/2*d*x+1/2*c)^3+2/5*tan(1/2*d*x+1/2*c)^2+9/16*tan(1/2*d*x+1/2*c)+2/5)/(1+tan(1/2*d*x+1/2*c)^2)^6+7/16*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. 393 vs. \(2 (94) = 188\).
time = 0.85, size = 393, normalized size = 3.78 \begin {gather*} \frac {\frac {\frac {135 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {96 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {445 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {960 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {330 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {445 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {480 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {135 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 96}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, 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))^2,x, algorithm="maxima")

[Out]

1/120*((135*sin(d*x + c)/(cos(d*x + c) + 1) + 96*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 445*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 + 960*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 330*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 960*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6 + 330*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 480*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 - 445*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 480*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 135*sin(d*x + c
)^11/(cos(d*x + c) + 1)^11 + 96)/(a^2 + 6*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^2*sin(d*x + c)^4/(cos
(d*x + c) + 1)^4 + 20*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6
*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) + 105*arctan(sin(d*x +
c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [A]
time = 0.36, size = 60, normalized size = 0.58 \begin {gather*} \frac {96 \, \cos \left (d x + c\right )^{5} + 105 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} - 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/240*(96*cos(d*x + c)^5 + 105*d*x - 5*(8*cos(d*x + c)^5 - 14*cos(d*x + c)^3 - 21*cos(d*x + c))*sin(d*x + c))/
(a^2*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2531 vs. \(2 (95) = 190\).
time = 78.33, size = 2531, normalized size = 24.34 \begin {gather*} \text {Too large to display} \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))**2,x)

[Out]

Piecewise((105*d*x*tan(c/2 + d*x/2)**12/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 +
3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**
2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 630*d*x*tan(c/2 + d*x/2)**10/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a
**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*t
an(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 1575*d*x*tan(c/2 + d*x/2)**8/(240*a**2*d*
tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/
2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 2100*d*x*tan
(c/2 + d*x/2)**6/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d
*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2
+ 240*a**2*d) + 1575*d*x*tan(c/2 + d*x/2)**4/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)*
*10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 14
40*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 630*d*x*tan(c/2 + d*x/2)**2/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1
440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**
2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 105*d*x/(240*a**2*d*tan(c/2 + d*x/2)
**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 +
3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) - 270*tan(c/2 + d*x/2)**11/(24
0*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2
*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 960
*tan(c/2 + d*x/2)**10/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/
2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/
2)**2 + 240*a**2*d) - 890*tan(c/2 + d*x/2)**9/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)*
*10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 14
40*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 960*tan(c/2 + d*x/2)**8/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*
a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*
tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 660*tan(c/2 + d*x/2)**7/(240*a**2*d*tan(
c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 +
d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 1920*tan(c/2 + d
*x/2)**6/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8
+ 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a
**2*d) - 660*tan(c/2 + d*x/2)**5/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a*
*2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan
(c/2 + d*x/2)**2 + 240*a**2*d) + 1920*tan(c/2 + d*x/2)**4/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c
/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*
x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 890*tan(c/2 + d*x/2)**3/(240*a**2*d*tan(c/2 + d*x/2)
**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 +
3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 192*tan(c/2 + d*x/2)**2/(240
*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*
d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 270*
tan(c/2 + d*x/2)/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d
*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2
+ 240*a**2*d) + 192/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2
+ d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2
)**2 + 240*a**2*d), Ne(d, 0)), (x*cos(c)**8/(a*sin(c) + a)**2, True))

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Giac [A]
time = 6.89, size = 179, normalized size = 1.72 \begin {gather*} \frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} - \frac {2 \, {\left (135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, 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))^2,x, algorithm="giac")

[Out]

1/240*(105*(d*x + c)/a^2 - 2*(135*tan(1/2*d*x + 1/2*c)^11 - 480*tan(1/2*d*x + 1/2*c)^10 + 445*tan(1/2*d*x + 1/
2*c)^9 - 480*tan(1/2*d*x + 1/2*c)^8 - 330*tan(1/2*d*x + 1/2*c)^7 - 960*tan(1/2*d*x + 1/2*c)^6 + 330*tan(1/2*d*
x + 1/2*c)^5 - 960*tan(1/2*d*x + 1/2*c)^4 - 445*tan(1/2*d*x + 1/2*c)^3 - 96*tan(1/2*d*x + 1/2*c)^2 - 135*tan(1
/2*d*x + 1/2*c) - 96)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^2))/d

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Mupad [B]
time = 8.22, size = 172, normalized size = 1.65 \begin {gather*} \frac {7\,x}{16\,a^2}+\frac {-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{5}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \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))^2,x)

[Out]

(7*x)/(16*a^2) + ((9*tan(c/2 + (d*x)/2))/8 + (4*tan(c/2 + (d*x)/2)^2)/5 + (89*tan(c/2 + (d*x)/2)^3)/24 + 8*tan
(c/2 + (d*x)/2)^4 - (11*tan(c/2 + (d*x)/2)^5)/4 + 8*tan(c/2 + (d*x)/2)^6 + (11*tan(c/2 + (d*x)/2)^7)/4 + 4*tan
(c/2 + (d*x)/2)^8 - (89*tan(c/2 + (d*x)/2)^9)/24 + 4*tan(c/2 + (d*x)/2)^10 - (9*tan(c/2 + (d*x)/2)^11)/8 + 4/5
)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^6)

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