3.627 \(\int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=97 \[ -\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac {\sin (c+d x) \cos (c+d x)}{16 a d}-\frac {x}{16 a} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2839, 2565, 30, 2568, 2635, 8} \[ -\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac {\sin (c+d x) \cos (c+d x)}{16 a d}-\frac {x}{16 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
 &&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2568

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

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 2839

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

Rubi steps

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

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Mathematica [B]  time = 5.08, size = 377, normalized size = 3.89 \[ -\frac {120 d x \sin \left (\frac {c}{2}\right )-120 \sin \left (\frac {c}{2}+d x\right )+120 \sin \left (\frac {3 c}{2}+d x\right )+15 \sin \left (\frac {3 c}{2}+2 d x\right )+15 \sin \left (\frac {5 c}{2}+2 d x\right )-60 \sin \left (\frac {5 c}{2}+3 d x\right )+60 \sin \left (\frac {7 c}{2}+3 d x\right )-15 \sin \left (\frac {7 c}{2}+4 d x\right )-15 \sin \left (\frac {9 c}{2}+4 d x\right )-12 \sin \left (\frac {9 c}{2}+5 d x\right )+12 \sin \left (\frac {11 c}{2}+5 d x\right )-5 \sin \left (\frac {11 c}{2}+6 d x\right )-5 \sin \left (\frac {13 c}{2}+6 d x\right )-30 \cos \left (\frac {c}{2}\right ) (5 c-4 d x)+120 \cos \left (\frac {c}{2}+d x\right )+120 \cos \left (\frac {3 c}{2}+d x\right )+15 \cos \left (\frac {3 c}{2}+2 d x\right )-15 \cos \left (\frac {5 c}{2}+2 d x\right )+60 \cos \left (\frac {5 c}{2}+3 d x\right )+60 \cos \left (\frac {7 c}{2}+3 d x\right )-15 \cos \left (\frac {7 c}{2}+4 d x\right )+15 \cos \left (\frac {9 c}{2}+4 d x\right )+12 \cos \left (\frac {9 c}{2}+5 d x\right )+12 \cos \left (\frac {11 c}{2}+5 d x\right )-5 \cos \left (\frac {11 c}{2}+6 d x\right )+5 \cos \left (\frac {13 c}{2}+6 d x\right )-150 c \sin \left (\frac {c}{2}\right )+300 \sin \left (\frac {c}{2}\right )}{1920 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

-1/1920*(-30*(5*c - 4*d*x)*Cos[c/2] + 120*Cos[c/2 + d*x] + 120*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] -
15*Cos[(5*c)/2 + 2*d*x] + 60*Cos[(5*c)/2 + 3*d*x] + 60*Cos[(7*c)/2 + 3*d*x] - 15*Cos[(7*c)/2 + 4*d*x] + 15*Cos
[(9*c)/2 + 4*d*x] + 12*Cos[(9*c)/2 + 5*d*x] + 12*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c
)/2 + 6*d*x] + 300*Sin[c/2] - 150*c*Sin[c/2] + 120*d*x*Sin[c/2] - 120*Sin[c/2 + d*x] + 120*Sin[(3*c)/2 + d*x]
+ 15*Sin[(3*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 2*d*x] - 60*Sin[(5*c)/2 + 3*d*x] + 60*Sin[(7*c)/2 + 3*d*x] - 15*S
in[(7*c)/2 + 4*d*x] - 15*Sin[(9*c)/2 + 4*d*x] - 12*Sin[(9*c)/2 + 5*d*x] + 12*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11
*c)/2 + 6*d*x] - 5*Sin[(13*c)/2 + 6*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))

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fricas [A]  time = 0.85, size = 60, normalized size = 0.62 \[ -\frac {48 \, \cos \left (d x + c\right )^{5} + 15 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/240*(48*cos(d*x + c)^5 + 15*d*x - 5*(8*cos(d*x + c)^5 - 2*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c))/(a
*d)

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giac [B]  time = 0.15, size = 179, normalized size = 1.85 \[ -\frac {\frac {15 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/240*(15*(d*x + c)/a + 2*(15*tan(1/2*d*x + 1/2*c)^11 + 240*tan(1/2*d*x + 1/2*c)^10 - 235*tan(1/2*d*x + 1/2*c
)^9 + 240*tan(1/2*d*x + 1/2*c)^8 + 390*tan(1/2*d*x + 1/2*c)^7 + 480*tan(1/2*d*x + 1/2*c)^6 - 390*tan(1/2*d*x +
 1/2*c)^5 + 480*tan(1/2*d*x + 1/2*c)^4 + 235*tan(1/2*d*x + 1/2*c)^3 + 48*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d
*x + 1/2*c) + 48)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a))/d

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maple [B]  time = 0.21, size = 415, normalized size = 4.28 \[ -\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {47 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {47 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

