Integrand size = 28, antiderivative size = 145 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \] Output:
256/3003*a^3*c^7*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+64/429*a^3*c^6*cos( f*x+e)^7/f/(c-c*sin(f*x+e))^(5/2)+24/143*a^3*c^5*cos(f*x+e)^7/f/(c-c*sin(f *x+e))^(3/2)+2/13*a^3*c^4*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(1/2)
Time = 13.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\frac {a^3 c^3 \cos ^6(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (5230-1890 \cos (2 (e+f x))-6377 \sin (e+f x)+231 \sin (3 (e+f x)))}{6006 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \] Input:
Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2),x]
Output:
(a^3*c^3*Cos[e + f*x]^6*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*S in[e + f*x]]*(5230 - 1890*Cos[2*(e + f*x)] - 6377*Sin[e + f*x] + 231*Sin[3 *(e + f*x)]))/(6006*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)
Time = 0.81 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3215, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3152}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{7/2}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \cos ^6(e+f x) \sqrt {c-c \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \cos (e+f x)^6 \sqrt {c-c \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \int \frac {\cos ^6(e+f x)}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \int \frac {\cos (e+f x)^6}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \left (\frac {8}{11} c \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \left (\frac {8}{11} c \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{3/2}}dx+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \left (\frac {8}{11} c \left (\frac {4}{9} c \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \left (\frac {8}{11} c \left (\frac {4}{9} c \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{5/2}}dx+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle a^3 c^3 \left (\frac {12}{13} c \left (\frac {8}{11} c \left (\frac {8 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2),x]
Output:
a^3*c^3*((2*c*Cos[e + f*x]^7)/(13*f*Sqrt[c - c*Sin[e + f*x]]) + (12*c*((2* c*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2)) + (8*c*((8*c^2*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) + (2*c*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^(5/2))))/11))/13)
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 - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
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] + Simp[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] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 38.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.56
| method | result | size |
| default | \(\frac {2 \left (-1+\sin \left (f x +e \right )\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (231 \sin \left (f x +e \right )^{3}-945 \sin \left (f x +e \right )^{2}+1421 \sin \left (f x +e \right )-835\right )}{3003 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(81\) |
| parts | \(\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (5 \sin \left (f x +e \right )^{3}-27 \sin \left (f x +e \right )^{2}+71 \sin \left (f x +e \right )-177\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (165 \sin \left (f x +e \right )^{6}-765 \sin \left (f x +e \right )^{5}+1565 \sin \left (f x +e \right )^{4}-2095 \sin \left (f x +e \right )^{3}+2514 \sin \left (f x +e \right )^{2}-3352 \sin \left (f x +e \right )+6704\right )}{2145 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (5 \sin \left (f x +e \right )^{4}-25 \sin \left (f x +e \right )^{3}+57 \sin \left (f x +e \right )^{2}-91 \sin \left (f x +e \right )+182\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (315 \sin \left (f x +e \right )^{5}-1505 \sin \left (f x +e \right )^{4}+3205 \sin \left (f x +e \right )^{3}-4539 \sin \left (f x +e \right )^{2}+6052 \sin \left (f x +e \right )-12104\right )}{1155 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(374\) |
Input:
int((a+sin(f*x+e)*a)^3*(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
Output:
2/3003*(-1+sin(f*x+e))*c^4*(1+sin(f*x+e))^4*a^3*(231*sin(f*x+e)^3-945*sin( f*x+e)^2+1421*sin(f*x+e)-835)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (129) = 258\).
Time = 0.10 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.83 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 \, {\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{7} - 21 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 28 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} - 40 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} - 128 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3} + {\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 252 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} + 280 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 320 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 384 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3003 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")
Output:
2/3003*(231*a^3*c^3*cos(f*x + e)^7 - 21*a^3*c^3*cos(f*x + e)^6 + 28*a^3*c^ 3*cos(f*x + e)^5 - 40*a^3*c^3*cos(f*x + e)^4 + 64*a^3*c^3*cos(f*x + e)^3 - 128*a^3*c^3*cos(f*x + e)^2 + 512*a^3*c^3*cos(f*x + e) + 1024*a^3*c^3 + (2 31*a^3*c^3*cos(f*x + e)^6 + 252*a^3*c^3*cos(f*x + e)^5 + 280*a^3*c^3*cos(f *x + e)^4 + 320*a^3*c^3*cos(f*x + e)^3 + 384*a^3*c^3*cos(f*x + e)^2 + 512* a^3*c^3*cos(f*x + e) + 1024*a^3*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)
Timed out. \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**3*(c-c*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3*(-c*sin(f*x + e) + c)^(7/2), x)
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.68 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (60060 \, a^{3} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15015 \, a^{3} c^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 9009 \, a^{3} c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2574 \, a^{3} c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2002 \, a^{3} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 273 \, a^{3} c^{3} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 231 \, a^{3} c^{3} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{96096 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")
Output:
-1/96096*sqrt(2)*(60060*a^3*c^3*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(sin(-1/ 4*pi + 1/2*f*x + 1/2*e)) + 15015*a^3*c^3*cos(-3/4*pi + 3/2*f*x + 3/2*e)*sg n(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 9009*a^3*c^3*cos(-5/4*pi + 5/2*f*x + 5 /2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 2574*a^3*c^3*cos(-7/4*pi + 7/2 *f*x + 7/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 2002*a^3*c^3*cos(-9/4* pi + 9/2*f*x + 9/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 273*a^3*c^3*co s(-11/4*pi + 11/2*f*x + 11/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 231* a^3*c^3*cos(-13/4*pi + 13/2*f*x + 13/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2* e)))*sqrt(c)/f
Timed out. \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \] Input:
int((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(7/2),x)
Output:
int((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(7/2), x)
\[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\sqrt {c}\, a^{3} c^{3} \left (\int \sqrt {-\sin \left (f x +e \right )+1}d x -\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{6}d x \right )+3 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right )-3 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right )\right ) \] Input:
int((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(7/2),x)
Output:
sqrt(c)*a**3*c**3*(int(sqrt( - sin(e + f*x) + 1),x) - int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**6,x) + 3*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x )**4,x) - 3*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2,x))