3.50 \(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx\)
Optimal. Leaf size=156 \[ \frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 c^2 (3 A-10 B) \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7}+\frac {2 a^3 c (3 A-10 B) \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8} \]
[Out]
1/13*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^10+1/143*a^3*(3*A-10*B)*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))
^9+2/1287*a^3*(3*A-10*B)*c*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+2/9009*a^3*(3*A-10*B)*cos(f*x+e)^7/f/(c-c*sin(f*x
+e))^7
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Rubi [A] time = 0.38, antiderivative size = 156, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used =
{2967, 2859, 2672, 2671} \[ \frac {a^3 c^2 (3 A-10 B) \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7}+\frac {2 a^3 c (3 A-10 B) \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x]
[Out]
(a^3*(A + B)*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (a^3*(3*A - 10*B)*c^2*Cos[e + f*x]^7)/(143*f
*(c - c*Sin[e + f*x])^9) + (2*a^3*(3*A - 10*B)*c*Cos[e + f*x]^7)/(1287*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(3*A
- 10*B)*Cos[e + f*x]^7)/(9009*f*(c - c*Sin[e + f*x])^7)
Rule 2671
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)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Rule 2672
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)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), 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] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Rule 2859
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m +
p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]
Rule 2967
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{10}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {1}{13} \left (a^3 (3 A-10 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^9} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {1}{143} \left (2 a^3 (3 A-10 B) c\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) c \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8}+\frac {\left (2 a^3 (3 A-10 B)\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx}{1287}\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) c \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8}+\frac {2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7}\\ \end {align*}
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Mathematica [B] time = 5.29, size = 339, normalized size = 2.17 \[ -\frac {a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6006 (9 A+5 B) \cos \left (\frac {1}{2} (e+f x)\right )-7722 (4 A+3 B) \cos \left (\frac {3}{2} (e+f x)\right )+48906 A \sin \left (\frac {1}{2} (e+f x)\right )+27027 A \sin \left (\frac {3}{2} (e+f x)\right )-6864 A \sin \left (\frac {5}{2} (e+f x)\right )-234 A \sin \left (\frac {9}{2} (e+f x)\right )+3 A \sin \left (\frac {13}{2} (e+f x)\right )-9009 A \cos \left (\frac {5}{2} (e+f x)\right )+858 A \cos \left (\frac {7}{2} (e+f x)\right )-39 A \cos \left (\frac {11}{2} (e+f x)\right )+47190 B \sin \left (\frac {1}{2} (e+f x)\right )+36036 B \sin \left (\frac {3}{2} (e+f x)\right )-19162 B \sin \left (\frac {5}{2} (e+f x)\right )-6006 B \sin \left (\frac {7}{2} (e+f x)\right )+780 B \sin \left (\frac {9}{2} (e+f x)\right )-10 B \sin \left (\frac {13}{2} (e+f x)\right )-12012 B \cos \left (\frac {5}{2} (e+f x)\right )+3146 B \cos \left (\frac {7}{2} (e+f x)\right )+130 B \cos \left (\frac {11}{2} (e+f x)\right )\right )}{144144 c^7 f (\sin (e+f x)-1)^7 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x]
[Out]
-1/144144*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(6006*(9*A + 5*B)*Cos[(e + f*x)/2] -
7722*(4*A + 3*B)*Cos[(3*(e + f*x))/2] - 9009*A*Cos[(5*(e + f*x))/2] - 12012*B*Cos[(5*(e + f*x))/2] + 858*A*Co
s[(7*(e + f*x))/2] + 3146*B*Cos[(7*(e + f*x))/2] - 39*A*Cos[(11*(e + f*x))/2] + 130*B*Cos[(11*(e + f*x))/2] +
48906*A*Sin[(e + f*x)/2] + 47190*B*Sin[(e + f*x)/2] + 27027*A*Sin[(3*(e + f*x))/2] + 36036*B*Sin[(3*(e + f*x))
/2] - 6864*A*Sin[(5*(e + f*x))/2] - 19162*B*Sin[(5*(e + f*x))/2] - 6006*B*Sin[(7*(e + f*x))/2] - 234*A*Sin[(9*
(e + f*x))/2] + 780*B*Sin[(9*(e + f*x))/2] + 3*A*Sin[(13*(e + f*x))/2] - 10*B*Sin[(13*(e + f*x))/2]))/(c^7*f*(
Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^7)
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fricas [B] time = 0.45, size = 475, normalized size = 3.04 \[ -\frac {2 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{7} - 12 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 49 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 7 \, {\left (30 \, A + 329 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 63 \, {\left (27 \, A + 53 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 252 \, {\left (19 \, A + 32 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 2772 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 5544 \, {\left (A + B\right )} a^{3} + {\left (2 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} + 14 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 35 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 63 \, {\left (5 \, A + 31 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 252 \, {\left (8 \, A + 21 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 2772 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 5544 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{9009 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f - {\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algorithm="fricas")
