\(\int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx\) [478]

Optimal result

Integrand size = 25, antiderivative size = 354 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx=\frac {d^3 \left (40 c^3-90 c^2 d+78 c d^2-23 d^3\right ) x}{2 a^3}+\frac {2 d \left (2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right ) \cos (e+f x)}{15 a^3 f}+\frac {d^2 \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \cos (e+f x) \sin (e+f x)}{30 a^3 f}+\frac {d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}-\frac {(c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a+a \sin (e+f x))^3} \] Output:

1/2*d^3*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*x/a^3+2/15*d*(2*c^5+18*c^4*d+107 
*c^3*d^2-472*c^2*d^3+456*c*d^4-136*d^5)*cos(f*x+e)/a^3/f+1/30*d^2*(4*c^4+3 
6*c^3*d+216*c^2*d^2-626*c*d^3+345*d^4)*cos(f*x+e)*sin(f*x+e)/a^3/f+1/15*d* 
(2*c^3+18*c^2*d+111*c*d^2-136*d^3)*cos(f*x+e)*(c+d*sin(f*x+e))^2/a^3/f-1/1 
5*(c-d)*(2*c^2+18*c*d+115*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^3/f/(a^3+a^3*si 
n(f*x+e))-1/15*(c-d)*(2*c+13*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/a/f/(a+a*sin 
(f*x+e))^2-1/5*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^5/f/(a+a*sin(f*x+e))^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.60 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (48 (c-d)^6 \sin \left (\frac {1}{2} (e+f x)\right )-24 (c-d)^6 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+32 (c-d)^5 (c+14 d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-16 (c-d)^5 (c+14 d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+16 (c-d)^4 \left (2 c^2+26 c d+197 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-60 d^3 \left (-40 c^3+90 c^2 d-78 c d^2+23 d^3\right ) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+10 d^6 \cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-45 d^4 \left (20 c^2-24 c d+9 d^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (e+f x)-i \sin (e+f x))-45 d^4 \left (20 c^2-24 c d+9 d^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (e+f x)+i \sin (e+f x))-45 i (2 c-d) d^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (2 (e+f x))-i \sin (2 (e+f x)))+45 i (2 c-d) d^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (\cos (2 (e+f x))+i \sin (2 (e+f x)))\right )}{120 a^3 f (1+\sin (e+f x))^3} \] Input:

Integrate[(c + d*Sin[e + f*x])^6/(a + a*Sin[e + f*x])^3,x]
 

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*(c - d)^6*Sin[(e + f*x)/2] - 24 
*(c - d)^6*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 32*(c - d)^5*(c + 14*d) 
*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 16*(c - d)^5*( 
c + 14*d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 16*(c - d)^4*(2*c^2 + 
26*c*d + 197*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 
 - 60*d^3*(-40*c^3 + 90*c^2*d - 78*c*d^2 + 23*d^3)*(e + f*x)*(Cos[(e + f*x 
)/2] + Sin[(e + f*x)/2])^5 + 10*d^6*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + S 
in[(e + f*x)/2])^5 - 45*d^4*(20*c^2 - 24*c*d + 9*d^2)*(Cos[(e + f*x)/2] + 
Sin[(e + f*x)/2])^5*(Cos[e + f*x] - I*Sin[e + f*x]) - 45*d^4*(20*c^2 - 24* 
c*d + 9*d^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(Cos[e + f*x] + I*Sin 
[e + f*x]) - (45*I)*(2*c - d)*d^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5* 
(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]) + (45*I)*(2*c - d)*d^5*(Cos[(e + f 
*x)/2] + Sin[(e + f*x)/2])^5*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])))/(12 
0*a^3*f*(1 + Sin[e + f*x])^3)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3244, 25, 3042, 3456, 3042, 3456, 27, 3042, 3232, 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.

