3.5.71 \(\int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx\) [471]

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

[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 = 0.41, 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 = {2844, 3056, 2813} \begin {gather*} -\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} \end {gather*}

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 2813

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 2844

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 3056

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] Leaf count is larger than twice the leaf count of optimal. \(992\) vs. \(2(278)=556\).
time = 6.86, size = 992, normalized size = 3.57 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (1200 c^4 d \cos \left (\frac {1}{2} (e+f x)\right )+4800 c^3 d^2 \cos \left (\frac {1}{2} (e+f x)\right )-21600 c^2 d^3 \cos \left (\frac {1}{2} (e+f x)\right )+22500 c d^4 \cos \left (\frac {1}{2} (e+f x)\right )-7560 d^5 \cos \left (\frac {1}{2} (e+f x)\right )+12000 c^2 d^3 (e+f x) \cos \left (\frac {1}{2} (e+f x)\right )-18000 c d^4 (e+f x) \cos \left (\frac {1}{2} (e+f x)\right )+7800 d^5 (e+f x) \cos \left (\frac {1}{2} (e+f x)\right )-160 c^5 \cos \left (\frac {3}{2} (e+f x)\right )-1200 c^4 d \cos \left (\frac {3}{2} (e+f x)\right )-3200 c^3 d^2 \cos \left (\frac {3}{2} (e+f x)\right )+18400 c^2 d^3 \cos \left (\frac {3}{2} (e+f x)\right )-24300 c d^4 \cos \left (\frac {3}{2} (e+f x)\right )+9230 d^5 \cos \left (\frac {3}{2} (e+f x)\right )-6000 c^2 d^3 (e+f x) \cos \left (\frac {3}{2} (e+f x)\right )+9000 c d^4 (e+f x) \cos \left (\frac {3}{2} (e+f x)\right )-3900 d^5 (e+f x) \cos \left (\frac {3}{2} (e+f x)\right )+1500 c d^4 \cos \left (\frac {5}{2} (e+f x)\right )-750 d^5 \cos \left (\frac {5}{2} (e+f x)\right )-1200 c^2 d^3 (e+f x) \cos \left (\frac {5}{2} (e+f x)\right )+1800 c d^4 (e+f x) \cos \left (\frac {5}{2} (e+f x)\right )-780 d^5 (e+f x) \cos \left (\frac {5}{2} (e+f x)\right )+300 c d^4 \cos \left (\frac {7}{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 )+320 c^5 \sin \left (\frac {1}{2} (e+f x)\right )+1200 c^4 d \sin \left (\frac {1}{2} (e+f x)\right )+6400 c^3 d^2 \sin \left (\frac {1}{2} (e+f x)\right )-29600 c^2 d^3 \sin \left (\frac {1}{2} (e+f x)\right )+35100 c d^4 \sin \left (\frac {1}{2} (e+f x)\right )-12760 d^5 \sin \left (\frac {1}{2} (e+f x)\right )+12000 c^2 d^3 (e+f x) \sin \left (\frac {1}{2} (e+f x)\right )-18000 c d^4 (e+f x) \sin \left (\frac {1}{2} (e+f x)\right )+7800 d^5 (e+f x) \sin \left (\frac {1}{2} (e+f x)\right )+2400 c^3 d^2 \sin \left (\frac {3}{2} (e+f x)\right )-7200 c^2 d^3 \sin \left (\frac {3}{2} (e+f x)\right )+4500 c d^4 \sin \left (\frac {3}{2} (e+f x)\right )-930 d^5 \sin \left (\frac {3}{2} (e+f x)\right )+6000 c^2 d^3 (e+f x) \sin \left (\frac {3}{2} (e+f x)\right )-9000 c d^4 (e+f x) \sin \left (\frac {3}{2} (e+f x)\right )+3900 d^5 (e+f x) \sin \left (\frac {3}{2} (e+f x)\right )-32 c^5 \sin \left (\frac {5}{2} (e+f x)\right )-240 c^4 d \sin \left (\frac {5}{2} (e+f x)\right )-1120 c^3 d^2 \sin \left (\frac {5}{2} (e+f x)\right )+5120 c^2 d^3 \sin \left (\frac {5}{2} (e+f x)\right )-7260 c d^4 \sin \left (\frac {5}{2} (e+f x)\right )+2782 d^5 \sin \left (\frac {5}{2} (e+f x)\right )-1200 c^2 d^3 (e+f x) \sin \left (\frac {5}{2} (e+f x)\right )+1800 c d^4 (e+f x) \sin \left (\frac {5}{2} (e+f x)\right )-780 d^5 (e+f x) \sin \left (\frac {5}{2} (e+f x)\right )+300 c d^4 \sin \left (\frac {7}{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 (a+a \sin (e+f x))^3} \end {gather*}

