3.471 \(\int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=278 \[ \frac{2 d \left (72 c^2 d^2+15 c^3 d+2 c^4-180 c d^3+76 d^4\right ) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac{d^2 \left (30 c^2 d+4 c^3+146 c d^2-195 d^3\right ) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left (20 c^2-30 c d+13 d^2\right )}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a \sin (e+f x)+a)^2} \]

[Out]

(d^3*(20*c^2 - 30*c*d + 13*d^2)*x)/(2*a^3) + (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*Cos[e +
f*x])/(15*a^3*f) + (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Cos[e + f*x]*Sin[e + f*x])/(30*a^3*f) - ((c
- d)*(2*c^2 + 15*c*d + 76*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)
*(2*c + 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c
+ d*Sin[e + f*x])^4)/(5*f*(a + a*Sin[e + f*x])^3)

________________________________________________________________________________________

Rubi [A]  time = 0.614009, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2765, 2977, 2734} \[ \frac{2 d \left (72 c^2 d^2+15 c^3 d+2 c^4-180 c d^3+76 d^4\right ) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac{d^2 \left (30 c^2 d+4 c^3+146 c d^2-195 d^3\right ) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left (20 c^2-30 c d+13 d^2\right )}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a \sin (e+f x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^3,x]

[Out]

(d^3*(20*c^2 - 30*c*d + 13*d^2)*x)/(2*a^3) + (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*Cos[e +
f*x])/(15*a^3*f) + (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Cos[e + f*x]*Sin[e + f*x])/(30*a^3*f) - ((c
- d)*(2*c^2 + 15*c*d + 76*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)
*(2*c + 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c
+ d*Sin[e + f*x])^4)/(5*f*(a + a*Sin[e + f*x])^3)

Rule 2765

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2977

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a+a \sin (e+f x))^3}-\frac{\int \frac{(c+d \sin (e+f x))^3 (-a (2 c-d) (c+4 d)+a (2 c-7 d) d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a+a \sin (e+f x))^3}-\frac{\int \frac{(c+d \sin (e+f x))^2 \left (-a^2 \left (2 c^3+9 c^2 d+37 c d^2-33 d^3\right )+a^2 d \left (4 c^2+24 c d-43 d^2\right ) \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{15 a^4}\\ &=-\frac{(c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a+a \sin (e+f x))^3}-\frac{\int (c+d \sin (e+f x)) \left (-a^3 d^2 \left (2 c^2+165 c d-152 d^2\right )+a^3 d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sin (e+f x)\right ) \, dx}{15 a^6}\\ &=\frac{d^3 \left (20 c^2-30 c d+13 d^2\right ) x}{2 a^3}+\frac{2 d \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right ) \cos (e+f x)}{15 a^3 f}+\frac{d^2 \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \cos (e+f x) \sin (e+f x)}{30 a^3 f}-\frac{(c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a+a \sin (e+f x))^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a+a \sin (e+f x))^3}\\ \end{align*}

