3.644 \(\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=129 \[ -\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac{23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3} \]

[Out]

(-23*x)/(16*a^3) - (4*Cos[c + d*x])/(a^3*d) + (7*Cos[c + d*x]^3)/(3*a^3*d) - (3*Cos[c + d*x]^5)/(5*a^3*d) + (2
3*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) + (23*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^3*d) + (Cos[c + d*x]*Sin[c +
d*x]^5)/(6*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.24209, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2869, 2757, 2633, 2635, 8} \[ -\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac{23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-23*x)/(16*a^3) - (4*Cos[c + d*x])/(a^3*d) + (7*Cos[c + d*x]^3)/(3*a^3*d) - (3*Cos[c + d*x]^5)/(5*a^3*d) + (2
3*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) + (23*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^3*d) + (Cos[c + d*x]*Sin[c +
d*x]^5)/(6*a^3*d)

Rule 2869

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

Rule 2757

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \sin ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (a^3 \sin ^3(c+d x)-3 a^3 \sin ^4(c+d x)+3 a^3 \sin ^5(c+d x)-a^3 \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \sin ^3(c+d x) \, dx}{a^3}-\frac{\int \sin ^6(c+d x) \, dx}{a^3}-\frac{3 \int \sin ^4(c+d x) \, dx}{a^3}+\frac{3 \int \sin ^5(c+d x) \, dx}{a^3}\\ &=\frac{3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac{5 \int \sin ^4(c+d x) \, dx}{6 a^3}-\frac{9 \int \sin ^2(c+d x) \, dx}{4 a^3}-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=-\frac{4 \cos (c+d x)}{a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{9 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac{23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac{5 \int \sin ^2(c+d x) \, dx}{8 a^3}-\frac{9 \int 1 \, dx}{8 a^3}\\ &=-\frac{9 x}{8 a^3}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac{23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac{5 \int 1 \, dx}{16 a^3}\\ &=-\frac{23 x}{16 a^3}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac{23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}\\ \end{align*}

Mathematica [B]  time = 2.18278, size = 366, normalized size = 2.84 \[ \frac{-2760 d x \sin \left (\frac{c}{2}\right )+2520 \sin \left (\frac{c}{2}+d x\right )-2520 \sin \left (\frac{3 c}{2}+d x\right )+945 \sin \left (\frac{3 c}{2}+2 d x\right )+945 \sin \left (\frac{5 c}{2}+2 d x\right )-380 \sin \left (\frac{5 c}{2}+3 d x\right )+380 \sin \left (\frac{7 c}{2}+3 d x\right )-135 \sin \left (\frac{7 c}{2}+4 d x\right )-135 \sin \left (\frac{9 c}{2}+4 d x\right )+36 \sin \left (\frac{9 c}{2}+5 d x\right )-36 \sin \left (\frac{11 c}{2}+5 d x\right )+5 \sin \left (\frac{11 c}{2}+6 d x\right )+5 \sin \left (\frac{13 c}{2}+6 d x\right )-3 \cos \left (\frac{c}{2}\right ) (920 d x+3)-2520 \cos \left (\frac{c}{2}+d x\right )-2520 \cos \left (\frac{3 c}{2}+d x\right )+945 \cos \left (\frac{3 c}{2}+2 d x\right )-945 \cos \left (\frac{5 c}{2}+2 d x\right )+380 \cos \left (\frac{5 c}{2}+3 d x\right )+380 \cos \left (\frac{7 c}{2}+3 d x\right )-135 \cos \left (\frac{7 c}{2}+4 d x\right )+135 \cos \left (\frac{9 c}{2}+4 d x\right )-36 \cos \left (\frac{9 c}{2}+5 d x\right )-36 \cos \left (\frac{11 c}{2}+5 d x\right )+5 \cos \left (\frac{11 c}{2}+6 d x\right )-5 \cos \left (\frac{13 c}{2}+6 d x\right )+9 \sin \left (\frac{c}{2}\right )}{1920 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-3*(3 + 920*d*x)*Cos[c/2] - 2520*Cos[c/2 + d*x] - 2520*Cos[(3*c)/2 + d*x] + 945*Cos[(3*c)/2 + 2*d*x] - 945*Co
s[(5*c)/2 + 2*d*x] + 380*Cos[(5*c)/2 + 3*d*x] + 380*Cos[(7*c)/2 + 3*d*x] - 135*Cos[(7*c)/2 + 4*d*x] + 135*Cos[
(9*c)/2 + 4*d*x] - 36*Cos[(9*c)/2 + 5*d*x] - 36*Cos[(11*c)/2 + 5*d*x] + 5*Cos[(11*c)/2 + 6*d*x] - 5*Cos[(13*c)
/2 + 6*d*x] + 9*Sin[c/2] - 2760*d*x*Sin[c/2] + 2520*Sin[c/2 + d*x] - 2520*Sin[(3*c)/2 + d*x] + 945*Sin[(3*c)/2
+ 2*d*x] + 945*Sin[(5*c)/2 + 2*d*x] - 380*Sin[(5*c)/2 + 3*d*x] + 380*Sin[(7*c)/2 + 3*d*x] - 135*Sin[(7*c)/2 +
4*d*x] - 135*Sin[(9*c)/2 + 4*d*x] + 36*Sin[(9*c)/2 + 5*d*x] - 36*Sin[(11*c)/2 + 5*d*x] + 5*Sin[(11*c)/2 + 6*d
*x] + 5*Sin[(13*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))

