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]

-5/8*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

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Rubi [A]  time = 0.22, antiderivative size = 147, normalized size of antiderivative = 1.00, 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 8

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

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 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 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]

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*}

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Mathematica [B]  time = 4.83, 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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fricas [A]  time = 0.45, size = 88, normalized size = 0.60 \[ \frac {120 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} + 1400 \, \cos \left (d x + c\right )^{3} - 525 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 1680 \, \cos \left (d x + c\right )}{840 \, a^{2} d} \]

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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giac [A]  time = 0.23, size = 166, normalized size = 1.13 \[ -\frac {\frac {525 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 3500 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 9905 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 24640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 9905 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 17472 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3500 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5824 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 832\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]

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

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maple [B]  time = 0.44, size = 381, normalized size = 2.59 \[ -\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {25 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {283 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {32 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {176 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {283 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {208 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {208 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {208}{105 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}} \]

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/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13-25/3/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+
1/2*c)^11-283/12/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9-32/3/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*t
an(1/2*d*x+1/2*c)^8-176/3/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6+283/12/a^2/d/(1+tan(1/2*d*x+1/
2*c)^2)^7*tan(1/2*d*x+1/2*c)^5-208/5/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4+25/3/a^2/d/(1+tan(1
/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3-208/15/a^2/d/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2+5/4/a^2/d
/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)-208/105/a^2/d/(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 = 0.49, size = 396, normalized size = 2.69 \[ \frac {\frac {\frac {525 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5824 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3500 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {17472 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {9905 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {24640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {9905 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3500 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {525 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 832}{a^{2} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {525 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \]

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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mupad [B]  time = 12.22, size = 160, normalized size = 1.09 \[ -\frac {5\,x}{8\,a^2}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+\frac {283\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {176\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {283\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {208}{105}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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