3.412 \(\int \frac{\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a} \]

[Out]

-x/(8*a) - Cos[c + d*x]^3/(3*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d
)

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Rubi [A]  time = 0.112174, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

-x/(8*a) - Cos[c + d*x]^3/(3*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d
)

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]

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 30

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

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 8

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

Rubi steps

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

Mathematica [B]  time = 1.65278, size = 219, normalized size = 3. \[ -\frac{24 d x \sin \left (\frac{c}{2}\right )-24 \sin \left (\frac{c}{2}+d x\right )+24 \sin \left (\frac{3 c}{2}+d x\right )-8 \sin \left (\frac{5 c}{2}+3 d x\right )+8 \sin \left (\frac{7 c}{2}+3 d x\right )-3 \sin \left (\frac{7 c}{2}+4 d x\right )-3 \sin \left (\frac{9 c}{2}+4 d x\right )-24 \cos \left (\frac{c}{2}\right ) (c-d x)+24 \cos \left (\frac{c}{2}+d x\right )+24 \cos \left (\frac{3 c}{2}+d x\right )+8 \cos \left (\frac{5 c}{2}+3 d x\right )+8 \cos \left (\frac{7 c}{2}+3 d x\right )-3 \cos \left (\frac{7 c}{2}+4 d x\right )+3 \cos \left (\frac{9 c}{2}+4 d x\right )-24 c \sin \left (\frac{c}{2}\right )+48 \sin \left (\frac{c}{2}\right )}{192 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

-(-24*(c - d*x)*Cos[c/2] + 24*Cos[c/2 + d*x] + 24*Cos[(3*c)/2 + d*x] + 8*Cos[(5*c)/2 + 3*d*x] + 8*Cos[(7*c)/2
+ 3*d*x] - 3*Cos[(7*c)/2 + 4*d*x] + 3*Cos[(9*c)/2 + 4*d*x] + 48*Sin[c/2] - 24*c*Sin[c/2] + 24*d*x*Sin[c/2] - 2
4*Sin[c/2 + d*x] + 24*Sin[(3*c)/2 + d*x] - 8*Sin[(5*c)/2 + 3*d*x] + 8*Sin[(7*c)/2 + 3*d*x] - 3*Sin[(7*c)/2 + 4
*d*x] - 3*Sin[(9*c)/2 + 4*d*x])/(192*a*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.06, size = 279, normalized size = 3.8 \begin{align*} -{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}+{\frac{7}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{7}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{2}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{1}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{2}{3\,da} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{1}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [B]  time = 1.6647, size = 347, normalized size = 4.75 \begin{align*} \frac{\frac{\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{21 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{24 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{24 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 8}{a + \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/12*((3*sin(d*x + c)/(cos(d*x + c) + 1) - 8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 21*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 - 24*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 24*sin(d*x + c
)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 8)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) +
1)^2 + 6*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8/(c
os(d*x + c) + 1)^8) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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Fricas [A]  time = 1.07232, size = 123, normalized size = 1.68 \begin{align*} -\frac{8 \, \cos \left (d x + c\right )^{3} + 3 \, d x - 3 \,{\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/24*(8*cos(d*x + c)^3 + 3*d*x - 3*(2*cos(d*x + c)^3 - cos(d*x + c))*sin(d*x + c))/(a*d)

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Sympy [A]  time = 23.9163, size = 1221, normalized size = 16.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

Piecewise((-15*d*x*tan(c/2 + d*x/2)**8/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*ta
n(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 60*d*x*tan(c/2 + d*x/2)**6/(120*a*d*tan(c/2 + d*x
/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) -
90*d*x*tan(c/2 + d*x/2)**4/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/
2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 60*d*x*tan(c/2 + d*x/2)**2/(120*a*d*tan(c/2 + d*x/2)**8 + 480
*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 15*d*x/(120*
a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)
**2 + 120*a*d) + 54*tan(c/2 + d*x/2)**8/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*t
an(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 30*tan(c/2 + d*x/2)**7/(120*a*d*tan(c/2 + d*x/2)
**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 24*
tan(c/2 + d*x/2)**6/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 +
480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 210*tan(c/2 + d*x/2)**5/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c
/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 84*tan(c/2 + d*x/2)**4
/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 +
d*x/2)**2 + 120*a*d) - 210*tan(c/2 + d*x/2)**3/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 72
0*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 136*tan(c/2 + d*x/2)**2/(120*a*d*tan(c/2
+ d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*
d) + 30*tan(c/2 + d*x/2)/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 + 720*a*d*tan(c/2 + d*x/2)
**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) - 26/(120*a*d*tan(c/2 + d*x/2)**8 + 480*a*d*tan(c/2 + d*x/2)**6 +
720*a*d*tan(c/2 + d*x/2)**4 + 480*a*d*tan(c/2 + d*x/2)**2 + 120*a*d), Ne(d, 0)), (x*sin(c)*cos(c)**4/(a*sin(c
) + a), True))

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Giac [A]  time = 1.60002, size = 171, normalized size = 2.34 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/24*(3*(d*x + c)/a + 2*(3*tan(1/2*d*x + 1/2*c)^7 + 24*tan(1/2*d*x + 1/2*c)^6 - 21*tan(1/2*d*x + 1/2*c)^5 + 2
4*tan(1/2*d*x + 1/2*c)^4 + 21*tan(1/2*d*x + 1/2*c)^3 + 8*tan(1/2*d*x + 1/2*c)^2 - 3*tan(1/2*d*x + 1/2*c) + 8)/
((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d