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 [A]
time = 0.58, size = 360, normalized size = 1.29

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (c^{5}-10 c^{2} d^{3}+15 c \,d^{4}-6 d^{5}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{5}+10 c^{4} d -20 c^{2} d^{3}+20 c \,d^{4}-6 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c^{5}-30 c^{4} d +40 c^{3} d^{2}-20 c^{2} d^{3}+2 d^{5}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 c^{5}+40 c^{4} d -80 c^{3} d^{2}+80 c^{2} d^{3}-40 c \,d^{4}+8 d^{5}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{5}-20 c^{4} d +40 c^{3} d^{2}-40 c^{2} d^{3}+20 c \,d^{4}-4 d^{5}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+2 d^{3} \left (\frac {\frac {d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (-5 c d +3 d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-5 c d +3 d^{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (20 c^{2}-30 c d +13 d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{3}}\)	\(360\)
default	\(\frac {-\frac {2 \left (c^{5}-10 c^{2} d^{3}+15 c \,d^{4}-6 d^{5}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{5}+10 c^{4} d -20 c^{2} d^{3}+20 c \,d^{4}-6 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c^{5}-30 c^{4} d +40 c^{3} d^{2}-20 c^{2} d^{3}+2 d^{5}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 c^{5}+40 c^{4} d -80 c^{3} d^{2}+80 c^{2} d^{3}-40 c \,d^{4}+8 d^{5}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{5}-20 c^{4} d +40 c^{3} d^{2}-40 c^{2} d^{3}+20 c \,d^{4}-4 d^{5}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+2 d^{3} \left (\frac {\frac {d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (-5 c d +3 d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-5 c d +3 d^{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (20 c^{2}-30 c d +13 d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{3}}\)	\(360\)
risch	\(\frac {10 d^{3} x \,c^{2}}{a^{3}}-\frac {15 d^{4} x c}{a^{3}}+\frac {13 d^{5} x}{2 a^{3}}+\frac {i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 f \,a^{3}}-\frac {5 d^{4} {\mathrm e}^{i \left (f x +e \right )} c}{2 f \,a^{3}}+\frac {3 d^{5} {\mathrm e}^{i \left (f x +e \right )}}{2 f \,a^{3}}-\frac {5 d^{4} {\mathrm e}^{-i \left (f x +e \right )} c}{2 f \,a^{3}}+\frac {3 d^{5} {\mathrm e}^{-i \left (f x +e \right )}}{2 f \,a^{3}}-\frac {i d^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 f \,a^{3}}+\frac {\frac {4 i c^{5} {\mathrm e}^{i \left (f x +e \right )}}{3}-60 c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+\frac {160 c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{3}+70 i d^{5} {\mathrm e}^{3 i \left (f x +e \right )}+280 c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-20 c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+10 c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+60 c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-\frac {740 c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{3}-\frac {194 i d^{5} {\mathrm e}^{i \left (f x +e \right )}}{3}+\frac {8 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{3}-\frac {4 c^{5}}{15}+\frac {254 d^{5}}{15}-40 i c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-200 i c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-10 i c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}+180 i c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+180 i c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}+10 i c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}+\frac {80 i c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}}{3}+\frac {128 c^{2} d^{3}}{3}-48 c \,d^{4}-\frac {28 c^{3} d^{2}}{3}-2 c^{4} d -\frac {460 i c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}}{3}+20 d^{5} {\mathrm e}^{4 i \left (f x +e \right )}-\frac {298 d^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{3}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\)	\(554\)
norman	\(\text {Expression too large to display}\)	\(1411\)

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,method=_RETURNVERBOSE)

[Out]

