\(\int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [420]

Optimal result

Integrand size = 29, antiderivative size = 147 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {5 x}{8 a^2}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 2713, 2715, 8} \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac {5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {5 x}{8 a^2} \]

[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 8

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

Rule 2713

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 2715

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] && Integ
erQ[2*n]

Rule 2836

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 2948

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]

Rubi steps \begin{align*} \text {integral}& = \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 {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(147)=294\).

Time = 3.83 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-210 (1+40 d x) \cos \left (\frac {c}{2}\right )-7875 \cos \left (\frac {c}{2}+d x\right )-7875 \cos \left (\frac {3 c}{2}+d x\right )+3150 \cos \left (\frac {3 c}{2}+2 d x\right )-3150 \cos \left (\frac {5 c}{2}+2 d x\right )+1435 \cos \left (\frac {5 c}{2}+3 d x\right )+1435 \cos \left (\frac {7 c}{2}+3 d x\right )-630 \cos \left (\frac {7 c}{2}+4 d x\right )+630 \cos \left (\frac {9 c}{2}+4 d x\right )-231 \cos \left (\frac {9 c}{2}+5 d x\right )-231 \cos \left (\frac {11 c}{2}+5 d x\right )+70 \cos \left (\frac {11 c}{2}+6 d x\right )-70 \cos \left (\frac {13 c}{2}+6 d x\right )+15 \cos \left (\frac {13 c}{2}+7 d x\right )+15 \cos \left (\frac {15 c}{2}+7 d x\right )+210 \sin \left (\frac {c}{2}\right )-8400 d x \sin \left (\frac {c}{2}\right )+7875 \sin \left (\frac {c}{2}+d x\right )-7875 \sin \left (\frac {3 c}{2}+d x\right )+3150 \sin \left (\frac {3 c}{2}+2 d x\right )+3150 \sin \left (\frac {5 c}{2}+2 d x\right )-1435 \sin \left (\frac {5 c}{2}+3 d x\right )+1435 \sin \left (\frac {7 c}{2}+3 d x\right )-630 \sin \left (\frac {7 c}{2}+4 d x\right )-630 \sin \left (\frac {9 c}{2}+4 d x\right )+231 \sin \left (\frac {9 c}{2}+5 d x\right )-231 \sin \left (\frac {11 c}{2}+5 d x\right )+70 \sin \left (\frac {11 c}{2}+6 d x\right )+70 \sin \left (\frac {13 c}{2}+6 d x\right )-15 \sin \left (\frac {13 c}{2}+7 d x\right )+15 \sin \left (\frac {15 c}{2}+7 d x\right )}{13440 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[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]))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61

method	result	size
parallelrisch	\(\frac {-4200 d x +1435 \cos \left (3 d x +3 c \right )-7875 \cos \left (d x +c \right )+15 \cos \left (7 d x +7 c \right )+70 \sin \left (6 d x +6 c \right )-231 \cos \left (5 d x +5 c \right )-630 \sin \left (4 d x +4 c \right )+3150 \sin \left (2 d x +2 c \right )-6656}{6720 d \,a^{2}}\)	\(89\)
risch	\(-\frac {5 x}{8 a^{2}}-\frac {75 \cos \left (d x +c \right )}{64 a^{2} d}+\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{2}}+\frac {\sin \left (6 d x +6 c \right )}{96 d \,a^{2}}-\frac {11 \cos \left (5 d x +5 c \right )}{320 d \,a^{2}}-\frac {3 \sin \left (4 d x +4 c \right )}{32 d \,a^{2}}+\frac {41 \cos \left (3 d x +3 c \right )}{192 d \,a^{2}}+\frac {15 \sin \left (2 d x +2 c \right )}{32 d \,a^{2}}\)	\(124\)
derivativedivides	\(\frac {\frac {64 \left (-\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {25 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {283 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {283 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {13 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {13}{420}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\)	\(168\)
default	\(\frac {\frac {64 \left (-\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {25 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {283 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {283 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {13 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}-\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {13}{420}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\)	\(168\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/6720*(-4200*d*x+1435*cos(3*d*x+3*c)-7875*cos(d*x+c)+15*cos(7*d*x+7*c)+70*sin(6*d*x+6*c)-231*cos(5*d*x+5*c)-6
30*sin(4*d*x+4*c)+3150*sin(2*d*x+2*c)-6656)/d/a^2

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {120 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} + 1400 \, \cos \left (d x + c\right )^{3} - 525 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 1680 \, \cos \left (d x + c\right )}{840 \, a^{2} d} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2895 vs. \(2 (138) = 276\).

Time = 81.11 (sec) , antiderivative size = 2895, normalized size of antiderivative = 19.69 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-525*d*x*tan(c/2 + d*x/2)**14/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 +
17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 176
40*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 3675*d*x*tan(c/2 + d*x/2)**12/
(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 2940
0*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2
*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 11025*d*x*tan(c/2 + d*x/2)**10/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*
a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**
2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1
8375*d*x*tan(c/2 + d*x/2)**8/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*
d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*ta
n(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 18375*d*x*tan(c/2 + d*x/2)**6/(840*a**2*d*
tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan
(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 +
d*x/2)**2 + 840*a**2*d) - 11025*d*x*tan(c/2 + d*x/2)**4/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/
2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 +
d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 3675*d*x*tan(c
/2 + d*x/2)**2/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*
x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)
**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 525*d*x/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*t
an(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(
c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1050*tan(
c/2 + d*x/2)**13/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 +
d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/
2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 7000*tan(c/2 + d*x/2)**11/(840*a**2*d*tan(c/2 + d*x/2)
**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8
+ 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840
*a**2*d) - 19810*tan(c/2 + d*x/2)**9/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 176
40*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a
**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 8960*tan(c/2 + d*x/2)**8/(840*a**2
*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*
tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/
2 + d*x/2)**2 + 840*a**2*d) - 49280*tan(c/2 + d*x/2)**6/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2
+ d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 +
d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) + 19810*tan(c/2 +
d*x/2)**5/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)
**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4
+ 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 34944*tan(c/2 + d*x/2)**4/(840*a**2*d*tan(c/2 + d*x/2)**14 +
5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 294
00*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*
d) + 7000*tan(c/2 + d*x/2)**3/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2
*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*t
an(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 11648*tan(c/2 + d*x/2)**2/(840*a**2*d*tan
(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/
2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*
x/2)**2 + 840*a**2*d) + 1050*tan(c/2 + d*x/2)/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)*
*12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6
+ 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1664/(840*a**2*d*tan(c/2
+ d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d
*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)*
*2 + 840*a**2*d), Ne(d, 0)), (x*sin(c)**5*cos(c)**4/(a*sin(c) + a)**2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (133) = 266\).

Time = 0.31 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.69 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\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} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {525 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 3500 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 9905 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 24640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 9905 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 17472 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3500 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5824 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 832\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 13.03 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {5\,x}{8\,a^2}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+\frac {283\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {176\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {283\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {208}{105}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

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