-1/8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11-2/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^
10+47/24/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9-2/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*
c)^8-13/4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7-4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2
*c)^6+13/4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5-4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/
2*c)^4-47/24/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3-2/5/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*
x+1/2*c)^2+1/8/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)-2/5/a/d/(1+tan(1/2*d*x+1/2*c)^2)^6-1/8/a/d*ar
ctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.52, size = 379, normalized size = 3.91 \[ \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {480 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {390 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {480 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {235 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {240 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 48}{a + \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/120*((15*sin(d*x + c)/(cos(d*x + c) + 1) - 48*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 235*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 - 480*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 390*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 480*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6 - 390*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 240*sin(d*x + c)^8/(cos(d*x + c) +
 1)^8 + 235*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 240*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 15*sin(d*x + c)^
11/(cos(d*x + c) + 1)^11 - 48)/(a + 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a*sin(d*x + c)^4/(cos(d*x + c
) + 1)^4 + 20*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a*sin(d*x +
 c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 15*arctan(sin(d*x + c)/(cos(d*x + c)
 + 1))/a)/d

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mupad [B]  time = 12.49, size = 173, normalized size = 1.78 \[ -\frac {x}{16\,a}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {2}{5}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*sin(c + d*x))/(a + a*sin(c + d*x)),x)

[Out]

- x/(16*a) - ((2*tan(c/2 + (d*x)/2)^2)/5 - tan(c/2 + (d*x)/2)/8 + (47*tan(c/2 + (d*x)/2)^3)/24 + 4*tan(c/2 + (
d*x)/2)^4 - (13*tan(c/2 + (d*x)/2)^5)/4 + 4*tan(c/2 + (d*x)/2)^6 + (13*tan(c/2 + (d*x)/2)^7)/4 + 2*tan(c/2 + (
d*x)/2)^8 - (47*tan(c/2 + (d*x)/2)^9)/24 + 2*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^11/8 + 2/5)/(a*d*(tan(
c/2 + (d*x)/2)^2 + 1)^6)

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sympy [A]  time = 49.69, size = 2307, normalized size = 23.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((-15*d*x*tan(c/2 + d*x/2)**12/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a
*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/
2)**2 + 240*a*d) - 90*d*x*tan(c/2 + d*x/2)**10/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*tan(c/2 + d*x/2)**10 +
 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2
 + d*x/2)**2 + 240*a*d) - 225*d*x*tan(c/2 + d*x/2)**8/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*tan(c/2 + d*x/2
)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**4 + 1440*a*d*
tan(c/2 + d*x/2)**2 + 240*a*d) - 300*d*x*tan(c/2 + d*x/2)**6/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*tan(c/2
+ d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**4 + 14
40*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 225*d*x*tan(c/2 + d*x/2)**4/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*t
an(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)*
*4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 90*d*x*tan(c/2 + d*x/2)**2/(240*a*d*tan(c/2 + d*x/2)**12 + 1440
*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d
*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 15*d*x/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*tan(c/2 +
 d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**4 + 144
0*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 30*tan(c/2 + d*x/2)**11/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*tan(c/
2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**4 +
1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 480*tan(c/2 + d*x/2)**10/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d*ta
n(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2)**
4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) + 470*tan(c/2 + d*x/2)**9/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*d
*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/2
)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 480*tan(c/2 + d*x/2)**8/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*
a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*
x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 780*tan(c/2 + d*x/2)**7/(240*a*d*tan(c/2 + d*x/2)**12 + 14
40*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 +
 d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 960*tan(c/2 + d*x/2)**6/(240*a*d*tan(c/2 + d*x/2)**12 +
 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/
2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) + 780*tan(c/2 + d*x/2)**5/(240*a*d*tan(c/2 + d*x/2)**1
2 + 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan
(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 960*tan(c/2 + d*x/2)**4/(240*a*d*tan(c/2 + d*x/2)
**12 + 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*
tan(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 470*tan(c/2 + d*x/2)**3/(240*a*d*tan(c/2 + d*x
/2)**12 + 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a
*d*tan(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 96*tan(c/2 + d*x/2)**2/(240*a*d*tan(c/2 + d
*x/2)**12 + 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600
*a*d*tan(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) + 30*tan(c/2 + d*x/2)/(240*a*d*tan(c/2 + d*
x/2)**12 + 1440*a*d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*
a*d*tan(c/2 + d*x/2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d) - 96/(240*a*d*tan(c/2 + d*x/2)**12 + 1440*a*
d*tan(c/2 + d*x/2)**10 + 3600*a*d*tan(c/2 + d*x/2)**8 + 4800*a*d*tan(c/2 + d*x/2)**6 + 3600*a*d*tan(c/2 + d*x/
2)**4 + 1440*a*d*tan(c/2 + d*x/2)**2 + 240*a*d), Ne(d, 0)), (x*sin(c)*cos(c)**6/(a*sin(c) + a), True))

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