\(\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx\) [259]
Optimal result
Integrand size = 26, antiderivative size = 132 \[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}+\frac {2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}
\]
[Out]
1/13*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^10+3/143*a^3*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^9+2/429*a^3*c*co
s(f*x+e)^7/f/(c-c*sin(f*x+e))^8+2/3003*a^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^7
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 2750}
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}+\frac {2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}
\]
[In]
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7,x]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (3*a^3*c^2*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x
])^9) + (2*a^3*c*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^8) + (2*a^3*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e
+ f*x])^7)
Rule 2750
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 2751
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 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*}
\text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{10}} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {1}{13} \left (3 a^3 c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^9} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {1}{143} \left (6 a^3 c\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}+\frac {1}{429} \left (2 a^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}+\frac {2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.90 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.19
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=\frac {a^3 \left (18018 \cos \left (\frac {1}{2} (e+f x)\right )-10296 \cos \left (\frac {3}{2} (e+f x)\right )-3003 \cos \left (\frac {5}{2} (e+f x)\right )+286 \cos \left (\frac {7}{2} (e+f x)\right )-13 \cos \left (\frac {11}{2} (e+f x)\right )+16302 \sin \left (\frac {1}{2} (e+f x)\right )+9009 \sin \left (\frac {3}{2} (e+f x)\right )-2288 \sin \left (\frac {5}{2} (e+f x)\right )-78 \sin \left (\frac {9}{2} (e+f x)\right )+\sin \left (\frac {13}{2} (e+f x)\right )\right )}{48048 c^7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{13}}
\]
[In]
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7,x]
[Out]
(a^3*(18018*Cos[(e + f*x)/2] - 10296*Cos[(3*(e + f*x))/2] - 3003*Cos[(5*(e + f*x))/2] + 286*Cos[(7*(e + f*x))/
2] - 13*Cos[(11*(e + f*x))/2] + 16302*Sin[(e + f*x)/2] + 9009*Sin[(3*(e + f*x))/2] - 2288*Sin[(5*(e + f*x))/2]
- 78*Sin[(9*(e + f*x))/2] + Sin[(13*(e + f*x))/2]))/(48048*c^7*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01
| | |
| method | result | size |
| | |
| risch |
\(-\frac {4 i a^{3} \left (9009 i {\mathrm e}^{8 i \left (f x +e \right )}+3003 \,{\mathrm e}^{9 i \left (f x +e \right )}-16302 i {\mathrm e}^{6 i \left (f x +e \right )}-18018 \,{\mathrm e}^{7 i \left (f x +e \right )}+2288 i {\mathrm e}^{4 i \left (f x +e \right )}+10296 \,{\mathrm e}^{5 i \left (f x +e \right )}+78 i {\mathrm e}^{2 i \left (f x +e \right )}-286 \,{\mathrm e}^{3 i \left (f x +e \right )}-i+13 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3003 f \,c^{7} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{13}}\) |
\(133\) |
| parallelrisch |
\(-\frac {2 \left (\frac {310}{3003}-\frac {263 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21}-\frac {79 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{231}+\frac {666 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+72 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-82 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {426 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {857 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21}+\frac {389 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{77}+\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )-3 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+17 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-33 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) a^{3}}{f \,c^{7} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}\) |
\(179\) |
| derivativedivides |
\(\frac {2 a^{3} \left (-\frac {50}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1148}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {1600}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {192}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8832}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {512}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {13112}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {256}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {2352}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {9}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {540}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {6752}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{7}}\) |
\(208\) |
| default |
\(\frac {2 a^{3} \left (-\frac {50}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1148}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {1600}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {192}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8832}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {512}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {13112}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {256}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {2352}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {9}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {540}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {6752}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{7}}\) |
\(208\) |
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[In]
int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x,method=_RETURNVERBOSE)
[Out]
-4/3003*I*a^3*(9009*I*exp(8*I*(f*x+e))+3003*exp(9*I*(f*x+e))-16302*I*exp(6*I*(f*x+e))-18018*exp(7*I*(f*x+e))+2
288*I*exp(4*I*(f*x+e))+10296*exp(5*I*(f*x+e))+78*I*exp(2*I*(f*x+e))-286*exp(3*I*(f*x+e))-I+13*exp(I*(f*x+e)))/
f/c^7/(exp(I*(f*x+e))-I)^13
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (128) = 256\).