\(\displaystyle \int \frac {(c+d \sin (e+f x))^6}{(a \sin (e+f x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d \sin (e+f x))^6}{(a \sin (e+f x)+a)^3}dx\)

\(\Big \downarrow \) 3244

\(\displaystyle -\frac {\int -\frac {(c+d \sin (e+f x))^4 \left (a \left (2 c^2+8 d c-5 d^2\right )-a (3 c-8 d) d \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^4 \left (a \left (2 c^2+8 d c-5 d^2\right )-a (3 c-8 d) d \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^4 \left (a \left (2 c^2+8 d c-5 d^2\right )-a (3 c-8 d) d \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^3 \left (a^2 \left (2 c^3+10 d c^2+55 d^2 c-52 d^3\right )-3 a^2 d \left (2 c^2+14 d c-21 d^2\right ) \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x))^3 \left (a^2 \left (2 c^3+10 d c^2+55 d^2 c-52 d^3\right )-3 a^2 d \left (2 c^2+14 d c-21 d^2\right ) \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\frac {\int 3 (c+d \sin (e+f x))^2 \left (a^3 d^2 \left (2 c^2+118 d c-115 d^2\right )-a^3 d \left (2 c^3+18 d c^2+111 d^2 c-136 d^3\right ) \sin (e+f x)\right )dx}{a^2}-\frac {a^2 (c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int (c+d \sin (e+f x))^2 \left (a^3 d^2 \left (2 c^2+118 d c-115 d^2\right )-a^3 d \left (2 c^3+18 d c^2+111 d^2 c-136 d^3\right ) \sin (e+f x)\right )dx}{a^2}-\frac {a^2 (c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \int (c+d \sin (e+f x))^2 \left (a^3 d^2 \left (2 c^2+118 d c-115 d^2\right )-a^3 d \left (2 c^3+18 d c^2+111 d^2 c-136 d^3\right ) \sin (e+f x)\right )dx}{a^2}-\frac {a^2 (c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^3 d^2 \left (2 c^3+318 d c^2-567 d^2 c+272 d^3\right )-a^3 d \left (4 c^4+36 d c^3+216 d^2 c^2-626 d^3 c+345 d^4\right ) \sin (e+f x)\right )dx+\frac {a^3 d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )}{a^2}-\frac {a^2 (c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^3 d^2 \left (2 c^3+318 d c^2-567 d^2 c+272 d^3\right )-a^3 d \left (4 c^4+36 d c^3+216 d^2 c^2-626 d^3 c+345 d^4\right ) \sin (e+f x)\right )dx+\frac {a^3 d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )}{a^2}-\frac {a^2 (c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {a^3 d \left (2 c^3+18 c^2 d+111 c d^2-136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}+\frac {1}{3} \left (\frac {15}{2} a^3 d^3 x \left (40 c^3-90 c^2 d+78 c d^2-23 d^3\right )+\frac {a^3 d^2 \left (4 c^4+36 c^3 d+216 c^2 d^2-626 c d^3+345 d^4\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {2 a^3 d \left (2 c^5+18 c^4 d+107 c^3 d^2-472 c^2 d^3+456 c d^4-136 d^5\right ) \cos (e+f x)}{f}\right )\right )}{a^2}-\frac {a^2 (c-d) \left (2 c^2+18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {a (c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}\)

Input:

Int[(c + d*Sin[e + f*x])^6/(a + a*Sin[e + f*x])^3,x]
 

Output:

-1/5*((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(f*(a + a*Sin[e + f*x]) 
^3) + (-1/3*(a*(c - d)*(2*c + 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/( 
f*(a + a*Sin[e + f*x])^2) + (-((a^2*(c - d)*(2*c^2 + 18*c*d + 115*d^2)*Cos 
[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))) + (3*((a^3*d*( 
2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^ 
2)/(3*f) + ((15*a^3*d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/2 + (2* 
a^3*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5) 
*Cos[e + f*x])/f + (a^3*d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 
345*d^4)*Cos[e + f*x]*Sin[e + f*x])/(2*f))/3))/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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 3232

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

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 = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.32

