3.3.60 \(\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx\) [260]
Optimal. Leaf size=166 \[ \frac {a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7} \]
[Out]
1/15*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^11+4/195*a^3*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^10+4/715*a^3*c*c
os(f*x+e)^7/f/(c-c*sin(f*x+e))^9+8/6435*a^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+8/45045*a^3*cos(f*x+e)^7/c/f/(c-
c*sin(f*x+e))^7
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751,
2750} \begin {gather*} \frac {a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac {8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (4*a^3*c^2*Cos[e + f*x]^7)/(195*f*(c - c*Sin[e + f*x
])^10) + (4*a^3*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (8*a^3*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e
+ f*x])^8) + (8*a^3*Cos[e + f*x]^7)/(45045*c*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*} \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{11}} \, dx\\ &=\frac {a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {1}{15} \left (4 a^3 c^2\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {1}{65} \left (4 a^3 c\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {1}{715} \left (8 a^3\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {\left (8 a^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx}{6435 c}\\ &=\frac {a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.20, size = 209, normalized size = 1.26 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a+a \sin (e+f x))^3 \left (115830 \cos \left (\frac {1}{2} (e+f x)\right )-65065 \cos \left (\frac {3}{2} (e+f x)\right )-18018 \cos \left (\frac {5}{2} (e+f x)\right )+1365 \cos \left (\frac {7}{2} (e+f x)\right )-105 \cos \left (\frac {11}{2} (e+f x)\right )+\cos \left (\frac {15}{2} (e+f x)\right )+109395 \sin \left (\frac {1}{2} (e+f x)\right )+60060 \sin \left (\frac {3}{2} (e+f x)\right )-15015 \sin \left (\frac {5}{2} (e+f x)\right )-455 \sin \left (\frac {9}{2} (e+f x)\right )+15 \sin \left (\frac {13}{2} (e+f x)\right )\right )}{360360 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]
[Out]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^3*(115830*Cos[(e + f*x)/2] - 65065*Cos[(3*(e + f*x
))/2] - 18018*Cos[(5*(e + f*x))/2] + 1365*Cos[(7*(e + f*x))/2] - 105*Cos[(11*(e + f*x))/2] + Cos[(15*(e + f*x)
)/2] + 109395*Sin[(e + f*x)/2] + 60060*Sin[(3*(e + f*x))/2] - 15015*Sin[(5*(e + f*x))/2] - 455*Sin[(9*(e + f*x
))/2] + 15*Sin[(13*(e + f*x))/2]))/(360360*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^8)
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 238, normalized size = 1.43
| | |
| method |
result |
size |
| | |
| risch |
\(-\frac {16 \left (105 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-1365 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+65065 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-115830 a^{3} {\mathrm e}^{8 i \left (f x +e \right )}+455 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-15 i a^{3} {\mathrm e}^{i \left (f x +e \right )}-109395 i a^{3} {\mathrm e}^{7 i \left (f x +e \right )}+15015 i a^{3} {\mathrm e}^{5 i \left (f x +e \right )}-a^{3}+60060 i a^{3} {\mathrm e}^{9 i \left (f x +e \right )}+18018 a^{3} {\mathrm e}^{10 i \left (f x +e \right )}\right )}{45045 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{15} f \,c^{8}}\) |
\(174\) |
| derivativedivides |
\(\frac {2 a^{3} \left (-\frac {512}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{14}}-\frac {13184}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {32288}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {47072}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {81344}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {276}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1024}{15 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}-\frac {4536}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2304}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {188}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {24320}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {84112}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {7352}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {10}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}\right )}{f \,c^{8}}\) |
\(238\) |
| default |
\(\frac {2 a^{3} \left (-\frac {512}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{14}}-\frac {13184}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {32288}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {47072}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {81344}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {276}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1024}{15 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}-\frac {4536}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2304}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {188}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {24320}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {84112}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {7352}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {10}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}\right )}{f \,c^{8}}\) |
\(238\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x,method=_RETURNVERBOSE)
[Out]
2/f*a^3/c^8*(-512/(tan(1/2*f*x+1/2*e)-1)^14-13184/3/(tan(1/2*f*x+1/2*e)-1)^12-1/(tan(1/2*f*x+1/2*e)-1)-32288/7
/(tan(1/2*f*x+1/2*e)-1)^7-47072/5/(tan(1/2*f*x+1/2*e)-1)^10-81344/11/(tan(1/2*f*x+1/2*e)-1)^11-276/(tan(1/2*f*
x+1/2*e)-1)^4-1024/15/(tan(1/2*f*x+1/2*e)-1)^15-4536/5/(tan(1/2*f*x+1/2*e)-1)^5-2304/(tan(1/2*f*x+1/2*e)-1)^6-
188/3/(tan(1/2*f*x+1/2*e)-1)^3-24320/13/(tan(1/2*f*x+1/2*e)-1)^13-84112/9/(tan(1/2*f*x+1/2*e)-1)^9-7352/(tan(1
/2*f*x+1/2*e)-1)^8-10/(tan(1/2*f*x+1/2*e)-1)^2)
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2642 vs.
