3.3.58 \(\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx\) [258]
Optimal. Leaf size=101 \[ \frac {a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \]
[Out]
1/11*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^9+2/99*a^3*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+2/693*a^3*c*cos(
f*x+e)^7/f/(c-c*sin(f*x+e))^7
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Rubi [A]
time = 0.13, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, 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)}{11 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c^2*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])
^8) + (2*a^3*c*Cos[e + f*x]^7)/(693*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))^6} \, dx &=\left (a^3 c^3\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)}{11 f (c-c \sin (e+f x))^9}+\frac {1}{11} \left (2 a^3 c^2\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)}{11 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {1}{99} \left (2 a^3 c\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)}{11 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7}\\ \end {align*}
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Mathematica [A]
time = 0.59, size = 145, normalized size = 1.44 \begin {gather*} -\frac {a^3 \left (-2541 \cos \left (\frac {1}{2} (e+f x)\right )+1485 \cos \left (\frac {3}{2} (e+f x)\right )+462 \cos \left (\frac {5}{2} (e+f x)\right )-55 \cos \left (\frac {7}{2} (e+f x)\right )+\cos \left (\frac {11}{2} (e+f x)\right )-2079 \sin \left (\frac {1}{2} (e+f x)\right )-1155 \sin \left (\frac {3}{2} (e+f x)\right )+297 \sin \left (\frac {5}{2} (e+f x)\right )+11 \sin \left (\frac {9}{2} (e+f x)\right )\right )}{5544 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]
[Out]
-1/5544*(a^3*(-2541*Cos[(e + f*x)/2] + 1485*Cos[(3*(e + f*x))/2] + 462*Cos[(5*(e + f*x))/2] - 55*Cos[(7*(e + f
*x))/2] + Cos[(11*(e + f*x))/2] - 2079*Sin[(e + f*x)/2] - 1155*Sin[(3*(e + f*x))/2] + 297*Sin[(5*(e + f*x))/2]
+ 11*Sin[(9*(e + f*x))/2]))/(c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11)
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Maple [A]
time = 0.54, size = 178, normalized size = 1.76
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {\frac {60 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}}{7}-12 i a^{3} {\mathrm e}^{5 i \left (f x +e \right )}-\frac {20 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{63}+\frac {12 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}}{7}+\frac {4 i a^{3} {\mathrm e}^{i \left (f x +e \right )}}{63}-\frac {44 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}}{3}+\frac {4 a^{3}}{693}+\frac {20 i a^{3} {\mathrm e}^{7 i \left (f x +e \right )}}{3}+\frac {8 a^{3} {\mathrm e}^{8 i \left (f x +e \right )}}{3}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} f \,c^{6}}\) |
\(143\) |
| derivativedivides |
\(\frac {2 a^{3} \left (-\frac {544}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {3008}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {4272}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {126}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {256}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1480}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {128}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {116}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{6}}\) |
\(178\) |
| default |
\(\frac {2 a^{3} \left (-\frac {544}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {3008}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {4272}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {126}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {256}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1480}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {128}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {116}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{6}}\) |
\(178\) |
| norman |
\(\frac {-\frac {158 a^{3}}{693 c f}-\frac {2 a^{3} \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {4 a^{3} \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {88 a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {32 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{63 c f}+\frac {128 a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {148 a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {332 a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {412 a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {1696 a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}+\frac {12980 a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 c f}+\frac {1856 a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {2104 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {6392 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{693 c f}-\frac {22024 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 c f}-\frac {16372 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{231 c f}-\frac {153080 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{693 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) |
\(395\) |
| | |
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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))^6,x,method=_RETURNVERBOSE)
[Out]
2/f*a^3/c^6*(-544/(tan(1/2*f*x+1/2*e)-1)^8-3008/9/(tan(1/2*f*x+1/2*e)-1)^9-1/(tan(1/2*f*x+1/2*e)-1)-4272/7/(ta
n(1/2*f*x+1/2*e)-1)^7-126/(tan(1/2*f*x+1/2*e)-1)^4-256/11/(tan(1/2*f*x+1/2*e)-1)^11-8/(tan(1/2*f*x+1/2*e)-1)^2
-1480/3/(tan(1/2*f*x+1/2*e)-1)^6-292/(tan(1/2*f*x+1/2*e)-1)^5-128/(tan(1/2*f*x+1/2*e)-1)^10-116/3/(tan(1/2*f*x
+1/2*e)-1)^3)
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1890 vs.
