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

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

[Out]

1/2*d^3*(20*c^2-30*c*d+13*d^2)*x/a^3+2/15*d*(2*c^4+15*c^3*d+72*c^2*d^2-180*c*d^3+76*d^4)*cos(f*x+e)/a^3/f+1/30
*d^2*(4*c^3+30*c^2*d+146*c*d^2-195*d^3)*cos(f*x+e)*sin(f*x+e)/a^3/f-1/15*(c-d)*(2*c^2+15*c*d+76*d^2)*cos(f*x+e
)*(c+d*sin(f*x+e))^2/f/(a^3+a^3*sin(f*x+e))-1/15*(c-d)*(2*c+11*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/a/f/(a+a*sin(f
*x+e))^2-1/5*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/f/(a+a*sin(f*x+e))^3

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Rubi [A]  time = 0.61, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 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]

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])

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*}

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Mathematica [B]  time = 7.90, 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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fricas [B]  time = 0.50, size = 653, normalized size = 2.35 \[ \frac {15 \, d^{5} \cos \left (f x + e\right )^{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 \, {\left (5 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{4} - {\left (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 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )^{3} - 60 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x + {\left (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 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (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 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right ) - {\left (15 \, d^{5} \cos \left (f x + e\right )^{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 \, {\left (10 \, c d^{4} - 3 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + 60 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x - {\left (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 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (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 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]

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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giac [B]  time = 0.25, size = 564, normalized size = 2.03 \[ \frac {\frac {15 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} {\left (f x + e\right )}}{a^{3}} + \frac {30 \, {\left (d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 10 \, c d^{4} + 6 \, d^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{3}} - \frac {4 \, {\left (15 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 150 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 225 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 90 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 75 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 750 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1050 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 405 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 75 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 200 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1450 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1800 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 665 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 75 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 100 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 950 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1200 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 445 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{5} + 15 \, c^{4} d + 20 \, c^{3} d^{2} - 220 \, c^{2} d^{3} + 285 \, c d^{4} - 107 \, d^{5}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{30 \, f} \]

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

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maple [B]  time = 0.29, size = 924, normalized size = 3.32 \[ \text {result too large to display} \]

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

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maxima [B]  time = 0.49, size = 1504, normalized size = 5.41 \[ \text {result too large to display} \]

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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mupad [B]  time = 9.54, size = 652, normalized size = 2.35 \[ \frac {d^3\,\mathrm {atan}\left (\frac {d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^2-30\,c\,d+13\,d^2\right )}{20\,c^2\,d^3-30\,c\,d^4+13\,d^5}\right )\,\left (20\,c^2-30\,c\,d+13\,d^2\right )}{a^3\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {28\,c^5}{3}+10\,c^4\,d+\frac {80\,c^3\,d^2}{3}-\frac {700\,c^2\,d^3}{3}+350\,c\,d^4-\frac {455\,d^5}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {36\,c^5}{5}+14\,c^4\,d+32\,c^3\,d^2-252\,c^2\,d^3+426\,c\,d^4-\frac {891\,d^5}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {32\,c^5}{3}+30\,c^4\,d+\frac {40\,c^3\,d^2}{3}-\frac {980\,c^2\,d^3}{3}+550\,c\,d^4-\frac {715\,d^5}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {28\,c^5}{3}+30\,c^4\,d+\frac {80\,c^3\,d^2}{3}-\frac {1060\,c^2\,d^3}{3}+610\,c\,d^4-\frac {761\,d^5}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {68\,c^5}{5}+22\,c^4\,d+56\,c^3\,d^2-436\,c^2\,d^3+698\,c\,d^4-\frac {1443\,d^5}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (4\,c^5+10\,c^4\,d-100\,c^2\,d^3+150\,c\,d^4-65\,d^5\right )+48\,c\,d^4+2\,c^4\,d+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,c^5-20\,c^2\,d^3+30\,c\,d^4-13\,d^5\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8\,c^5}{3}+10\,c^4\,d+\frac {40\,c^3\,d^2}{3}-\frac {380\,c^2\,d^3}{3}+210\,c\,d^4-\frac {265\,d^5}{3}\right )+\frac {14\,c^5}{15}-\frac {304\,d^5}{15}-\frac {88\,c^2\,d^3}{3}+\frac {8\,c^3\,d^2}{3}}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+12\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+20\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+26\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+26\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+20\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+12\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^3*atan((d^3*tan(e/2 + (f*x)/2)*(20*c^2 - 30*c*d + 13*d^2))/(13*d^5 - 30*c*d^4 + 20*c^2*d^3))*(20*c^2 - 30*c
*d + 13*d^2))/(a^3*f) - (tan(e/2 + (f*x)/2)^6*(350*c*d^4 + 10*c^4*d + (28*c^5)/3 - (455*d^5)/3 - (700*c^2*d^3)
/3 + (80*c^3*d^2)/3) + tan(e/2 + (f*x)/2)^2*(426*c*d^4 + 14*c^4*d + (36*c^5)/5 - (891*d^5)/5 - 252*c^2*d^3 + 3
2*c^3*d^2) + tan(e/2 + (f*x)/2)^5*(550*c*d^4 + 30*c^4*d + (32*c^5)/3 - (715*d^5)/3 - (980*c^2*d^3)/3 + (40*c^3
*d^2)/3) + tan(e/2 + (f*x)/2)^3*(610*c*d^4 + 30*c^4*d + (28*c^5)/3 - (761*d^5)/3 - (1060*c^2*d^3)/3 + (80*c^3*
d^2)/3) + tan(e/2 + (f*x)/2)^4*(698*c*d^4 + 22*c^4*d + (68*c^5)/5 - (1443*d^5)/5 - 436*c^2*d^3 + 56*c^3*d^2) +
tan(e/2 + (f*x)/2)^7*(150*c*d^4 + 10*c^4*d + 4*c^5 - 65*d^5 - 100*c^2*d^3) + 48*c*d^4 + 2*c^4*d + tan(e/2 + (
f*x)/2)^8*(30*c*d^4 + 2*c^5 - 13*d^5 - 20*c^2*d^3) + tan(e/2 + (f*x)/2)*(210*c*d^4 + 10*c^4*d + (8*c^5)/3 - (2
65*d^5)/3 - (380*c^2*d^3)/3 + (40*c^3*d^2)/3) + (14*c^5)/15 - (304*d^5)/15 - (88*c^2*d^3)/3 + (8*c^3*d^2)/3)/(
f*(12*a^3*tan(e/2 + (f*x)/2)^2 + 20*a^3*tan(e/2 + (f*x)/2)^3 + 26*a^3*tan(e/2 + (f*x)/2)^4 + 26*a^3*tan(e/2 +
(f*x)/2)^5 + 20*a^3*tan(e/2 + (f*x)/2)^6 + 12*a^3*tan(e/2 + (f*x)/2)^7 + 5*a^3*tan(e/2 + (f*x)/2)^8 + a^3*tan(
e/2 + (f*x)/2)^9 + a^3 + 5*a^3*tan(e/2 + (f*x)/2)))

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

[Out]

Timed out

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