Time = 0.27 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.92
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=-\frac {2 \, a^{3} \cos \left (f x + e\right )^{7} - 12 \, a^{3} \cos \left (f x + e\right )^{6} - 49 \, a^{3} \cos \left (f x + e\right )^{5} + 70 \, a^{3} \cos \left (f x + e\right )^{4} - 567 \, a^{3} \cos \left (f x + e\right )^{3} - 1596 \, a^{3} \cos \left (f x + e\right )^{2} + 924 \, a^{3} \cos \left (f x + e\right ) + 1848 \, a^{3} + {\left (2 \, a^{3} \cos \left (f x + e\right )^{6} + 14 \, a^{3} \cos \left (f x + e\right )^{5} - 35 \, a^{3} \cos \left (f x + e\right )^{4} - 105 \, a^{3} \cos \left (f x + e\right )^{3} - 672 \, a^{3} \cos \left (f x + e\right )^{2} + 924 \, a^{3} \cos \left (f x + e\right ) + 1848 \, a^{3}\right )} \sin \left (f x + e\right )}{3003 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f - {\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}}
\]
[In]
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm="fricas")
[Out]
-1/3003*(2*a^3*cos(f*x + e)^7 - 12*a^3*cos(f*x + e)^6 - 49*a^3*cos(f*x + e)^5 + 70*a^3*cos(f*x + e)^4 - 567*a^
3*cos(f*x + e)^3 - 1596*a^3*cos(f*x + e)^2 + 924*a^3*cos(f*x + e) + 1848*a^3 + (2*a^3*cos(f*x + e)^6 + 14*a^3*
cos(f*x + e)^5 - 35*a^3*cos(f*x + e)^4 - 105*a^3*cos(f*x + e)^3 - 672*a^3*cos(f*x + e)^2 + 924*a^3*cos(f*x + e
) + 1848*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))
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3451 vs. \(2 (121) = 242\).
Time = 71.09 (sec) , antiderivative size = 3451, normalized size of antiderivative = 26.14
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**7,x)
[Out]
Piecewise((-6006*a**3*tan(e/2 + f*x/2)**12/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**
12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)
**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2
)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2
)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) + 18018*a**3*tan(e/2 +
f*x/2)**11/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*
x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 +
f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 +
f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 +
f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) - 102102*a**3*tan(e/2 + f*x/2)**10/(3003*c**7*f*tan(e
/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e
/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan
(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*ta
n(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e
/2 + f*x/2) - 3003*c**7*f) + 198198*a**3*tan(e/2 + f*x/2)**9/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*
tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*
f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7
*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7
*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) - 43
2432*a**3*tan(e/2 + f*x/2)**8/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c
**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861
*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 386486
1*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234
*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) + 492492*a**3*tan(e/2 + f*x/2)**7/(
3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 85
8858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5
153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 -
2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 3
9039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) - 571428*a**3*tan(e/2 + f*x/2)**6/(3003*c**7*f*tan(e/2 + f*x/2)**1
3 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**1
0 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)*
*7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)
**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) -
3003*c**7*f) + 365508*a**3*tan(e/2 + f*x/2)**5/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/
2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*
x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f
*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f
*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) - 245102*a**3*tan(
e/2 + f*x/2)**4/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2
+ f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/
2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e
/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/
2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) + 75218*a**3*tan(e/2 + f*x/2)**3/(3003*c**7*f*tan
(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan
(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*t
an(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*
tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan
(e/2 + f*x/2) - 3003*c**7*f) - 30342*a**3*tan(e/2 + f*x/2)**2/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f
*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7
*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**
7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**
7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) + 2
054*a**3*tan(e/2 + f*x/2)/(3003*c**7*f*tan(e/2 + f*x/2)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*
f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**
7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c*
*7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**
7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2) - 3003*c**7*f) - 620*a**3/(3003*c**7*f*tan(e/2 + f*x/2
)**13 - 39039*c**7*f*tan(e/2 + f*x/2)**12 + 234234*c**7*f*tan(e/2 + f*x/2)**11 - 858858*c**7*f*tan(e/2 + f*x/2
)**10 + 2147145*c**7*f*tan(e/2 + f*x/2)**9 - 3864861*c**7*f*tan(e/2 + f*x/2)**8 + 5153148*c**7*f*tan(e/2 + f*x
/2)**7 - 5153148*c**7*f*tan(e/2 + f*x/2)**6 + 3864861*c**7*f*tan(e/2 + f*x/2)**5 - 2147145*c**7*f*tan(e/2 + f*
x/2)**4 + 858858*c**7*f*tan(e/2 + f*x/2)**3 - 234234*c**7*f*tan(e/2 + f*x/2)**2 + 39039*c**7*f*tan(e/2 + f*x/2
) - 3003*c**7*f), Ne(f, 0)), (x*(a*sin(e) + a)**3/(-c*sin(e) + c)**7, True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2078 vs. \(2 (128) = 256\).