\(2 (171) = 342\).
time = 0.47, size = 2642, normalized size = 15.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm="maxima")
[Out]
2/45045*(3*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 342160*sin
(f*x + e)^3/(cos(f*x + e) + 1)^3 - 891345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(f*
x + e) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 36
87255*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x + e)
^10/(cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195195*sin(f*x + e)^12/(cos(f*x + e
) + 1)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1)
+ 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5
005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*
x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos
(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15)
- 7*a^3*(7845*sin(f*x + e)/(cos(f*x + e) + 1) - 54915*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 222950*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 - 668850*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1444443*sin(f*x + e)^5/(cos(f*x + e) +
1)^5 - 2407405*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3063060*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3063060*si
n(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2357355*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1414413*sin(f*x + e)^10/(cos
(f*x + e) + 1)^10 + 630630*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 210210*sin(f*x + e)^12/(cos(f*x + e) + 1)^1
2 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 6435*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 952)/(c^8 - 15*
c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(co
s(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^
5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin
(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(co
s(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) +
1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin
(f*x + e)^15/(cos(f*x + e) + 1)^15) - 12*a^3*(1740*sin(f*x + e)/(cos(f*x + e) + 1) - 12180*sin(f*x + e)^2/(cos
(f*x + e) + 1)^2 + 37765*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 113295*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 20
4204*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 340340*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 373230*sin(f*x + e)^7/
(cos(f*x + e) + 1)^7 - 373230*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 240240*sin(f*x + e)^9/(cos(f*x + e) + 1)^9
- 144144*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 45045*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 15015*sin(f*x
+ e)^12/(cos(f*x + e) + 1)^12 - 116)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^
4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin
(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos
(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1
)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*
sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) + 6*a^3*(675*sin(f*x + e)/(
cos(f*x + e) + 1) - 4725*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 20475*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 464
10*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 102102*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 130130*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 + 167310*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 122265*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 +
95095*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 33033*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 15015*sin(f*x + e)^
11/(cos(f*x + e) + 1)^11 - 45)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 30
03*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x +
e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 +
455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 ...
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 472 vs.
\(2 (171) = 342\).