[Out]
-1/9009*(2*(3*A - 10*B)*a^3*cos(f*x + e)^7 - 12*(3*A - 10*B)*a^3*cos(f*x + e)^6 - 49*(3*A - 10*B)*a^3*cos(f*x
+ e)^5 + 7*(30*A + 329*B)*a^3*cos(f*x + e)^4 - 63*(27*A + 53*B)*a^3*cos(f*x + e)^3 - 252*(19*A + 32*B)*a^3*cos
(f*x + e)^2 + 2772*(A + B)*a^3*cos(f*x + e) + 5544*(A + B)*a^3 + (2*(3*A - 10*B)*a^3*cos(f*x + e)^6 + 14*(3*A
- 10*B)*a^3*cos(f*x + e)^5 - 35*(3*A - 10*B)*a^3*cos(f*x + e)^4 - 63*(5*A + 31*B)*a^3*cos(f*x + e)^3 - 252*(8*
A + 21*B)*a^3*cos(f*x + e)^2 + 2772*(A + B)*a^3*cos(f*x + e) + 5544*(A + B)*a^3)*sin(f*x + e))/(c^7*f*cos(f*x
+ e)^7 + 7*c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^5 - 56*c^7*f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^3
+ 112*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f - (c^7*f*cos(f*x + e)^6 - 6*c^7*f*cos(f*x + e)^5
- 24*c^7*f*cos(f*x + e)^4 + 32*c^7*f*cos(f*x + e)^3 + 80*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^
7*f)*sin(f*x + e))
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giac [B] time = 0.36, size = 445, normalized size = 2.85 \[ -\frac {2 \, {\left (9009 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 27027 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 9009 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 153153 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} + 3003 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 297297 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 69069 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 648648 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 9009 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 738738 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 150150 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 857142 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 16302 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 548262 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 115830 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 367653 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 286 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 112827 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30745 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45513 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1443 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3081 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1261 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 930 \, A a^{3} - 97 \, B a^{3}\right )}}{9009 \, c^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algorithm="giac")
[Out]
-2/9009*(9009*A*a^3*tan(1/2*f*x + 1/2*e)^12 - 27027*A*a^3*tan(1/2*f*x + 1/2*e)^11 + 9009*B*a^3*tan(1/2*f*x + 1
/2*e)^11 + 153153*A*a^3*tan(1/2*f*x + 1/2*e)^10 + 3003*B*a^3*tan(1/2*f*x + 1/2*e)^10 - 297297*A*a^3*tan(1/2*f*
x + 1/2*e)^9 + 69069*B*a^3*tan(1/2*f*x + 1/2*e)^9 + 648648*A*a^3*tan(1/2*f*x + 1/2*e)^8 - 9009*B*a^3*tan(1/2*f
*x + 1/2*e)^8 - 738738*A*a^3*tan(1/2*f*x + 1/2*e)^7 + 150150*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 857142*A*a^3*tan(1
/2*f*x + 1/2*e)^6 - 16302*B*a^3*tan(1/2*f*x + 1/2*e)^6 - 548262*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 115830*B*a^3*ta
n(1/2*f*x + 1/2*e)^5 + 367653*A*a^3*tan(1/2*f*x + 1/2*e)^4 - 286*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 112827*A*a^3*t
an(1/2*f*x + 1/2*e)^3 + 30745*B*a^3*tan(1/2*f*x + 1/2*e)^3 + 45513*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 1443*B*a^3*t
an(1/2*f*x + 1/2*e)^2 - 3081*A*a^3*tan(1/2*f*x + 1/2*e) + 1261*B*a^3*tan(1/2*f*x + 1/2*e) + 930*A*a^3 - 97*B*a
^3)/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)
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maple [A] time = 0.57, size = 293, normalized size = 1.88 \[ \frac {2 a^{3} \left (-\frac {16000 A +14720 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {512 A +512 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {6888 A +3928 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {150 A +34 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {13112 A +8840 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {3072 A +3072 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {2700 A +1240 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {18816 A +14464 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {768 A +264 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8832 A +8576 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {18 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {20256 A +17248 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x)
[Out]
2/f*a^3/c^7*(-1/10*(16000*A+14720*B)/(tan(1/2*f*x+1/2*e)-1)^10-1/13*(512*A+512*B)/(tan(1/2*f*x+1/2*e)-1)^13-1/