Time = 0.27 (sec) , antiderivative size = 2078, normalized size of antiderivative = 15.74
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm="maxima")
[Out]
-2/15015*(2*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 - 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 - 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*si
n(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/(co
s(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) + 5*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 + 1
80180*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)^1
1/(cos(f*x + e) + 1)^11 - 3003*sin(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/(co
s(f*x + e) + 1)^13) - 35*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*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*s
in(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/(c
os(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) - 154*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*s
in(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*s
in(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/(c
os(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))/f
Giac [A] (verification not implemented)
none
Time = 0.38 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.64
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=-\frac {2 \, {\left (3003 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 9009 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 51051 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 99099 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 216216 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 246246 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 285714 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 182754 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 122551 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 37609 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15171 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1027 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 310 \, a^{3}\right )}}{3003 \, c^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{13}}
\]
[In]
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm="giac")
[Out]
-2/3003*(3003*a^3*tan(1/2*f*x + 1/2*e)^12 - 9009*a^3*tan(1/2*f*x + 1/2*e)^11 + 51051*a^3*tan(1/2*f*x + 1/2*e)^
10 - 99099*a^3*tan(1/2*f*x + 1/2*e)^9 + 216216*a^3*tan(1/2*f*x + 1/2*e)^8 - 246246*a^3*tan(1/2*f*x + 1/2*e)^7
+ 285714*a^3*tan(1/2*f*x + 1/2*e)^6 - 182754*a^3*tan(1/2*f*x + 1/2*e)^5 + 122551*a^3*tan(1/2*f*x + 1/2*e)^4 -
37609*a^3*tan(1/2*f*x + 1/2*e)^3 + 15171*a^3*tan(1/2*f*x + 1/2*e)^2 - 1027*a^3*tan(1/2*f*x + 1/2*e) + 310*a^3)
/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)
Mupad [B] (verification not implemented)
Time = 10.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.25
\[
\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx=-\frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8993\,\cos \left (e+f\,x\right )}{4}+\frac {57915\,\sin \left (e+f\,x\right )}{8}+\frac {73423\,\cos \left (2\,e+2\,f\,x\right )}{16}-\frac {15365\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {6943\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {937\,\cos \left (5\,e+5\,f\,x\right )}{16}+\frac {77\,\cos \left (6\,e+6\,f\,x\right )}{16}-\frac {6435\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {27027\,\sin \left (3\,e+3\,f\,x\right )}{16}+\frac {5005\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {1079\,\sin \left (5\,e+5\,f\,x\right )}{16}-\frac {39\,\sin \left (6\,e+6\,f\,x\right )}{8}-\frac {93061}{16}\right )}{192192\,c^7\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^{13}}
\]
[In]
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^7,x)
[Out]
-(2^(1/2)*a^3*cos(e/2 + (f*x)/2)*((8993*cos(e + f*x))/4 + (57915*sin(e + f*x))/8 + (73423*cos(2*e + 2*f*x))/16
- (15365*cos(3*e + 3*f*x))/16 - (6943*cos(4*e + 4*f*x))/16 + (937*cos(5*e + 5*f*x))/16 + (77*cos(6*e + 6*f*x)
)/16 - (6435*sin(2*e + 2*f*x))/4 - (27027*sin(3*e + 3*f*x))/16 + (5005*sin(4*e + 4*f*x))/16 + (1079*sin(5*e +
5*f*x))/16 - (39*sin(6*e + 6*f*x))/8 - 93061/16))/(192192*c^7*f*cos(e/2 + pi/4 + (f*x)/2)^13)