________________________________________________________________________________________

Maple [B]  time = 0.117, size = 381, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x)

[Out]

-23/8/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11-391/24/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d
*x+1/2*c)^9-4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8-75/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(
1/2*d*x+1/2*c)^7-136/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6+75/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^
2)^6*tan(1/2*d*x+1/2*c)^5-64/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4+391/24/d/a^3/(1+tan(1/2*d*x
+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3-136/5/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2+23/8/d/a^3/(1+ta
n(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)-68/15/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6-23/8/d/a^3*arctan(tan(1/2*d*x+
1/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.56375, size = 504, normalized size = 3.91 \begin{align*} \frac{\frac{\frac{345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3264 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1955 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{7680 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{2250 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{5440 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{2250 \, \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{1955 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{345 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 544}{a^{3} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{345 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/120*((345*sin(d*x + c)/(cos(d*x + c) + 1) - 3264*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1955*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 - 7680*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2250*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 54
40*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 2250*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 480*sin(d*x + c)^8/(cos(d*
x + c) + 1)^8 - 1955*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 345*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 544)/(a
^3 + 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a^3*sin(d*x +
c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a^3*sin(d*x + c)^10/(cos(d*x + c)
+ 1)^10 + a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 345*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

________________________________________________________________________________________

Fricas [A]  time = 1.09525, size = 219, normalized size = 1.7 \begin{align*} -\frac{144 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} + 345 \, d x - 5 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 62 \, \cos \left (d x + c\right )^{3} + 123 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/240*(144*cos(d*x + c)^5 - 560*cos(d*x + c)^3 + 345*d*x - 5*(8*cos(d*x + c)^5 - 62*cos(d*x + c)^3 + 123*cos(
d*x + c))*sin(d*x + c) + 960*cos(d*x + c))/(a^3*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.31117, size = 224, normalized size = 1.74 \begin{align*} -\frac{\frac{345 \,{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (345 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1955 \, \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} + 2250 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5440 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 2250 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7680 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1955 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3264 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 345 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 544\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/240*(345*(d*x + c)/a^3 + 2*(345*tan(1/2*d*x + 1/2*c)^11 + 1955*tan(1/2*d*x + 1/2*c)^9 + 480*tan(1/2*d*x + 1
/2*c)^8 + 2250*tan(1/2*d*x + 1/2*c)^7 + 5440*tan(1/2*d*x + 1/2*c)^6 - 2250*tan(1/2*d*x + 1/2*c)^5 + 7680*tan(1
/2*d*x + 1/2*c)^4 - 1955*tan(1/2*d*x + 1/2*c)^3 + 3264*tan(1/2*d*x + 1/2*c)^2 - 345*tan(1/2*d*x + 1/2*c) + 544
)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^3))/d