2/f/a^3*(-(c^5-10*c^2*d^3+15*c*d^4-6*d^5)/(tan(1/2*f*x+1/2*e)+1)-1/2*(-4*c^5+10*c^4*d-20*c^2*d^3+20*c*d^4-6*d^
5)/(tan(1/2*f*x+1/2*e)+1)^2-1/3*(8*c^5-30*c^4*d+40*c^3*d^2-20*c^2*d^3+2*d^5)/(tan(1/2*f*x+1/2*e)+1)^3-1/4*(-8*
c^5+40*c^4*d-80*c^3*d^2+80*c^2*d^3-40*c*d^4+8*d^5)/(tan(1/2*f*x+1/2*e)+1)^4-1/5*(4*c^5-20*c^4*d+40*c^3*d^2-40*
c^2*d^3+20*c*d^4-4*d^5)/(tan(1/2*f*x+1/2*e)+1)^5+d^3*((1/2*d^2*tan(1/2*f*x+1/2*e)^3+(-5*c*d+3*d^2)*tan(1/2*f*x
+1/2*e)^2-1/2*d^2*tan(1/2*f*x+1/2*e)-5*c*d+3*d^2)/(1+tan(1/2*f*x+1/2*e)^2)^2+1/2*(20*c^2-30*c*d+13*d^2)*arctan
(tan(1/2*f*x+1/2*e))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1636 vs. \(2 (278) = 556\).
time = 0.53, size = 1636, normalized size = 5.88 \begin {gather*} \text {Too large to display} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (278) = 556\).
time = 0.37, size = 669, normalized size = 2.41 \begin {gather*} \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 )}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 15553 vs. \(2 (264) = 528\).
time = 31.07, size = 15553, normalized size = 55.95 \begin {gather*} \text {Too large to display} \end {gather*}

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]

Piecewise((-60*c**5*tan(e/2 + f*x/2)**8/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2 + f*x/2)**8 + 360*
a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(
e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2
) + 30*a**3*f) - 120*c**5*tan(e/2 + f*x/2)**7/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2 + f*x/2)**8
+ 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5 + 780*a**3*
f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 +
f*x/2) + 30*a**3*f) - 280*c**5*tan(e/2 + f*x/2)**6/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2 + f*x/
2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5 + 780
*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan
(e/2 + f*x/2) + 30*a**3*f) - 320*c**5*tan(e/2 + f*x/2)**5/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2
+ f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5
+ 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3
*f*tan(e/2 + f*x/2) + 30*a**3*f) - 408*c**5*tan(e/2 + f*x/2)**4/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*ta
n(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x
/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 15
0*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 280*c**5*tan(e/2 + f*x/2)**3/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**
3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2
+ f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**
2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 216*c**5*tan(e/2 + f*x/2)**2/(30*a**3*f*tan(e/2 + f*x/2)**9 + 1
50*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*t
an(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*
x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 80*c**5*tan(e/2 + f*x/2)/(30*a**3*f*tan(e/2 + f*x/2)**9 +
150*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f
*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 +
f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 28*c**5/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan
(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/
2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 150
*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 300*c**4*d*tan(e/2 + f*x/2)**7/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a*
*3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/
2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)*
*2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 300*c**4*d*tan(e/2 + f*x/2)**6/(30*a**3*f*tan(e/2 + f*x/2)**9
+ 150*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 780*a**3*
f*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*tan(e/2 +
f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 900*c**4*d*tan(e/2 + f*x/2)**5/(30*a**3*f*tan(e/2 + f*
x/2)**9 + 150*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2)**6 + 7
80*a**3*f*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*a**3*f*t
an(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 660*c**4*d*tan(e/2 + f*x/2)**4/(30*a**3*f*tan(
e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2 + f*x/2
)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**3 + 360*
a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 900*c**4*d*tan(e/2 + f*x/2)**3/(30*a**
3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f*tan(e/2
+ f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 + f*x/2)**
3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 420*c**4*d*tan(e/2 + f*x/2)**2
/(30*a**3*f*tan(e/2 + f*x/2)**9 + 150*a**3*f*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)**7 + 600*a**3*f
*tan(e/2 + f*x/2)**6 + 780*a**3*f*tan(e/2 + f*x/2)**5 + 780*a**3*f*tan(e/2 + f*x/2)**4 + 600*a**3*f*tan(e/2 +
f*x/2)**3 + 360*a**3*f*tan(e/2 + f*x/2)**2 + 15...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (278) = 556\).
time = 0.45, size = 564, normalized size = 2.03 \begin {gather*} \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} \end {gather*}

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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Mupad [B]
time = 9.54, size = 652, normalized size = 2.35 \begin {gather*} \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 )} \end {gather*}

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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