3.420 \(\int \frac{\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=147 \[ \frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac{5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{5 x}{8 a^2} \]

[Out]

(-5*x)/(8*a^2) - (2*Cos[c + d*x])/(a^2*d) + (5*Cos[c + d*x]^3)/(3*a^2*d) - (4*Cos[c + d*x]^5)/(5*a^2*d) + Cos[
c + d*x]^7/(7*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a^2*d) +
(Cos[c + d*x]*Sin[c + d*x]^5)/(3*a^2*d)

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Rubi [A]  time = 0.223546, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac{5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{5 x}{8 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-5*x)/(8*a^2) - (2*Cos[c + d*x])/(a^2*d) + (5*Cos[c + d*x]^3)/(3*a^2*d) - (4*Cos[c + d*x]^5)/(5*a^2*d) + Cos[
c + d*x]^7/(7*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a^2*d) +
(Cos[c + d*x]*Sin[c + d*x]^5)/(3*a^2*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 ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \sin ^5(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \sin ^5(c+d x)-2 a^2 \sin ^6(c+d x)+a^2 \sin ^7(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sin ^5(c+d x) \, dx}{a^2}+\frac{\int \sin ^7(c+d x) \, dx}{a^2}-\frac{2 \int \sin ^6(c+d x) \, dx}{a^2}\\ &=\frac{\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}-\frac{5 \int \sin ^4(c+d x) \, dx}{3 a^2}-\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \cos (c+d x)}{a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}-\frac{5 \int \sin ^2(c+d x) \, dx}{4 a^2}\\ &=-\frac{2 \cos (c+d x)}{a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{5 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}-\frac{5 \int 1 \, dx}{8 a^2}\\ &=-\frac{5 x}{8 a^2}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{5 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac{\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}\\ \end{align*}

Mathematica [B]  time = 4.70445, size = 418, normalized size = 2.84 \[ \frac{-8400 d x \sin \left (\frac{c}{2}\right )+7875 \sin \left (\frac{c}{2}+d x\right )-7875 \sin \left (\frac{3 c}{2}+d x\right )+3150 \sin \left (\frac{3 c}{2}+2 d x\right )+3150 \sin \left (\frac{5 c}{2}+2 d x\right )-1435 \sin \left (\frac{5 c}{2}+3 d x\right )+1435 \sin \left (\frac{7 c}{2}+3 d x\right )-630 \sin \left (\frac{7 c}{2}+4 d x\right )-630 \sin \left (\frac{9 c}{2}+4 d x\right )+231 \sin \left (\frac{9 c}{2}+5 d x\right )-231 \sin \left (\frac{11 c}{2}+5 d x\right )+70 \sin \left (\frac{11 c}{2}+6 d x\right )+70 \sin \left (\frac{13 c}{2}+6 d x\right )-15 \sin \left (\frac{13 c}{2}+7 d x\right )+15 \sin \left (\frac{15 c}{2}+7 d x\right )-210 \cos \left (\frac{c}{2}\right ) (40 d x+1)-7875 \cos \left (\frac{c}{2}+d x\right )-7875 \cos \left (\frac{3 c}{2}+d x\right )+3150 \cos \left (\frac{3 c}{2}+2 d x\right )-3150 \cos \left (\frac{5 c}{2}+2 d x\right )+1435 \cos \left (\frac{5 c}{2}+3 d x\right )+1435 \cos \left (\frac{7 c}{2}+3 d x\right )-630 \cos \left (\frac{7 c}{2}+4 d x\right )+630 \cos \left (\frac{9 c}{2}+4 d x\right )-231 \cos \left (\frac{9 c}{2}+5 d x\right )-231 \cos \left (\frac{11 c}{2}+5 d x\right )+70 \cos \left (\frac{11 c}{2}+6 d x\right )-70 \cos \left (\frac{13 c}{2}+6 d x\right )+15 \cos \left (\frac{13 c}{2}+7 d x\right )+15 \cos \left (\frac{15 c}{2}+7 d x\right )+210 \sin \left (\frac{c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-210*(1 + 40*d*x)*Cos[c/2] - 7875*Cos[c/2 + d*x] - 7875*Cos[(3*c)/2 + d*x] + 3150*Cos[(3*c)/2 + 2*d*x] - 3150
*Cos[(5*c)/2 + 2*d*x] + 1435*Cos[(5*c)/2 + 3*d*x] + 1435*Cos[(7*c)/2 + 3*d*x] - 630*Cos[(7*c)/2 + 4*d*x] + 630
*Cos[(9*c)/2 + 4*d*x] - 231*Cos[(9*c)/2 + 5*d*x] - 231*Cos[(11*c)/2 + 5*d*x] + 70*Cos[(11*c)/2 + 6*d*x] - 70*C
os[(13*c)/2 + 6*d*x] + 15*Cos[(13*c)/2 + 7*d*x] + 15*Cos[(15*c)/2 + 7*d*x] + 210*Sin[c/2] - 8400*d*x*Sin[c/2]
+ 7875*Sin[c/2 + d*x] - 7875*Sin[(3*c)/2 + d*x] + 3150*Sin[(3*c)/2 + 2*d*x] + 3150*Sin[(5*c)/2 + 2*d*x] - 1435
*Sin[(5*c)/2 + 3*d*x] + 1435*Sin[(7*c)/2 + 3*d*x] - 630*Sin[(7*c)/2 + 4*d*x] - 630*Sin[(9*c)/2 + 4*d*x] + 231*
Sin[(9*c)/2 + 5*d*x] - 231*Sin[(11*c)/2 + 5*d*x] + 70*Sin[(11*c)/2 + 6*d*x] + 70*Sin[(13*c)/2 + 6*d*x] - 15*Si
n[(13*c)/2 + 7*d*x] + 15*Sin[(15*c)/2 + 7*d*x])/(13440*a^2*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.121, size = 381, normalized size = 2.6 \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)^4*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x)

