\(\int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx\) [479]

Optimal result

Integrand size = 25, antiderivative size = 278 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {d^3 \left (20 c^2-30 c d+13 d^2\right ) x}{2 a^3}+\frac {2 d \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right ) \cos (e+f x)}{15 a^3 f}+\frac {d^2 \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \cos (e+f x) \sin (e+f x)}{30 a^3 f}-\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a+a \sin (e+f x))^3} \] Output:

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
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(992\) vs. \(2(278)=556\).

Time = 8.41 (sec) , antiderivative size = 992, normalized size of antiderivative = 3.57 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((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*C 
os[(e + f*x)/2] - 7560*d^5*Cos[(e + f*x)/2] + 12000*c^2*d^3*(e + f*x)*Cos[ 
(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*Co 
s[(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] - 2960 
0*c^2*d^3*Sin[(e + f*x)/2] + 35100*c*d^4*Sin[(e + f*x)/2] - 12760*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] + 7800*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] + 4500*c*d^4*Si 
n[(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] + 390...
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3244, 25, 3042, 3456, 3042, 3456, 3042, 3213}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3244

\(\displaystyle -\frac {\int -\frac {(c+d \sin (e+f x))^3 (a (2 c-d) (c+4 d)-a (2 c-7 d) d \sin (e+f x))}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^3 (a (2 c-d) (c+4 d)-a (2 c-7 d) d \sin (e+f x))}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^3 (a (2 c-d) (c+4 d)-a (2 c-7 d) d \sin (e+f x))}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^2 \left (a^2 \left (2 c^3+9 d c^2+37 d^2 c-33 d^3\right )-a^2 d \left (4 c^2+24 d c-43 d^2\right ) \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^2 \left (a^2 \left (2 c^3+9 d c^2+37 d^2 c-33 d^3\right )-a^2 d \left (4 c^2+24 d c-43 d^2\right ) \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\frac {\int (c+d \sin (e+f x)) \left (a^3 d^2 \left (2 c^2+165 d c-152 d^2\right )-a^3 d \left (4 c^3+30 d c^2+146 d^2 c-195 d^3\right ) \sin (e+f x)\right )dx}{a^2}-\frac {a^2 (c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int (c+d \sin (e+f x)) \left (a^3 d^2 \left (2 c^2+165 d c-152 d^2\right )-a^3 d \left (4 c^3+30 d c^2+146 d^2 c-195 d^3\right ) \sin (e+f x)\right )dx}{a^2}-\frac {a^2 (c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {\frac {\frac {15}{2} a^3 d^3 x \left (20 c^2-30 c d+13 d^2\right )+\frac {a^3 d^2 \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {2 a^3 d \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right ) \cos (e+f x)}{f}}{a^2}-\frac {a^2 (c-d) \left (2 c^2+15 c d+76 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3213

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)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3244

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + Simp[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 3456

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(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] - Simp[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] /; Fre 
eQ[{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] && (In 
tegerQ[2*n] || EqQ[c, 0])
 

Maple [A] (verified)