\[\frac {-\frac {2 \left (c^{6}-20 c^{3} d^{3}+45 c^{2} d^{4}-36 c \,d^{5}+10 d^{6}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{6}+12 c^{5} d -40 c^{3} d^{3}+60 c^{2} d^{4}-36 c \,d^{5}+8 d^{6}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (8 c^{6}-36 c^{5} d +60 c^{4} d^{2}-40 c^{3} d^{3}+12 c \,d^{5}-4 d^{6}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-8 c^{6}+48 c^{5} d -120 c^{4} d^{2}+160 c^{3} d^{3}-120 c^{2} d^{4}+48 c \,d^{5}-8 d^{6}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{6}-24 c^{5} d +60 c^{4} d^{2}-80 c^{3} d^{3}+60 c^{2} d^{4}-24 c \,d^{5}+4 d^{6}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+2 d^{3} \left (\frac {\left (3 c \,d^{2}-\frac {3}{2} d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (-15 c^{2} d +18 c \,d^{2}-6 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-30 c^{2} d +36 c \,d^{2}-14 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-3 c \,d^{2}+\frac {3}{2} d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-15 c^{2} d +18 c \,d^{2}-\frac {20 d^{3}}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}+\frac {\left (40 c^{3}-90 c^{2} d +78 c \,d^{2}-23 d^{3}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{a^{3} f}\]

Input:

int((c+d*sin(f*x+e))^6/(a+sin(f*x+e)*a)^3,x)
 

Output:

2/f/a^3*(-(c^6-20*c^3*d^3+45*c^2*d^4-36*c*d^5+10*d^6)/(tan(1/2*f*x+1/2*e)+ 
1)-1/2*(-4*c^6+12*c^5*d-40*c^3*d^3+60*c^2*d^4-36*c*d^5+8*d^6)/(tan(1/2*f*x 
+1/2*e)+1)^2-1/3*(8*c^6-36*c^5*d+60*c^4*d^2-40*c^3*d^3+12*c*d^5-4*d^6)/(ta 
n(1/2*f*x+1/2*e)+1)^3-1/4*(-8*c^6+48*c^5*d-120*c^4*d^2+160*c^3*d^3-120*c^2 
*d^4+48*c*d^5-8*d^6)/(tan(1/2*f*x+1/2*e)+1)^4-1/5*(4*c^6-24*c^5*d+60*c^4*d 
^2-80*c^3*d^3+60*c^2*d^4-24*c*d^5+4*d^6)/(tan(1/2*f*x+1/2*e)+1)^5+d^3*(((3 
*c*d^2-3/2*d^3)*tan(1/2*f*x+1/2*e)^5+(-15*c^2*d+18*c*d^2-6*d^3)*tan(1/2*f* 
x+1/2*e)^4+(-30*c^2*d+36*c*d^2-14*d^3)*tan(1/2*f*x+1/2*e)^2+(-3*c*d^2+3/2* 
d^3)*tan(1/2*f*x+1/2*e)-15*c^2*d+18*c*d^2-20/3*d^3)/(1+tan(1/2*f*x+1/2*e)^ 
2)^3+1/2*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*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. 823 vs. \(2 (340) = 680\).

Time = 0.13 (sec) , antiderivative size = 823, normalized size of antiderivative = 2.32 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))^6/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
 

Output:

1/30*(10*d^6*cos(f*x + e)^6 + 6*c^6 - 36*c^5*d + 90*c^4*d^2 - 120*c^3*d^3 
+ 90*c^2*d^4 - 36*c*d^5 + 6*d^6 + 15*(6*c*d^5 - d^6)*cos(f*x + e)^5 - 10*( 
45*c^2*d^4 - 36*c*d^5 + 14*d^6)*cos(f*x + e)^4 - (4*c^6 + 36*c^5*d + 210*c 
^4*d^2 - 1280*c^3*d^3 + 3510*c^2*d^4 - 2694*c*d^5 + 839*d^6 - 15*(40*c^3*d 
^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e)^3 - 60*(40*c^3*d^3 
- 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (8*c^6 + 72*c^5*d - 30*c^4*d^2 - 7 
60*c^3*d^3 + 2520*c^2*d^4 - 2148*c*d^5 + 668*d^6 + 45*(40*c^3*d^3 - 90*c^2 
*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e)^2 + 6*(3*c^6 + 12*c^5*d + 45*c 
^4*d^2 - 360*c^3*d^3 + 945*c^2*d^4 - 768*c*d^5 + 233*d^6 - 5*(40*c^3*d^3 - 
 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e) + (10*d^6*cos(f*x + e)^ 
5 - 6*c^6 + 36*c^5*d - 90*c^4*d^2 + 120*c^3*d^3 - 90*c^2*d^4 + 36*c*d^5 - 
6*d^6 - 5*(18*c*d^5 - 5*d^6)*cos(f*x + e)^4 - 5*(90*c^2*d^4 - 54*c*d^5 + 2 
3*d^6)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f 
*x + (4*c^6 + 36*c^5*d + 210*c^4*d^2 - 1280*c^3*d^3 + 3060*c^2*d^4 - 2424* 
c*d^5 + 724*d^6 + 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*co 
s(f*x + e)^2 + 6*(2*c^6 + 18*c^5*d + 30*c^4*d^2 - 340*c^3*d^3 + 930*c^2*d^ 
4 - 762*c*d^5 + 232*d^6 - 5*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)* 
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*c 
os(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. 28065 vs. \(2 (340) = 680\).