6*(6888*A+3928*B)/(tan(1/2*f*x+1/2*e)-1)^6-A/(tan(1/2*f*x+1/2*e)-1)-1/3*(150*A+34*B)/(tan(1/2*f*x+1/2*e)-1)^3-
1/7*(13112*A+8840*B)/(tan(1/2*f*x+1/2*e)-1)^7-1/12*(3072*A+3072*B)/(tan(1/2*f*x+1/2*e)-1)^12-1/5*(2700*A+1240*
B)/(tan(1/2*f*x+1/2*e)-1)^5-1/8*(18816*A+14464*B)/(tan(1/2*f*x+1/2*e)-1)^8-1/4*(768*A+264*B)/(tan(1/2*f*x+1/2*
e)-1)^4-1/11*(8832*A+8576*B)/(tan(1/2*f*x+1/2*e)-1)^11-1/2*(18*A+2*B)/(tan(1/2*f*x+1/2*e)-1)^2-1/9*(20256*A+17
248*B)/(tan(1/2*f*x+1/2*e)-1)^9)
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maxima [B] time = 0.57, size = 4078, normalized size = 26.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algorithm="maxima")
[Out]
-2/45045*(6*A*a^3*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*si
n(f*x + e)^3/(cos(f*x + e) + 1)^3 - 187330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*
x + e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 2072
07*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(co
s(f*x + e) + 1)^10 - 367)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*
sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(
cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)
^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin
(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) + 6*B*a^3*(4771*sin(f*x + e)/(
cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 18
7330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 353496*sin(f*x + e)^6/
(cos(f*x + e) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 207207*sin(f*x + e)^8/(cos(f*x + e) + 1)^8
+ 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 367)/(c^7 - 13*c^
7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f
*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 +
1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x
+ e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x
+ e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c
^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) + 15*A*a^3*(3796*sin(f*x + e)/(cos(f*x + e) + 1) - 22776*sin(f*x + e
)^2/(cos(f*x + e) + 1)^2 + 77506*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 193765*sin(f*x + e)^4/(cos(f*x + e) + 1
)^4 + 339768*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 453024*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 444444*sin(f*x
+ e)^7/(cos(f*x + e) + 1)^7 - 333333*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 180180*sin(f*x + e)^9/(cos(f*x + e
) + 1)^9 - 72072*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 18018*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 3003*si
n(f*x + e)^12/(cos(f*x + e) + 1)^12 - 523)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)
^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7
*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(
cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1
)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 105*A*a^3*(
611*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 8723*sin(f*x + e)^3/(cos(f*x
+ e) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 40326*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 27027*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*si
n(f*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^
2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*
sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(c
os(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)
^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 35*B*a^3*(61
1*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 8723*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 40326*sin(
f*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 27027*sin(f*x + e)^8/(cos(f*x +
e) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*sin(
f*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/
(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)
^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*si
n(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos
(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^1
1 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) + 8*B*a^3*(559*s
in(f*x + e)/(cos(f*x + e) + 1) - 3354*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12298*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 30745*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 37323*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 49764*sin(f*
x + e)^6/(cos(f*x + e) + 1)^6 + 24024*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 18018*sin(f*x + e)^8/(cos(f*x + e)
+ 1)^8 - 43)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286
*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e)
+ 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c^7
*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12
/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 462*A*a^3*(13*sin(f*x + e)/(cos(f*x + e)
+ 1) - 78*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 286*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 520*sin(f*x + e)^4/
(cos(f*x + e) + 1)^4 + 936*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 858*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 858