Time = 84.61 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (c^{5}-10 c^{2} d^{3}+15 c \,d^{4}-6 d^{5}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{5}+10 c^{4} d -20 c^{2} d^{3}+20 c \,d^{4}-6 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c^{5}-30 c^{4} d +40 c^{3} d^{2}-20 c^{2} d^{3}+2 d^{5}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 c^{5}+40 c^{4} d -80 c^{3} d^{2}+80 c^{2} d^{3}-40 c \,d^{4}+8 d^{5}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{5}-20 c^{4} d +40 c^{3} d^{2}-40 c^{2} d^{3}+20 c \,d^{4}-4 d^{5}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+2 d^{3} \left (\frac {\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2}+\left (-5 c d +3 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-5 c d +3 d^{2}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2}}+\frac {\left (20 c^{2}-30 c d +13 d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{3}}\)	\(360\)
default	\(\frac {-\frac {2 \left (c^{5}-10 c^{2} d^{3}+15 c \,d^{4}-6 d^{5}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{5}+10 c^{4} d -20 c^{2} d^{3}+20 c \,d^{4}-6 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c^{5}-30 c^{4} d +40 c^{3} d^{2}-20 c^{2} d^{3}+2 d^{5}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 c^{5}+40 c^{4} d -80 c^{3} d^{2}+80 c^{2} d^{3}-40 c \,d^{4}+8 d^{5}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{5}-20 c^{4} d +40 c^{3} d^{2}-40 c^{2} d^{3}+20 c \,d^{4}-4 d^{5}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+2 d^{3} \left (\frac {\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2}+\left (-5 c d +3 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-5 c d +3 d^{2}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2}}+\frac {\left (20 c^{2}-30 c d +13 d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{3}}\)	\(360\)
parallelrisch	\(\frac {\left (\left (7800 f x +14760\right ) d^{5}+\left (-18000 f x -35100\right ) c \,d^{4}+12000 \left (f x +\frac {9}{5}\right ) c^{2} d^{3}-2400 c^{3} d^{2}-1200 c^{4} d -720 c^{5}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-780 f x -2982\right ) d^{5}+\left (1800 f x +7260\right ) c \,d^{4}+\left (-1200 f x -4320\right ) c^{2} d^{3}+720 c^{3} d^{2}+240 c^{4} d +72 c^{5}\right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\left (7800 f x +9560\right ) d^{5}+\left (-18000 f x -22500\right ) c \,d^{4}+12000 \left (f x +\frac {17}{15}\right ) c^{2} d^{3}-800 c^{3} d^{2}-1200 c^{4} d -400 c^{5}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (3900 f x +10230\right ) d^{5}+\left (-9000 f x -24300\right ) c \,d^{4}+6000 \left (f x +\frac {12}{5}\right ) c^{2} d^{3}-1200 c^{3} d^{2}-1200 c^{4} d -360 c^{5}\right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (-3900 f x -1930\right ) d^{5}+\left (9000 f x +4500\right ) c \,d^{4}+\left (-6000 f x -3200\right ) c^{2} d^{3}+400 c^{3} d^{2}+200 c^{5}\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (-780 f x +550\right ) d^{5}+\left (1800 f x -1500\right ) c \,d^{4}+\left (-1200 f x +800\right ) c^{2} d^{3}-400 c^{3} d^{2}+40 c^{5}\right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+300 d^{4} \left (\left (c -\frac {7 d}{20}\right ) \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\left (c -\frac {7 d}{20}\right ) \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )-\frac {d \left (\cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )-\sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )\right )}{20}\right )}{120 f \,a^{3} \left (10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\)	\(511\)
risch	\(\frac {10 d^{3} x \,c^{2}}{a^{3}}-\frac {15 d^{4} x c}{a^{3}}+\frac {13 d^{5} x}{2 a^{3}}+\frac {i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{3} f}-\frac {5 d^{4} {\mathrm e}^{i \left (f x +e \right )} c}{2 a^{3} f}+\frac {3 d^{5} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{3} f}-\frac {5 d^{4} {\mathrm e}^{-i \left (f x +e \right )} c}{2 a^{3} f}+\frac {3 d^{5} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{3} f}-\frac {i d^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} f}+\frac {\frac {254 d^{5}}{15}-200 i c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+180 i c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+\frac {80 i c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}}{3}-\frac {460 i c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}}{3}+180 i c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}+10 i c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}-2 c^{4} d +\frac {128 c^{2} d^{3}}{3}-48 c \,d^{4}-\frac {28 c^{3} d^{2}}{3}-10 i c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}-40 i c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-\frac {4 c^{5}}{15}-60 c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+280 c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+70 i d^{5} {\mathrm e}^{3 i \left (f x +e \right )}-\frac {740 c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{3}-20 c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+\frac {160 c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{3}+10 c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+60 c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-\frac {194 i d^{5} {\mathrm e}^{i \left (f x +e \right )}}{3}+\frac {4 i c^{5} {\mathrm e}^{i \left (f x +e \right )}}{3}-\frac {298 d^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{3}+\frac {8 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{3}+20 d^{5} {\mathrm e}^{4 i \left (f x +e \right )}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\)	\(554\)
norman	\(\text {Expression too large to display}\)	\(1411\)

Input:

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

Output:

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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (266) = 532\).

