3.3.36 \(\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5 \, dx\) [236]

Optimal. Leaf size=152 \[ \frac {9}{16} a^2 c^5 x+\frac {3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac {9 a^2 c^5 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {3 a^2 c^5 \cos ^3(e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f} \]

[Out]

9/16*a^2*c^5*x+3/10*a^2*c^5*cos(f*x+e)^5/f+9/16*a^2*c^5*cos(f*x+e)*sin(f*x+e)/f+3/8*a^2*c^5*cos(f*x+e)^3*sin(f
*x+e)/f+1/7*a^2*c^3*cos(f*x+e)^5*(c-c*sin(f*x+e))^2/f+3/14*a^2*cos(f*x+e)^5*(c^5-c^5*sin(f*x+e))/f

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Rubi [A]
time = 0.14, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2757, 2748, 2715, 8} \begin {gather*} \frac {3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f}+\frac {3 a^2 c^5 \sin (e+f x) \cos ^3(e+f x)}{8 f}+\frac {9 a^2 c^5 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {9}{16} a^2 c^5 x+\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5,x]

[Out]

(9*a^2*c^5*x)/16 + (3*a^2*c^5*Cos[e + f*x]^5)/(10*f) + (9*a^2*c^5*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (3*a^2*c
^5*Cos[e + f*x]^3*Sin[e + f*x])/(8*f) + (a^2*c^3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(7*f) + (3*a^2*Cos[e +
f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(14*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2757

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(
g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[a*((2*m + p - 1)/(m + p)), Int
[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2,
0] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2815

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {1}{7} \left (9 a^2 c^3\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f}+\frac {1}{2} \left (3 a^2 c^4\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f}+\frac {1}{2} \left (3 a^2 c^5\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac {3 a^2 c^5 \cos ^3(e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f}+\frac {1}{8} \left (9 a^2 c^5\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac {9 a^2 c^5 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {3 a^2 c^5 \cos ^3(e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f}+\frac {1}{16} \left (9 a^2 c^5\right ) \int 1 \, dx\\ &=\frac {9}{16} a^2 c^5 x+\frac {3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac {9 a^2 c^5 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {3 a^2 c^5 \cos ^3(e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac {3 a^2 \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{14 f}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 89, normalized size = 0.59 \begin {gather*} \frac {a^2 c^5 (1260 e+1260 f x+945 \cos (e+f x)+455 \cos (3 (e+f x))+77 \cos (5 (e+f x))-5 \cos (7 (e+f x))+665 \sin (2 (e+f x))-35 \sin (4 (e+f x))-35 \sin (6 (e+f x)))}{2240 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5,x]

[Out]

(a^2*c^5*(1260*e + 1260*f*x + 945*Cos[e + f*x] + 455*Cos[3*(e + f*x)] + 77*Cos[5*(e + f*x)] - 5*Cos[7*(e + f*x
)] + 665*Sin[2*(e + f*x)] - 35*Sin[4*(e + f*x)] - 35*Sin[6*(e + f*x)]))/(2240*f)

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Maple [A]
time = 0.46, size = 255, normalized size = 1.68

method	result	size
risch	\(\frac {9 a^{2} c^{5} x}{16}+\frac {27 c^{5} a^{2} \cos \left (f x +e \right )}{64 f}-\frac {c^{5} a^{2} \cos \left (7 f x +7 e \right )}{448 f}-\frac {c^{5} a^{2} \sin \left (6 f x +6 e \right )}{64 f}+\frac {11 c^{5} a^{2} \cos \left (5 f x +5 e \right )}{320 f}-\frac {c^{5} a^{2} \sin \left (4 f x +4 e \right )}{64 f}+\frac {13 c^{5} a^{2} \cos \left (3 f x +3 e \right )}{64 f}+\frac {19 c^{5} a^{2} \sin \left (2 f x +2 e \right )}{64 f}\)	\(148\)
derivativedivides	\(\frac {\frac {c^{5} a^{2} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}+3 c^{5} a^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {c^{5} a^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-5 c^{5} a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {5 c^{5} a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+c^{5} a^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 c^{5} a^{2} \cos \left (f x +e \right )+c^{5} a^{2} \left (f x +e \right )}{f}\)	\(255\)
default	\(\frac {\frac {c^{5} a^{2} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}+3 c^{5} a^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {c^{5} a^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-5 c^{5} a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {5 c^{5} a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+c^{5} a^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 c^{5} a^{2} \cos \left (f x +e \right )+c^{5} a^{2} \left (f x +e \right )}{f}\)	\(255\)
norman	\(\frac {\frac {6 c^{5} a^{2} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {16 c^{5} a^{2} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {32 c^{5} a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {46 c^{5} a^{2}}{35 f}+\frac {9 a^{2} c^{5} x}{16}+\frac {14 c^{5} a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {16 c^{5} a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}+\frac {58 c^{5} a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}+\frac {63 a^{2} c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {189 a^{2} c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {315 a^{2} c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {315 a^{2} c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {189 a^{2} c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {63 a^{2} c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {9 a^{2} c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {7 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {17 c^{5} a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {13 c^{5} a^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {13 c^{5} a^{2} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {17 c^{5} a^{2} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {7 c^{5} a^{2} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{7}}\)	\(440\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