time = 0.33, size = 472, normalized size = 2.84 \begin {gather*} \frac {8 \, a^{3} \cos \left (f x + e\right )^{8} + 64 \, a^{3} \cos \left (f x + e\right )^{7} - 196 \, a^{3} \cos \left (f x + e\right )^{6} - 672 \, a^{3} \cos \left (f x + e\right )^{5} + 735 \, a^{3} \cos \left (f x + e\right )^{4} - 7161 \, a^{3} \cos \left (f x + e\right )^{3} - 20328 \, a^{3} \cos \left (f x + e\right )^{2} + 12012 \, a^{3} \cos \left (f x + e\right ) + 24024 \, a^{3} - {\left (8 \, a^{3} \cos \left (f x + e\right )^{7} - 56 \, a^{3} \cos \left (f x + e\right )^{6} - 252 \, a^{3} \cos \left (f x + e\right )^{5} + 420 \, a^{3} \cos \left (f x + e\right )^{4} + 1155 \, a^{3} \cos \left (f x + e\right )^{3} + 8316 \, a^{3} \cos \left (f x + e\right )^{2} - 12012 \, a^{3} \cos \left (f x + e\right ) - 24024 \, a^{3}\right )} \sin \left (f x + e\right )}{45045 \, {\left (c^{8} f \cos \left (f x + e\right )^{8} - 7 \, c^{8} f \cos \left (f x + e\right )^{7} - 32 \, c^{8} f \cos \left (f x + e\right )^{6} + 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 160 \, c^{8} f \cos \left (f x + e\right )^{4} - 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 256 \, c^{8} f \cos \left (f x + e\right )^{2} + 64 \, c^{8} f \cos \left (f x + e\right ) + 128 \, c^{8} f + {\left (c^{8} f \cos \left (f x + e\right )^{7} + 8 \, c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} - 80 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} + 192 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f \cos \left (f x + e\right ) - 128 \, c^{8} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm="fricas")
[Out]
1/45045*(8*a^3*cos(f*x + e)^8 + 64*a^3*cos(f*x + e)^7 - 196*a^3*cos(f*x + e)^6 - 672*a^3*cos(f*x + e)^5 + 735*
a^3*cos(f*x + e)^4 - 7161*a^3*cos(f*x + e)^3 - 20328*a^3*cos(f*x + e)^2 + 12012*a^3*cos(f*x + e) + 24024*a^3 -
(8*a^3*cos(f*x + e)^7 - 56*a^3*cos(f*x + e)^6 - 252*a^3*cos(f*x + e)^5 + 420*a^3*cos(f*x + e)^4 + 1155*a^3*co
s(f*x + e)^3 + 8316*a^3*cos(f*x + e)^2 - 12012*a^3*cos(f*x + e) - 24024*a^3)*sin(f*x + e))/(c^8*f*cos(f*x + e)
^8 - 7*c^8*f*cos(f*x + e)^7 - 32*c^8*f*cos(f*x + e)^6 + 56*c^8*f*cos(f*x + e)^5 + 160*c^8*f*cos(f*x + e)^4 - 1
12*c^8*f*cos(f*x + e)^3 - 256*c^8*f*cos(f*x + e)^2 + 64*c^8*f*cos(f*x + e) + 128*c^8*f + (c^8*f*cos(f*x + e)^7
+ 8*c^8*f*cos(f*x + e)^6 - 24*c^8*f*cos(f*x + e)^5 - 80*c^8*f*cos(f*x + e)^4 + 80*c^8*f*cos(f*x + e)^3 + 192*
c^8*f*cos(f*x + e)^2 - 64*c^8*f*cos(f*x + e) - 128*c^8*f)*sin(f*x + e))
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 4542 vs.
\(2 (151) = 302\).
time = 148.30, size = 4542, normalized size = 27.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**8,x)
[Out]
Piecewise((-90090*a**3*tan(e/2 + f*x/2)**14/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2
)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 +
f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*
tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*
c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 47297
25*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 360360*a**3*tan(e/2 + f*x/2)*
*13/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)*
*13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 +
f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*t
an(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c*
*8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c
**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 2132130*a**3*tan(e/2 + f*x/2)**12/(45045*c**8*f*tan(e/2 + f*x/2)**15
- 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)*
*12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2
+ f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*t
an(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**
8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f)
+ 5405400*a**3*tan(e/2 + f*x/2)**11/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 +
4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)*
*11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2
+ f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*t
an(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*
f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 13351338*a**3*tan(e/2 + f*x/2)**10/(4
5045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 -
20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2
)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2
+ f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*t
an(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*
tan(e/2 + f*x/2) - 45045*c**8*f) + 20420400*a**3*tan(e/2 + f*x/2)**9/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 6756
75*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 +
61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/
2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2
+ f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*ta
n(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 2824
9650*a**3*tan(e/2 + f*x/2)**8/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 472972
5*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 -
135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/
2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2
+ f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(
e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 26357760*a**3*tan(e/2 + f*x/2)**7/(45045*c*
*8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 2049547
5*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 +
225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/
2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2
+ f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2
+ f*x/2) - 45045*c**8*f) - 22052030*a**3*tan(e...