*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 351*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 - 1)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 2
86*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x +
e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x +
e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c
^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^
12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 1386*B*a^3*(13*sin(f*x + e)/(cos(f*x +
e) + 1) - 78*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 286*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 520*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 + 936*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 858*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 +
858*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 351*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^9/(cos(f*
x + e) + 1)^9 - 1)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2
- 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x
+ e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 28
6*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x +
e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13))/f
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mupad [B] time = 13.79, size = 500, normalized size = 3.21 \[ \frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2363\,B\,a^3}{32}-\frac {279183\,A\,a^3}{16}+\frac {220269\,A\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{16}-\frac {46095\,A\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {20829\,A\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {2811\,A\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{16}+\frac {231\,A\,a^3\,\cos \left (6\,e+6\,f\,x\right )}{16}-\frac {8995\,B\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{64}+\frac {497\,B\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{16}+\frac {3725\,B\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{32}-\frac {361\,B\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{16}-\frac {77\,B\,a^3\,\cos \left (6\,e+6\,f\,x\right )}{64}-\frac {19305\,A\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {81081\,A\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{16}+\frac {15015\,A\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {3237\,A\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{16}-\frac {117\,A\,a^3\,\sin \left (6\,e+6\,f\,x\right )}{8}+\frac {77649\,B\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {27027\,B\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{32}-\frac {1001\,B\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{8}-\frac {559\,B\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{32}+\frac {117\,B\,a^3\,\sin \left (6\,e+6\,f\,x\right )}{64}+\frac {26979\,A\,a^3\,\cos \left (e+f\,x\right )}{4}+40\,B\,a^3\,\cos \left (e+f\,x\right )+\frac {173745\,A\,a^3\,\sin \left (e+f\,x\right )}{8}-\frac {80223\,B\,a^3\,\sin \left (e+f\,x\right )}{16}\right )}{9009\,c^7\,f\,\left (\frac {1287\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{64}-\frac {429\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{16}+\frac {715\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{64}-\frac {143\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{32}-\frac {39\,\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{32}+\frac {13\,\sqrt {2}\,\cos \left (\frac {11\,e}{2}-\frac {\pi }{4}+\frac {11\,f\,x}{2}\right )}{64}+\frac {\sqrt {2}\,\cos \left (\frac {13\,e}{2}+\frac {\pi }{4}+\frac {13\,f\,x}{2}\right )}{64}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^7,x)
[Out]
(2*cos(e/2 + (f*x)/2)*((2363*B*a^3)/32 - (279183*A*a^3)/16 + (220269*A*a^3*cos(2*e + 2*f*x))/16 - (46095*A*a^3
*cos(3*e + 3*f*x))/16 - (20829*A*a^3*cos(4*e + 4*f*x))/16 + (2811*A*a^3*cos(5*e + 5*f*x))/16 + (231*A*a^3*cos(
6*e + 6*f*x))/16 - (8995*B*a^3*cos(2*e + 2*f*x))/64 + (497*B*a^3*cos(3*e + 3*f*x))/16 + (3725*B*a^3*cos(4*e +
4*f*x))/32 - (361*B*a^3*cos(5*e + 5*f*x))/16 - (77*B*a^3*cos(6*e + 6*f*x))/64 - (19305*A*a^3*sin(2*e + 2*f*x))
/4 - (81081*A*a^3*sin(3*e + 3*f*x))/16 + (15015*A*a^3*sin(4*e + 4*f*x))/16 + (3237*A*a^3*sin(5*e + 5*f*x))/16
- (117*A*a^3*sin(6*e + 6*f*x))/8 + (77649*B*a^3*sin(2*e + 2*f*x))/64 + (27027*B*a^3*sin(3*e + 3*f*x))/32 - (10
01*B*a^3*sin(4*e + 4*f*x))/8 - (559*B*a^3*sin(5*e + 5*f*x))/32 + (117*B*a^3*sin(6*e + 6*f*x))/64 + (26979*A*a^
3*cos(e + f*x))/4 + 40*B*a^3*cos(e + f*x) + (173745*A*a^3*sin(e + f*x))/8 - (80223*B*a^3*sin(e + f*x))/16))/(9
009*c^7*f*((1287*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/64 - (429*2^(1/2)*cos(e/2 + pi/4 + (f*x)/2))/16 + (7
15*2^(1/2)*cos((5*e)/2 + pi/4 + (5*f*x)/2))/64 - (143*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/32 - (39*2^(1/2
)*cos((9*e)/2 + pi/4 + (9*f*x)/2))/32 + (13*2^(1/2)*cos((11*e)/2 - pi/4 + (11*f*x)/2))/64 + (2^(1/2)*cos((13*e
)/2 + pi/4 + (13*f*x)/2))/64))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**7,x)
[Out]
Timed out
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