[Out]

-5/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13-25/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+
1/2*c)^11-283/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9-32/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*t
an(1/2*d*x+1/2*c)^8-176/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6+283/12/d/a^2/(1+tan(1/2*d*x+1/
2*c)^2)^7*tan(1/2*d*x+1/2*c)^5-208/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4+25/3/d/a^2/(1+tan(1
/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3-208/15/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2+5/4/d/a^2
/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)-208/105/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7-5/4/d/a^2*arctan(tan(1
/2*d*x+1/2*c))

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Maxima [B]  time = 1.68619, size = 535, normalized size = 3.64 \begin{align*} \frac{\frac{\frac{525 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{5824 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3500 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{17472 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{9905 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{24640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{4480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{9905 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{3500 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{525 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 832}{a^{2} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac{a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac{525 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/420*((525*sin(d*x + c)/(cos(d*x + c) + 1) - 5824*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3500*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 - 17472*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 9905*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 2
4640*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 4480*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 9905*sin(d*x + c)^9/(cos
(d*x + c) + 1)^9 - 3500*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 525*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 83
2)/(a^2 + 7*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 21*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 35*a^2*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6 + 35*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 21*a^2*sin(d*x + c)^10/(cos(d*x
+ c) + 1)^10 + 7*a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14) - 525
*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [A]  time = 1.11901, size = 248, normalized size = 1.69 \begin{align*} \frac{120 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} + 1400 \, \cos \left (d x + c\right )^{3} - 525 \, d x + 35 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 1680 \, \cos \left (d x + c\right )}{840 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/840*(120*cos(d*x + c)^7 - 672*cos(d*x + c)^5 + 1400*cos(d*x + c)^3 - 525*d*x + 35*(8*cos(d*x + c)^5 - 26*cos
(d*x + c)^3 + 33*cos(d*x + c))*sin(d*x + c) - 1680*cos(d*x + c))/(a^2*d)

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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)**4*sin(d*x+c)**5/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.30012, size = 224, normalized size = 1.52 \begin{align*} -\frac{\frac{525 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (525 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 3500 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 9905 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4480 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 24640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 9905 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 17472 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3500 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5824 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 525 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 832\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/840*(525*(d*x + c)/a^2 + 2*(525*tan(1/2*d*x + 1/2*c)^13 + 3500*tan(1/2*d*x + 1/2*c)^11 + 9905*tan(1/2*d*x +
1/2*c)^9 + 4480*tan(1/2*d*x + 1/2*c)^8 + 24640*tan(1/2*d*x + 1/2*c)^6 - 9905*tan(1/2*d*x + 1/2*c)^5 + 17472*t
an(1/2*d*x + 1/2*c)^4 - 3500*tan(1/2*d*x + 1/2*c)^3 + 5824*tan(1/2*d*x + 1/2*c)^2 - 525*tan(1/2*d*x + 1/2*c) +
832)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*a^2))/d