Mathematica [B]  time = 8.02663, size = 992, normalized size = 3.57 \[ \frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-160 \cos \left (\frac{3}{2} (e+f x)\right ) c^5+320 \sin \left (\frac{1}{2} (e+f x)\right ) c^5-32 \sin \left (\frac{5}{2} (e+f x)\right ) c^5+1200 d \cos \left (\frac{1}{2} (e+f x)\right ) c^4-1200 d \cos \left (\frac{3}{2} (e+f x)\right ) c^4+1200 d \sin \left (\frac{1}{2} (e+f x)\right ) c^4-240 d \sin \left (\frac{5}{2} (e+f x)\right ) c^4+4800 d^2 \cos \left (\frac{1}{2} (e+f x)\right ) c^3-3200 d^2 \cos \left (\frac{3}{2} (e+f x)\right ) c^3+6400 d^2 \sin \left (\frac{1}{2} (e+f x)\right ) c^3+2400 d^2 \sin \left (\frac{3}{2} (e+f x)\right ) c^3-1120 d^2 \sin \left (\frac{5}{2} (e+f x)\right ) c^3-21600 d^3 \cos \left (\frac{1}{2} (e+f x)\right ) c^2+12000 d^3 (e+f x) \cos \left (\frac{1}{2} (e+f x)\right ) c^2+18400 d^3 \cos \left (\frac{3}{2} (e+f x)\right ) c^2-6000 d^3 (e+f x) \cos \left (\frac{3}{2} (e+f x)\right ) c^2-1200 d^3 (e+f x) \cos \left (\frac{5}{2} (e+f x)\right ) c^2-29600 d^3 \sin \left (\frac{1}{2} (e+f x)\right ) c^2+12000 d^3 (e+f x) \sin \left (\frac{1}{2} (e+f x)\right ) c^2-7200 d^3 \sin \left (\frac{3}{2} (e+f x)\right ) c^2+6000 d^3 (e+f x) \sin \left (\frac{3}{2} (e+f x)\right ) c^2+5120 d^3 \sin \left (\frac{5}{2} (e+f x)\right ) c^2-1200 d^3 (e+f x) \sin \left (\frac{5}{2} (e+f x)\right ) c^2+22500 d^4 \cos \left (\frac{1}{2} (e+f x)\right ) c-18000 d^4 (e+f x) \cos \left (\frac{1}{2} (e+f x)\right ) c-24300 d^4 \cos \left (\frac{3}{2} (e+f x)\right ) c+9000 d^4 (e+f x) \cos \left (\frac{3}{2} (e+f x)\right ) c+1500 d^4 \cos \left (\frac{5}{2} (e+f x)\right ) c+1800 d^4 (e+f x) \cos \left (\frac{5}{2} (e+f x)\right ) c+300 d^4 \cos \left (\frac{7}{2} (e+f x)\right ) c+35100 d^4 \sin \left (\frac{1}{2} (e+f x)\right ) c-18000 d^4 (e+f x) \sin \left (\frac{1}{2} (e+f x)\right ) c+4500 d^4 \sin \left (\frac{3}{2} (e+f x)\right ) c-9000 d^4 (e+f x) \sin \left (\frac{3}{2} (e+f x)\right ) c-7260 d^4 \sin \left (\frac{5}{2} (e+f x)\right ) c+1800 d^4 (e+f x) \sin \left (\frac{5}{2} (e+f x)\right ) c+300 d^4 \sin \left (\frac{7}{2} (e+f x)\right ) c-7560 d^5 \cos \left (\frac{1}{2} (e+f x)\right )+7800 d^5 (e+f x) \cos \left (\frac{1}{2} (e+f x)\right )+9230 d^5 \cos \left (\frac{3}{2} (e+f x)\right )-3900 d^5 (e+f x) \cos \left (\frac{3}{2} (e+f x)\right )-750 d^5 \cos \left (\frac{5}{2} (e+f x)\right )-780 d^5 (e+f x) \cos \left (\frac{5}{2} (e+f x)\right )-105 d^5 \cos \left (\frac{7}{2} (e+f x)\right )-15 d^5 \cos \left (\frac{9}{2} (e+f x)\right )-12760 d^5 \sin \left (\frac{1}{2} (e+f x)\right )+7800 d^5 (e+f x) \sin \left (\frac{1}{2} (e+f x)\right )-930 d^5 \sin \left (\frac{3}{2} (e+f x)\right )+3900 d^5 (e+f x) \sin \left (\frac{3}{2} (e+f x)\right )+2782 d^5 \sin \left (\frac{5}{2} (e+f x)\right )-780 d^5 (e+f x) \sin \left (\frac{5}{2} (e+f x)\right )-105 d^5 \sin \left (\frac{7}{2} (e+f x)\right )+15 d^5 \sin \left (\frac{9}{2} (e+f x)\right )\right )}{480 f (\sin (e+f x) a+a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^3,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(1200*c^4*d*Cos[(e + f*x)/2] + 4800*c^3*d^2*Cos[(e + f*x)/2] - 21600*c^
2*d^3*Cos[(e + f*x)/2] + 22500*c*d^4*Cos[(e + f*x)/2] - 7560*d^5*Cos[(e + f*x)/2] + 12000*c^2*d^3*(e + f*x)*Co
s[(e + f*x)/2] - 18000*c*d^4*(e + f*x)*Cos[(e + f*x)/2] + 7800*d^5*(e + f*x)*Cos[(e + f*x)/2] - 160*c^5*Cos[(3
*(e + f*x))/2] - 1200*c^4*d*Cos[(3*(e + f*x))/2] - 3200*c^3*d^2*Cos[(3*(e + f*x))/2] + 18400*c^2*d^3*Cos[(3*(e
+ f*x))/2] - 24300*c*d^4*Cos[(3*(e + f*x))/2] + 9230*d^5*Cos[(3*(e + f*x))/2] - 6000*c^2*d^3*(e + f*x)*Cos[(3
*(e + f*x))/2] + 9000*c*d^4*(e + f*x)*Cos[(3*(e + f*x))/2] - 3900*d^5*(e + f*x)*Cos[(3*(e + f*x))/2] + 1500*c*
d^4*Cos[(5*(e + f*x))/2] - 750*d^5*Cos[(5*(e + f*x))/2] - 1200*c^2*d^3*(e + f*x)*Cos[(5*(e + f*x))/2] + 1800*c
*d^4*(e + f*x)*Cos[(5*(e + f*x))/2] - 780*d^5*(e + f*x)*Cos[(5*(e + f*x))/2] + 300*c*d^4*Cos[(7*(e + f*x))/2]
- 105*d^5*Cos[(7*(e + f*x))/2] - 15*d^5*Cos[(9*(e + f*x))/2] + 320*c^5*Sin[(e + f*x)/2] + 1200*c^4*d*Sin[(e +
f*x)/2] + 6400*c^3*d^2*Sin[(e + f*x)/2] - 29600*c^2*d^3*Sin[(e + f*x)/2] + 35100*c*d^4*Sin[(e + f*x)/2] - 1276
0*d^5*Sin[(e + f*x)/2] + 12000*c^2*d^3*(e + f*x)*Sin[(e + f*x)/2] - 18000*c*d^4*(e + f*x)*Sin[(e + f*x)/2] + 7
800*d^5*(e + f*x)*Sin[(e + f*x)/2] + 2400*c^3*d^2*Sin[(3*(e + f*x))/2] - 7200*c^2*d^3*Sin[(3*(e + f*x))/2] + 4
500*c*d^4*Sin[(3*(e + f*x))/2] - 930*d^5*Sin[(3*(e + f*x))/2] + 6000*c^2*d^3*(e + f*x)*Sin[(3*(e + f*x))/2] -
9000*c*d^4*(e + f*x)*Sin[(3*(e + f*x))/2] + 3900*d^5*(e + f*x)*Sin[(3*(e + f*x))/2] - 32*c^5*Sin[(5*(e + f*x))
/2] - 240*c^4*d*Sin[(5*(e + f*x))/2] - 1120*c^3*d^2*Sin[(5*(e + f*x))/2] + 5120*c^2*d^3*Sin[(5*(e + f*x))/2] -
7260*c*d^4*Sin[(5*(e + f*x))/2] + 2782*d^5*Sin[(5*(e + f*x))/2] - 1200*c^2*d^3*(e + f*x)*Sin[(5*(e + f*x))/2]
+ 1800*c*d^4*(e + f*x)*Sin[(5*(e + f*x))/2] - 780*d^5*(e + f*x)*Sin[(5*(e + f*x))/2] + 300*c*d^4*Sin[(7*(e +
f*x))/2] - 105*d^5*Sin[(7*(e + f*x))/2] + 15*d^5*Sin[(9*(e + f*x))/2]))/(480*f*(a + a*Sin[e + f*x])^3)