1/f*(1/7*c^5*a^2*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)+3*c^5*a^2*(-1/6*(sin(f*x+e)^
5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+1/5*c^5*a^2*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2
)*cos(f*x+e)-5*c^5*a^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-5/3*c^5*a^2*(2+sin(f*x+e)
^2)*cos(f*x+e)+c^5*a^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)+3*c^5*a^2*cos(f*x+e)+c^5*a^2*(f*x+e))

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Maxima [A]
time = 0.29, size = 276, normalized size = 1.82 \begin {gather*} -\frac {192 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} a^{2} c^{5} - 448 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c^{5} - 11200 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{5} - 105 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{5} + 1050 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{5} - 1680 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{5} - 6720 \, {\left (f x + e\right )} a^{2} c^{5} - 20160 \, a^{2} c^{5} \cos \left (f x + e\right )}{6720 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-1/6720*(192*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*a^2*c^5 - 448*(3*cos
(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*c^5 - 11200*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^2*c^5 -
105*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^2*c^5 + 1050*(12*f*x
+ 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c^5 - 1680*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^5 - 6720
*(f*x + e)*a^2*c^5 - 20160*a^2*c^5*cos(f*x + e))/f

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Fricas [A]
time = 0.33, size = 109, normalized size = 0.72 \begin {gather*} -\frac {80 \, a^{2} c^{5} \cos \left (f x + e\right )^{7} - 448 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 315 \, a^{2} c^{5} f x + 35 \, {\left (8 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 6 \, a^{2} c^{5} \cos \left (f x + e\right )^{3} - 9 \, a^{2} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{560 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/560*(80*a^2*c^5*cos(f*x + e)^7 - 448*a^2*c^5*cos(f*x + e)^5 - 315*a^2*c^5*f*x + 35*(8*a^2*c^5*cos(f*x + e)^
5 - 6*a^2*c^5*cos(f*x + e)^3 - 9*a^2*c^5*cos(f*x + e))*sin(f*x + e))/f

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (144) = 288\).
time = 0.77, size = 629, normalized size = 4.14 \begin {gather*} \begin {cases} \frac {15 a^{2} c^{5} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {45 a^{2} c^{5} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} - \frac {15 a^{2} c^{5} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {45 a^{2} c^{5} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} - \frac {15 a^{2} c^{5} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{2} c^{5} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {15 a^{2} c^{5} x \cos ^{6}{\left (e + f x \right )}}{16} - \frac {15 a^{2} c^{5} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a^{2} c^{5} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{5} x + \frac {a^{2} c^{5} \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {33 a^{2} c^{5} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} + \frac {2 a^{2} c^{5} \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {a^{2} c^{5} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{2} c^{5} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{2 f} + \frac {25 a^{2} c^{5} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {8 a^{2} c^{5} \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {4 a^{2} c^{5} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {5 a^{2} c^{5} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {15 a^{2} c^{5} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} + \frac {15 a^{2} c^{5} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a^{2} c^{5} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {16 a^{2} c^{5} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {8 a^{2} c^{5} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {10 a^{2} c^{5} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 a^{2} c^{5} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right )^{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**2*(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((15*a**2*c**5*x*sin(e + f*x)**6/16 + 45*a**2*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/16 - 15*a**2*c**
5*x*sin(e + f*x)**4/8 + 45*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/16 - 15*a**2*c**5*x*sin(e + f*x)**2*cos
(e + f*x)**2/4 + a**2*c**5*x*sin(e + f*x)**2/2 + 15*a**2*c**5*x*cos(e + f*x)**6/16 - 15*a**2*c**5*x*cos(e + f*
x)**4/8 + a**2*c**5*x*cos(e + f*x)**2/2 + a**2*c**5*x + a**2*c**5*sin(e + f*x)**6*cos(e + f*x)/f - 33*a**2*c**
5*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 2*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f + a**2*c**5*sin(e + f*x)
**4*cos(e + f*x)/f - 5*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) + 25*a**2*c**5*sin(e + f*x)**3*cos(e +
f*x)/(8*f) + 8*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**3/(
3*f) - 5*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)/f - 15*a**2*c**5*sin(e + f*x)*cos(e + f*x)**5/(16*f) + 15*a**2
*c**5*sin(e + f*x)*cos(e + f*x)**3/(8*f) - a**2*c**5*sin(e + f*x)*cos(e + f*x)/(2*f) + 16*a**2*c**5*cos(e + f*
x)**7/(35*f) + 8*a**2*c**5*cos(e + f*x)**5/(15*f) - 10*a**2*c**5*cos(e + f*x)**3/(3*f) + 3*a**2*c**5*cos(e + f
*x)/f, Ne(f, 0)), (x*(a*sin(e) + a)**2*(-c*sin(e) + c)**5, True))