Time = 0.11 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.35 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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 - 3 
0*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*co 
s(f*x + e) - 4*a^3*f)*sin(f*x + e))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15553 vs. \(2 (264) = 528\).

Time = 27.57 (sec) , antiderivative size = 15553, normalized size of antiderivative = 55.95 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((-60*c**5*tan(e/2 + f*x/2)**8/(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*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 + 36 
0*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*t 
an(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*tan(e/2 + f*x/2)**8 + 360*a**3*f*tan(e/2 + f*x/2)...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1504 vs. \(2 (266) = 532\).

Time = 0.15 (sec) , antiderivative size = 1504, normalized size of antiderivative = 5.41 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/15*(d^5*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(co 
s(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/(cos(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*si 
n(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/(cos(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*sin(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)/(cos(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)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (266) = 532\).

Time = 0.17 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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*tan(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
 

Mupad [B] (verification not implemented)

Time = 18.32 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.35 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(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 + 32*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 - (265*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 
)))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.74 \[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*cos(e + f*x)*sin(e + f*x)**4*d**5 + 150*cos(e + f*x)*sin(e + f*x)**3*c 
*d**4 - 45*cos(e + f*x)*sin(e + f*x)**3*d**5 + 2*cos(e + f*x)*sin(e + f*x) 
**2*c**5 + 100*cos(e + f*x)*sin(e + f*x)**2*c**3*d**2 + 300*cos(e + f*x)*s 
in(e + f*x)**2*c**2*d**3*f*x - 320*cos(e + f*x)*sin(e + f*x)**2*c**2*d**3 
- 450*cos(e + f*x)*sin(e + f*x)**2*c*d**4*f*x + 630*cos(e + f*x)*sin(e + f 
*x)**2*c*d**4 + 195*cos(e + f*x)*sin(e + f*x)**2*d**5*f*x - 253*cos(e + f* 
x)*sin(e + f*x)**2*d**5 + 8*cos(e + f*x)*sin(e + f*x)*c**5 + 30*cos(e + f* 
x)*sin(e + f*x)*c**4*d + 40*cos(e + f*x)*sin(e + f*x)*c**3*d**2 + 600*cos( 
e + f*x)*sin(e + f*x)*c**2*d**3*f*x - 380*cos(e + f*x)*sin(e + f*x)*c**2*d 
**3 - 900*cos(e + f*x)*sin(e + f*x)*c*d**4*f*x + 630*cos(e + f*x)*sin(e + 
f*x)*c*d**4 + 390*cos(e + f*x)*sin(e + f*x)*d**5*f*x - 265*cos(e + f*x)*si 
n(e + f*x)*d**5 + 12*cos(e + f*x)*c**5 + 300*cos(e + f*x)*c**2*d**3*f*x - 
120*cos(e + f*x)*c**2*d**3 - 450*cos(e + f*x)*c*d**4*f*x + 180*cos(e + f*x 
)*c*d**4 + 195*cos(e + f*x)*d**5*f*x - 78*cos(e + f*x)*d**5 + 15*sin(e + f 
*x)**5*d**5 + 150*sin(e + f*x)**4*c*d**4 - 60*sin(e + f*x)**4*d**5 + 6*sin 
(e + f*x)**3*c**5 + 60*sin(e + f*x)**3*c**4*d + 180*sin(e + f*x)**3*c**3*d 
**2 - 300*sin(e + f*x)**3*c**2*d**3*f*x - 960*sin(e + f*x)**3*c**2*d**3 + 
450*sin(e + f*x)**3*c*d**4*f*x + 1560*sin(e + f*x)**3*c*d**4 - 195*sin(e + 
 f*x)**3*d**5*f*x - 660*sin(e + f*x)**3*d**5 + 14*sin(e + f*x)**2*c**5 + 1 
50*sin(e + f*x)**2*c**4*d + 100*sin(e + f*x)**2*c**3*d**2 - 900*sin(e +...