3.7.83 \(\int \frac {\cos ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [683]

Optimal. Leaf size=68 \[ -\frac {(a-a \sin (c+d x))^4}{a^5 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^6}{6 a^7 d} \]

[Out]

-(a-a*sin(d*x+c))^4/a^5/d+4/5*(a-a*sin(d*x+c))^5/a^6/d-1/6*(a-a*sin(d*x+c))^6/a^7/d

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Rubi [A]
time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45} \begin {gather*} -\frac {(a-a \sin (c+d x))^6}{6 a^7 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^4}{a^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x]),x]

[Out]

-((a - a*Sin[c + d*x])^4/(a^5*d)) + (4*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - (a - a*Sin[c + d*x])^6/(6*a^7*d)

Rule 45

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])

Rule 2746

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])

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 66, normalized size = 0.97 \begin {gather*} -\frac {\sin (c+d x) \left (-30+15 \sin (c+d x)+20 \sin ^2(c+d x)-15 \sin ^3(c+d x)-6 \sin ^4(c+d x)+5 \sin ^5(c+d x)\right )}{30 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x]),x]

[Out]

-1/30*(Sin[c + d*x]*(-30 + 15*Sin[c + d*x] + 20*Sin[c + d*x]^2 - 15*Sin[c + d*x]^3 - 6*Sin[c + d*x]^4 + 5*Sin[
c + d*x]^5))/(a*d)

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Maple [A]
time = 0.23, size = 65, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a}\)	\(65\)
default	\(\frac {-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a}\)	\(65\)
risch	\(\frac {5 \sin \left (d x +c \right )}{8 a d}+\frac {\cos \left (6 d x +6 c \right )}{192 a d}+\frac {\sin \left (5 d x +5 c \right )}{80 a d}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {5 \sin \left (3 d x +3 c \right )}{48 a d}+\frac {5 \cos \left (2 d x +2 c \right )}{64 a d}\)	\(101\)
norman	\(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {14 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {14 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {14 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {42 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}+\frac {42 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}+\frac {196 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {196 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {212 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {212 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(-1/6*sin(d*x+c)^6+1/5*sin(d*x+c)^5+1/2*sin(d*x+c)^4-2/3*sin(d*x+c)^3-1/2*sin(d*x+c)^2+sin(d*x+c))

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Maxima [A]
time = 0.29, size = 67, normalized size = 0.99 \begin {gather*} -\frac {5 \, \sin \left (d x + c\right )^{6} - 6 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right )}{30 \, a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/30*(5*sin(d*x + c)^6 - 6*sin(d*x + c)^5 - 15*sin(d*x + c)^4 + 20*sin(d*x + c)^3 + 15*sin(d*x + c)^2 - 30*si
n(d*x + c))/(a*d)

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Fricas [A]
time = 0.38, size = 49, normalized size = 0.72 \begin {gather*} \frac {5 \, \cos \left (d x + c\right )^{6} + 2 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{30 \, a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/30*(5*cos(d*x + c)^6 + 2*(3*cos(d*x + c)^4 + 4*cos(d*x + c)^2 + 8)*sin(d*x + c))/(a*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (54) = 108\).
time = 24.65, size = 1096, normalized size = 16.12 \begin {gather*} \begin {cases} \frac {30 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {30 \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {70 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {156 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {100 \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {156 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {70 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {30 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {30 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{7}{\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((30*tan(c/2 + d*x/2)**11/(15*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/
2 + d*x/2)**8 + 300*a*d*tan(c/2 + d*x/2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*
d) - 30*tan(c/2 + d*x/2)**10/(15*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*
x/2)**8 + 300*a*d*tan(c/2 + d*x/2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) + 7
0*tan(c/2 + d*x/2)**9/(15*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8
+ 300*a*d*tan(c/2 + d*x/2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) + 156*tan(
c/2 + d*x/2)**7/(15*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8 + 300
*a*d*tan(c/2 + d*x/2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) - 100*tan(c/2 +
d*x/2)**6/(15*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8 + 300*a*d*t
an(c/2 + d*x/2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) + 156*tan(c/2 + d*x/2)
**5/(15*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8 + 300*a*d*tan(c/2
+ d*x/2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) + 70*tan(c/2 + d*x/2)**3/(15
*a*d*tan(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8 + 300*a*d*tan(c/2 + d*x/
2)**6 + 225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) - 30*tan(c/2 + d*x/2)**2/(15*a*d*ta
n(c/2 + d*x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8 + 300*a*d*tan(c/2 + d*x/2)**6 +
225*a*d*tan(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d) + 30*tan(c/2 + d*x/2)/(15*a*d*tan(c/2 + d*
x/2)**12 + 90*a*d*tan(c/2 + d*x/2)**10 + 225*a*d*tan(c/2 + d*x/2)**8 + 300*a*d*tan(c/2 + d*x/2)**6 + 225*a*d*t
an(c/2 + d*x/2)**4 + 90*a*d*tan(c/2 + d*x/2)**2 + 15*a*d), Ne(d, 0)), (x*cos(c)**7/(a*sin(c) + a), True))

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Giac [A]
time = 0.45, size = 67, normalized size = 0.99 \begin {gather*} -\frac {5 \, \sin \left (d x + c\right )^{6} - 6 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right )}{30 \, a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/30*(5*sin(d*x + c)^6 - 6*sin(d*x + c)^5 - 15*sin(d*x + c)^4 + 20*sin(d*x + c)^3 + 15*sin(d*x + c)^2 - 30*si
n(d*x + c))/(a*d)

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Mupad [B]
time = 9.14, size = 80, normalized size = 1.18 \begin {gather*} \frac {\frac {\sin \left (c+d\,x\right )}{a}-\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{3\,a}+\frac {{\sin \left (c+d\,x\right )}^4}{2\,a}+\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7/(a + a*sin(c + d*x)),x)

[Out]

(sin(c + d*x)/a - sin(c + d*x)^2/(2*a) - (2*sin(c + d*x)^3)/(3*a) + sin(c + d*x)^4/(2*a) + sin(c + d*x)^5/(5*a
) - sin(c + d*x)^6/(6*a))/d

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