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Giac [A]
time = 0.50, size = 154, normalized size = 1.01 \begin {gather*} \frac {9}{16} \, a^{2} c^{5} x - \frac {a^{2} c^{5} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {11 \, a^{2} c^{5} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {13 \, a^{2} c^{5} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {27 \, a^{2} c^{5} \cos \left (f x + e\right )}{64 \, f} - \frac {a^{2} c^{5} \sin \left (6 \, f x + 6 \, e\right )}{64 \, f} - \frac {a^{2} c^{5} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {19 \, a^{2} c^{5} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

9/16*a^2*c^5*x - 1/448*a^2*c^5*cos(7*f*x + 7*e)/f + 11/320*a^2*c^5*cos(5*f*x + 5*e)/f + 13/64*a^2*c^5*cos(3*f*
x + 3*e)/f + 27/64*a^2*c^5*cos(f*x + e)/f - 1/64*a^2*c^5*sin(6*f*x + 6*e)/f - 1/64*a^2*c^5*sin(4*f*x + 4*e)/f
+ 19/64*a^2*c^5*sin(2*f*x + 2*e)/f

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Mupad [B]
time = 9.23, size = 452, normalized size = 2.97 \begin {gather*} \frac {9\,a^2\,c^5\,x}{16}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{80}-\frac {a^2\,c^5\,\left (2205\,e+2205\,f\,x+1792\right )}{560}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (\frac {a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{80}-\frac {a^2\,c^5\,\left (2205\,e+2205\,f\,x+3360\right )}{560}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {3\,a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{80}-\frac {a^2\,c^5\,\left (6615\,e+6615\,f\,x+6496\right )}{560}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (\frac {3\,a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{80}-\frac {a^2\,c^5\,\left (6615\,e+6615\,f\,x+8960\right )}{560}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{16}-\frac {a^2\,c^5\,\left (11025\,e+11025\,f\,x+7840\right )}{560}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{16}-\frac {a^2\,c^5\,\left (11025\,e+11025\,f\,x+17920\right )}{560}\right )-\frac {17\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{2}+\frac {13\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{8}-\frac {13\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{8}+\frac {17\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{2}+\frac {7\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{8}+\frac {a^2\,c^5\,\left (315\,e+315\,f\,x\right )}{560}-\frac {a^2\,c^5\,\left (315\,e+315\,f\,x+736\right )}{560}-\frac {7\,a^2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5,x)

[Out]

(9*a^2*c^5*x)/16 - (tan(e/2 + (f*x)/2)^2*((a^2*c^5*(315*e + 315*f*x))/80 - (a^2*c^5*(2205*e + 2205*f*x + 1792)
)/560) + tan(e/2 + (f*x)/2)^12*((a^2*c^5*(315*e + 315*f*x))/80 - (a^2*c^5*(2205*e + 2205*f*x + 3360))/560) + t
an(e/2 + (f*x)/2)^4*((3*a^2*c^5*(315*e + 315*f*x))/80 - (a^2*c^5*(6615*e + 6615*f*x + 6496))/560) + tan(e/2 +
(f*x)/2)^10*((3*a^2*c^5*(315*e + 315*f*x))/80 - (a^2*c^5*(6615*e + 6615*f*x + 8960))/560) + tan(e/2 + (f*x)/2)
^8*((a^2*c^5*(315*e + 315*f*x))/16 - (a^2*c^5*(11025*e + 11025*f*x + 7840))/560) + tan(e/2 + (f*x)/2)^6*((a^2*
c^5*(315*e + 315*f*x))/16 - (a^2*c^5*(11025*e + 11025*f*x + 17920))/560) - (17*a^2*c^5*tan(e/2 + (f*x)/2)^3)/2
+ (13*a^2*c^5*tan(e/2 + (f*x)/2)^5)/8 - (13*a^2*c^5*tan(e/2 + (f*x)/2)^9)/8 + (17*a^2*c^5*tan(e/2 + (f*x)/2)^
11)/2 + (7*a^2*c^5*tan(e/2 + (f*x)/2)^13)/8 + (a^2*c^5*(315*e + 315*f*x))/560 - (a^2*c^5*(315*e + 315*f*x + 73
6))/560 - (7*a^2*c^5*tan(e/2 + (f*x)/2))/8)/(f*(tan(e/2 + (f*x)/2)^2 + 1)^7)

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