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Maple [B]  time = 0.101, size = 924, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x)

[Out]

6/f/a^3*d^5/(1+tan(1/2*f*x+1/2*e)^2)^2+13/f/a^3*d^5*arctan(tan(1/2*f*x+1/2*e))-2/f/a^3/(tan(1/2*f*x+1/2*e)+1)*
c^5+12/f/a^3/(tan(1/2*f*x+1/2*e)+1)*d^5+4/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^5+6/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*
d^5-16/3/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*c^5-4/3/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*d^5+4/f/a^3/(tan(1/2*f*x+1/2*e)
+1)^4*c^5-30/f/a^3/(tan(1/2*f*x+1/2*e)+1)*c*d^4-8/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c*d^4+1/f/a^3*d^5/(1+tan(1/2*
f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^3+6/f/a^3*d^5/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^2-1/f/a^3*d^5/(
1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)-10/f/a^3*d^4/(1+tan(1/2*f*x+1/2*e)^2)^2*c+20/f/a^3*d^3*arctan(tan
(1/2*f*x+1/2*e))*c^2-30/f/a^3*d^4*arctan(tan(1/2*f*x+1/2*e))*c-4/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*d^5-8/5/f/a^3/
(tan(1/2*f*x+1/2*e)+1)^5*c^5+8/5/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*d^5-10/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^4*d+20
/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c^2*d^3-20/f/a^3/(tan(1/2*f*x+1/2*e)+1)^2*c*d^4+20/f/a^3/(tan(1/2*f*x+1/2*e)+1
)^3*c^4*d-80/3/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*c^3*d^2+40/3/f/a^3/(tan(1/2*f*x+1/2*e)+1)^3*c^2*d^3-20/f/a^3/(ta
n(1/2*f*x+1/2*e)+1)^4*c^4*d+40/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^3*d^2-40/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c^2*d^
3+20/f/a^3/(tan(1/2*f*x+1/2*e)+1)^4*c*d^4+8/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c^4*d-16/f/a^3/(tan(1/2*f*x+1/2*e)+
1)^5*c^3*d^2+16/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5*c^2*d^3+20/f/a^3/(tan(1/2*f*x+1/2*e)+1)*c^2*d^3-10/f/a^3*d^4/(1
+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^2*c

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Maxima [B]  time = 2.56399, size = 2030, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