________________________________________________________________________________________
Giac [A]
time = 0.50, size = 264, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (45045 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 180180 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 1066065 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 2702700 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 6675669 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 10210200 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 14124825 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 13178880 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 11026015 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 6066060 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3088995 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 864500 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 265335 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 18600 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4243 \, a^{3}\right )}}{45045 \, c^{8} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm="giac")
[Out]
-2/45045*(45045*a^3*tan(1/2*f*x + 1/2*e)^14 - 180180*a^3*tan(1/2*f*x + 1/2*e)^13 + 1066065*a^3*tan(1/2*f*x + 1
/2*e)^12 - 2702700*a^3*tan(1/2*f*x + 1/2*e)^11 + 6675669*a^3*tan(1/2*f*x + 1/2*e)^10 - 10210200*a^3*tan(1/2*f*
x + 1/2*e)^9 + 14124825*a^3*tan(1/2*f*x + 1/2*e)^8 - 13178880*a^3*tan(1/2*f*x + 1/2*e)^7 + 11026015*a^3*tan(1/
2*f*x + 1/2*e)^6 - 6066060*a^3*tan(1/2*f*x + 1/2*e)^5 + 3088995*a^3*tan(1/2*f*x + 1/2*e)^4 - 864500*a^3*tan(1/
2*f*x + 1/2*e)^3 + 265335*a^3*tan(1/2*f*x + 1/2*e)^2 - 18600*a^3*tan(1/2*f*x + 1/2*e) + 4243*a^3)/(c^8*f*(tan(
1/2*f*x + 1/2*e) - 1)^15)
________________________________________________________________________________________
Mupad [B]
time = 11.20, size = 187, normalized size = 1.13 \begin {gather*} \frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3497111\,\cos \left (3\,e+3\,f\,x\right )}{128}-\frac {25501905\,\sin \left (e+f\,x\right )}{128}-\frac {257861\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {5734111\,\cos \left (e+f\,x\right )}{128}+\frac {72047\,\cos \left (4\,e+4\,f\,x\right )}{4}-\frac {378579\,\cos \left (5\,e+5\,f\,x\right )}{128}-\frac {1059\,\cos \left (6\,e+6\,f\,x\right )}{2}+\frac {4251\,\cos \left (7\,e+7\,f\,x\right )}{128}+\frac {2633345\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {7210775\,\sin \left (3\,e+3\,f\,x\right )}{128}-\frac {89375\,\sin \left (4\,e+4\,f\,x\right )}{8}-\frac {504205\,\sin \left (5\,e+5\,f\,x\right )}{128}+\frac {29765\,\sin \left (6\,e+6\,f\,x\right )}{64}+\frac {4235\,\sin \left (7\,e+7\,f\,x\right )}{128}+\frac {544369}{4}\right )}{5765760\,c^8\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^8,x)
[Out]
(2^(1/2)*a^3*cos(e/2 + (f*x)/2)*((3497111*cos(3*e + 3*f*x))/128 - (25501905*sin(e + f*x))/128 - (257861*cos(2*
e + 2*f*x))/2 - (5734111*cos(e + f*x))/128 + (72047*cos(4*e + 4*f*x))/4 - (378579*cos(5*e + 5*f*x))/128 - (105
9*cos(6*e + 6*f*x))/2 + (4251*cos(7*e + 7*f*x))/128 + (2633345*sin(2*e + 2*f*x))/64 + (7210775*sin(3*e + 3*f*x
))/128 - (89375*sin(4*e + 4*f*x))/8 - (504205*sin(5*e + 5*f*x))/128 + (29765*sin(6*e + 6*f*x))/64 + (4235*sin(
7*e + 7*f*x))/128 + 544369/4))/(5765760*c^8*f*cos(e/2 + pi/4 + (f*x)/2)^15)
________________________________________________________________________________________