Time = 45.20 (sec) , antiderivative size = 28065, normalized size of antiderivative = 79.28 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))**6/(a+a*sin(f*x+e))**3,x)
 

Output:

Piecewise((-60*c**6*tan(e/2 + f*x/2)**10/(30*a**3*f*tan(e/2 + f*x/2)**11 + 
 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a* 
*3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*t 
an(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 
 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*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**6*tan(e/2 + f*x/2 
)**9/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 3 
90*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3 
*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan 
(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + 
f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 
 30*a**3*f) - 340*c**6*tan(e/2 + f*x/2)**8/(30*a**3*f*tan(e/2 + f*x/2)**11 
 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750* 
a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f 
*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e 
/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x 
/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 440*c**6*tan(e/2 + f*x 
/2)**7/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 
 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a* 
*3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1993 vs. \(2 (340) = 680\).

Time = 0.16 (sec) , antiderivative size = 1993, normalized size of antiderivative = 5.63 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))^6/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
 

Output:

-1/15*(d^6*((2375*sin(f*x + e)/(cos(f*x + e) + 1) + 5347*sin(f*x + e)^2/(c 
os(f*x + e) + 1)^2 + 9230*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12622*sin( 
f*x + e)^4/(cos(f*x + e) + 1)^4 + 13340*sin(f*x + e)^5/(cos(f*x + e) + 1)^ 
5 + 11684*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 8050*sin(f*x + e)^7/(cos(f 
*x + e) + 1)^7 + 4370*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1725*sin(f*x + 
 e)^9/(cos(f*x + e) + 1)^9 + 345*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 5 
44)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 13*a^3*sin(f*x + e)^2/( 
cos(f*x + e) + 1)^2 + 25*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 38*a^3* 
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 46*a^3*sin(f*x + e)^5/(cos(f*x + e) 
+ 1)^5 + 46*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 38*a^3*sin(f*x + e)^ 
7/(cos(f*x + e) + 1)^7 + 25*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 13*a 
^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5*a^3*sin(f*x + e)^10/(cos(f*x + 
e) + 1)^10 + a^3*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 345*arctan(sin(f 
*x + e)/(cos(f*x + e) + 1))/a^3) - 6*c*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/(cos(f*x + 
 e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*si 
n(f*x + e)^2/(cos(f*x + e) + 1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (340) = 680\).

Time = 0.17 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))^6/(a+a*sin(f*x+e))^3,x, algorithm="giac")
 

Output:

1/30*(15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*(f*x + e)/a^3 + 10* 
(18*c*d^5*tan(1/2*f*x + 1/2*e)^5 - 9*d^6*tan(1/2*f*x + 1/2*e)^5 - 90*c^2*d 
^4*tan(1/2*f*x + 1/2*e)^4 + 108*c*d^5*tan(1/2*f*x + 1/2*e)^4 - 36*d^6*tan( 
1/2*f*x + 1/2*e)^4 - 180*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 216*c*d^5*tan(1/ 
2*f*x + 1/2*e)^2 - 84*d^6*tan(1/2*f*x + 1/2*e)^2 - 18*c*d^5*tan(1/2*f*x + 
1/2*e) + 9*d^6*tan(1/2*f*x + 1/2*e) - 90*c^2*d^4 + 108*c*d^5 - 40*d^6)/((t 
an(1/2*f*x + 1/2*e)^2 + 1)^3*a^3) - 4*(15*c^6*tan(1/2*f*x + 1/2*e)^4 - 300 
*c^3*d^3*tan(1/2*f*x + 1/2*e)^4 + 675*c^2*d^4*tan(1/2*f*x + 1/2*e)^4 - 540 
*c*d^5*tan(1/2*f*x + 1/2*e)^4 + 150*d^6*tan(1/2*f*x + 1/2*e)^4 + 30*c^6*ta 
n(1/2*f*x + 1/2*e)^3 + 90*c^5*d*tan(1/2*f*x + 1/2*e)^3 - 1500*c^3*d^3*tan( 
1/2*f*x + 1/2*e)^3 + 3150*c^2*d^4*tan(1/2*f*x + 1/2*e)^3 - 2430*c*d^5*tan( 
1/2*f*x + 1/2*e)^3 + 660*d^6*tan(1/2*f*x + 1/2*e)^3 + 40*c^6*tan(1/2*f*x + 
 1/2*e)^2 + 90*c^5*d*tan(1/2*f*x + 1/2*e)^2 + 300*c^4*d^2*tan(1/2*f*x + 1/ 
2*e)^2 - 2900*c^3*d^3*tan(1/2*f*x + 1/2*e)^2 + 5400*c^2*d^4*tan(1/2*f*x + 
1/2*e)^2 - 3990*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 1060*d^6*tan(1/2*f*x + 1/2* 
e)^2 + 20*c^6*tan(1/2*f*x + 1/2*e) + 90*c^5*d*tan(1/2*f*x + 1/2*e) + 150*c 
^4*d^2*tan(1/2*f*x + 1/2*e) - 1900*c^3*d^3*tan(1/2*f*x + 1/2*e) + 3600*c^2 
*d^4*tan(1/2*f*x + 1/2*e) - 2670*c*d^5*tan(1/2*f*x + 1/2*e) + 710*d^6*tan( 
1/2*f*x + 1/2*e) + 7*c^6 + 18*c^5*d + 30*c^4*d^2 - 440*c^3*d^3 + 855*c^2*d 
^4 - 642*c*d^5 + 172*d^6)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f
 

Mupad [B] (verification not implemented)

Time = 18.50 (sec) , antiderivative size = 898, normalized size of antiderivative = 2.54 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

int((c + d*sin(e + f*x))^6/(a + a*sin(e + f*x))^3,x)
 

Output:

(d^3*atan((d^3*tan(e/2 + (f*x)/2)*(78*c*d^2 - 90*c^2*d + 40*c^3 - 23*d^3)) 
/(78*c*d^5 - 23*d^6 - 90*c^2*d^4 + 40*c^3*d^3))*(78*c*d^2 - 90*c^2*d + 40* 
c^3 - 23*d^3))/(a^3*f) - (tan(e/2 + (f*x)/2)^9*(12*c^5*d - 390*c*d^5 + 4*c 
^6 + 115*d^6 + 450*c^2*d^4 - 200*c^3*d^3) - (608*c*d^5)/5 + (12*c^5*d)/5 + 
 tan(e/2 + (f*x)/2)^10*(2*c^6 - 78*c*d^5 + 23*d^6 + 90*c^2*d^4 - 40*c^3*d^ 
3) + tan(e/2 + (f*x)/2)*(12*c^5*d - 530*c*d^5 + (8*c^6)/3 + (475*d^6)/3 + 
630*c^2*d^4 - (760*c^3*d^3)/3 + 20*c^4*d^2) + (14*c^6)/15 + (544*d^6)/15 + 
 144*c^2*d^4 - (176*c^3*d^3)/3 + 4*c^4*d^2 + tan(e/2 + (f*x)/2)^8*(12*c^5* 
d - 988*c*d^5 + (34*c^6)/3 + (874*d^6)/3 + 1140*c^2*d^4 - (1520*c^3*d^3)/3 
 + 40*c^4*d^2) + tan(e/2 + (f*x)/2)^3*(48*c^5*d - 2052*c*d^5 + 12*c^6 + (1 
846*d^6)/3 + 2460*c^2*d^4 - 960*c^3*d^3 + 60*c^4*d^2) + tan(e/2 + (f*x)/2) 
^7*(48*c^5*d - 1820*c*d^5 + (44*c^6)/3 + (1610*d^6)/3 + 2100*c^2*d^4 - (25 
60*c^3*d^3)/3 + 20*c^4*d^2) + tan(e/2 + (f*x)/2)^5*(72*c^5*d - 2952*c*d^5 
+ 20*c^6 + (2668*d^6)/3 + 3480*c^2*d^4 - 1360*c^3*d^3 + 60*c^4*d^2) + tan( 
e/2 + (f*x)/2)^2*((96*c^5*d)/5 - (5954*c*d^5)/5 + (122*c^6)/15 + (5347*d^6 
)/15 + 1422*c^2*d^4 - (1688*c^3*d^3)/3 + 52*c^4*d^2) + tan(e/2 + (f*x)/2)^ 
4*((216*c^5*d)/5 - (14004*c*d^5)/5 + (104*c^6)/5 + (12622*d^6)/15 + 3372*c 
^2*d^4 - 1376*c^3*d^3 + 132*c^4*d^2) + tan(e/2 + (f*x)/2)^6*((192*c^5*d)/5 
 - (13208*c*d^5)/5 + (344*c^6)/15 + (11684*d^6)/15 + 3144*c^2*d^4 - (4016* 
c^3*d^3)/3 + 124*c^4*d^2))/(f*(13*a^3*tan(e/2 + (f*x)/2)^2 + 25*a^3*tan...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1379, normalized size of antiderivative = 3.90 \[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