1/15*(d^5*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3805*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 + 4329*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f*x + e)^5/(cos(f*x + e) + 1)^5
+ 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^8/(co
s(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 26*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*a^3*sin(f*
x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12*a^3*sin(f*x + e)^7/(cos(f*x +
e) + 1)^7 + 5*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 195*arctan(
sin(f*x + e)/(cos(f*x + e) + 1))/a^3) - 30*c*d^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 200*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 75*si
n(f*x + e)^5/(cos(f*x + e) + 1)^5 + 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(co
s(f*x + e) + 1) + 11*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15
*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/(
cos(f*x + e) + 1)^6 + a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^
3) + 20*c^2*d^3*((95*sin(f*x + e)/(cos(f*x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x +
e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(
f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(cos(f*x +
e) + 1))/a^3) - 2*c^5*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f
*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 7)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*
x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 40*c^3*d^2*(5*sin(f*x + e)/(c
os(f*x + e) + 1) + 10*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) +
10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4
/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 30*c^4*d*(5*sin(f*x + e)/(cos(f*x + e) + 1)
+ 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e
)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3
+ 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5))/f

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Fricas [B]  time = 1.83992, size = 1548, normalized size = 5.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

1/30*(15*d^5*cos(f*x + e)^5 + 6*c^5 - 30*c^4*d + 60*c^3*d^2 - 60*c^2*d^3 + 30*c*d^4 - 6*d^5 - 30*(5*c*d^4 - 2*
d^5)*cos(f*x + e)^4 - (4*c^5 + 30*c^4*d + 140*c^3*d^2 - 640*c^2*d^3 + 1170*c*d^4 - 449*d^5 - 15*(20*c^2*d^3 -
30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e)^3 - 60*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x + (8*c^5 + 60*c^4*d - 20*c^3*
d^2 - 380*c^2*d^3 + 840*c*d^4 - 358*d^5 + 45*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e)^2 + 6*(3*c^5 +
10*c^4*d + 30*c^3*d^2 - 180*c^2*d^3 + 315*c*d^4 - 128*d^5 - 5*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x +
e) - (15*d^5*cos(f*x + e)^4 + 6*c^5 - 30*c^4*d + 60*c^3*d^2 - 60*c^2*d^3 + 30*c*d^4 - 6*d^5 + 15*(10*c*d^4 -
3*d^5)*cos(f*x + e)^3 + 60*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x - (4*c^5 + 30*c^4*d + 140*c^3*d^2 - 640*c^2*d^
3 + 1020*c*d^4 - 404*d^5 + 15*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e)^2 - 6*(2*c^5 + 15*c^4*d + 20*
c^3*d^2 - 170*c^2*d^3 + 310*c*d^4 - 127*d^5 - 5*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e))*sin(f*x +
e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 -
2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))

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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((c+d*sin(f*x+e))**5/(a+a*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.33615, size = 761, normalized size = 2.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/30*(15*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*(f*x + e)/a^3 + 30*(d^5*tan(1/2*f*x + 1/2*e)^3 - 10*c*d^4*tan(1/2*f*
x + 1/2*e)^2 + 6*d^5*tan(1/2*f*x + 1/2*e)^2 - d^5*tan(1/2*f*x + 1/2*e) - 10*c*d^4 + 6*d^5)/((tan(1/2*f*x + 1/2
*e)^2 + 1)^2*a^3) - 4*(15*c^5*tan(1/2*f*x + 1/2*e)^4 - 150*c^2*d^3*tan(1/2*f*x + 1/2*e)^4 + 225*c*d^4*tan(1/2*
f*x + 1/2*e)^4 - 90*d^5*tan(1/2*f*x + 1/2*e)^4 + 30*c^5*tan(1/2*f*x + 1/2*e)^3 + 75*c^4*d*tan(1/2*f*x + 1/2*e)
^3 - 750*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 + 1050*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 405*d^5*tan(1/2*f*x + 1/2*e)^3 +
40*c^5*tan(1/2*f*x + 1/2*e)^2 + 75*c^4*d*tan(1/2*f*x + 1/2*e)^2 + 200*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 - 1450*c
^2*d^3*tan(1/2*f*x + 1/2*e)^2 + 1800*c*d^4*tan(1/2*f*x + 1/2*e)^2 - 665*d^5*tan(1/2*f*x + 1/2*e)^2 + 20*c^5*ta
n(1/2*f*x + 1/2*e) + 75*c^4*d*tan(1/2*f*x + 1/2*e) + 100*c^3*d^2*tan(1/2*f*x + 1/2*e) - 950*c^2*d^3*tan(1/2*f*
x + 1/2*e) + 1200*c*d^4*tan(1/2*f*x + 1/2*e) - 445*d^5*tan(1/2*f*x + 1/2*e) + 7*c^5 + 15*c^4*d + 20*c^3*d^2 -
220*c^2*d^3 + 285*c*d^4 - 107*d^5)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f