int((c+d*sin(f*x+e))^6/(a+a*sin(f*x+e))^3,x)
 

Output:

(10*cos(e + f*x)*sin(e + f*x)**5*d**6 + 90*cos(e + f*x)*sin(e + f*x)**4*c* 
d**5 - 15*cos(e + f*x)*sin(e + f*x)**4*d**6 + 450*cos(e + f*x)*sin(e + f*x 
)**3*c**2*d**4 - 270*cos(e + f*x)*sin(e + f*x)**3*c*d**5 + 95*cos(e + f*x) 
*sin(e + f*x)**3*d**6 + 2*cos(e + f*x)*sin(e + f*x)**2*c**6 + 150*cos(e + 
f*x)*sin(e + f*x)**2*c**4*d**2 + 600*cos(e + f*x)*sin(e + f*x)**2*c**3*d** 
3*f*x - 640*cos(e + f*x)*sin(e + f*x)**2*c**3*d**3 - 1350*cos(e + f*x)*sin 
(e + f*x)**2*c**2*d**4*f*x + 1890*cos(e + f*x)*sin(e + f*x)**2*c**2*d**4 + 
 1170*cos(e + f*x)*sin(e + f*x)**2*c*d**5*f*x - 1518*cos(e + f*x)*sin(e + 
f*x)**2*c*d**5 - 345*cos(e + f*x)*sin(e + f*x)**2*d**6*f*x + 463*cos(e + f 
*x)*sin(e + f*x)**2*d**6 + 8*cos(e + f*x)*sin(e + f*x)*c**6 + 36*cos(e + f 
*x)*sin(e + f*x)*c**5*d + 60*cos(e + f*x)*sin(e + f*x)*c**4*d**2 + 1200*co 
s(e + f*x)*sin(e + f*x)*c**3*d**3*f*x - 760*cos(e + f*x)*sin(e + f*x)*c**3 
*d**3 - 2700*cos(e + f*x)*sin(e + f*x)*c**2*d**4*f*x + 1890*cos(e + f*x)*s 
in(e + f*x)*c**2*d**4 + 2340*cos(e + f*x)*sin(e + f*x)*c*d**5*f*x - 1590*c 
os(e + f*x)*sin(e + f*x)*c*d**5 - 690*cos(e + f*x)*sin(e + f*x)*d**6*f*x + 
 475*cos(e + f*x)*sin(e + f*x)*d**6 + 12*cos(e + f*x)*c**6 + 600*cos(e + f 
*x)*c**3*d**3*f*x - 240*cos(e + f*x)*c**3*d**3 - 1350*cos(e + f*x)*c**2*d* 
*4*f*x + 540*cos(e + f*x)*c**2*d**4 + 1170*cos(e + f*x)*c*d**5*f*x - 468*c 
os(e + f*x)*c*d**5 - 345*cos(e + f*x)*d**6*f*x + 138*cos(e + f*x)*d**6 + 1 
0*sin(e + f*x)**6*d**6 + 90*sin(e + f*x)**5*c*d**5 - 25*sin(e + f*x)**5...