\(2 (104) = 208\).
time = 0.39, size = 1890, normalized size = 18.71 \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))^6,x, algorithm="maxima")
[Out]
-2/3465*(5*a^3*(913*sin(f*x + e)/(cos(f*x + e) + 1) - 4565*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12540*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 - 25080*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33726*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 - 33726*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 23100*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 11550*sin(f*x
+ e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 693*sin(f*x + e)^10/(cos(f*x + e) +
1)^10 - 146)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*
c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5
/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1
)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*
x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 9*a^3*(671*sin(f*x + e)/(cos(f*
x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 10890*sin(
f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 12936*sin(f*x + e)^6/(cos(f*x +
e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1155*sin(f*x
+ e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 -
462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x +
e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e)
+ 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 2*a^3*(34
1*sin(f*x + e)/(cos(f*x + e) + 1) - 1705*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5115*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3 - 6765*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9471*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 4851*sin(f*x
+ e)^6/(cos(f*x + e) + 1)^6 + 3465*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 31)/(c^6 - 11*c^6*sin(f*x + e)/(cos(
f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330
*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^
6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) +
1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x
+ e)^11/(cos(f*x + e) + 1)^11) + 12*a^3*(253*sin(f*x + e)/(cos(f*x + e) + 1) - 1265*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5313*sin(f*
x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310*sin(f*x + e)^7/(cos(f*x + e) +
1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*
sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6
- 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x +
e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e)
+ 1)^11))/f
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs.
\(2 (104) = 208\).
time = 0.33, size = 356, normalized size = 3.52 \begin {gather*} \frac {2 \, a^{3} \cos \left (f x + e\right )^{6} + 12 \, a^{3} \cos \left (f x + e\right )^{5} - 25 \, a^{3} \cos \left (f x + e\right )^{4} + 161 \, a^{3} \cos \left (f x + e\right )^{3} + 448 \, a^{3} \cos \left (f x + e\right )^{2} - 252 \, a^{3} \cos \left (f x + e\right ) - 504 \, a^{3} - {\left (2 \, a^{3} \cos \left (f x + e\right )^{5} - 10 \, a^{3} \cos \left (f x + e\right )^{4} - 35 \, a^{3} \cos \left (f x + e\right )^{3} - 196 \, a^{3} \cos \left (f x + e\right )^{2} + 252 \, a^{3} \cos \left (f x + e\right ) + 504 \, a^{3}\right )} \sin \left (f x + e\right )}{693 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} 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))^6,x, algorithm="fricas")
[Out]
1/693*(2*a^3*cos(f*x + e)^6 + 12*a^3*cos(f*x + e)^5 - 25*a^3*cos(f*x + e)^4 + 161*a^3*cos(f*x + e)^3 + 448*a^3
*cos(f*x + e)^2 - 252*a^3*cos(f*x + e) - 504*a^3 - (2*a^3*cos(f*x + e)^5 - 10*a^3*cos(f*x + e)^4 - 35*a^3*cos(
f*x + e)^3 - 196*a^3*cos(f*x + e)^2 + 252*a^3*cos(f*x + e) + 504*a^3)*sin(f*x + e))/(c^6*f*cos(f*x + e)^6 - 5*
c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*
cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*
cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2509 vs.
\(2 (92) = 184\).
time = 54.89, size = 2509, normalized size = 24.84 \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))**6,x)
[Out]
Piecewise((-1386*a**3*tan(e/2 + f*x/2)**10/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10
+ 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 -
320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 11
4345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f
) + 2772*a**3*tan(e/2 + f*x/2)**9/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*
c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c*
*6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6
*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 16170
*a**3*tan(e/2 + f*x/2)**8/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*t
an(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan
(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e
/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 21252*a**3*ta
n(e/2 + f*x/2)**7/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 +
f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f
*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x
/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 42504*a**3*tan(e/2 +
f*x/2)**6/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)*
*9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6
+ 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 -
38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 30492*a**3*tan(e/2 + f*x/2)**
5/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114
345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 32016
6*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c
**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 30888*a**3*tan(e/2 + f*x/2)**4/(693*c
**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6
*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f
*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*ta
n(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 9900*a**3*tan(e/2 + f*x/2)**3/(693*c**6*f*tan
(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/
2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2
+ f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f
*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 5918*a**3*tan(e/2 + f*x/2)**2/(693*c**6*f*tan(e/2 + f*
x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2
)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)*
*5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2
+ 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 352*a**3*tan(e/2 + f*x/2)/(693*c**6*f*tan(e/2 + f*x/2)**11 - 76
23*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690
*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c
**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f
*tan(e/2 + f*x/2) - 693*c**6*f) - 158*a**3/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10
+ 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 -
320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 11
4345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f
), Ne(f, 0)), (x*(a*sin(e) + a)**3/(-c*sin(e) + c)**6, True))
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Giac [A]
time = 0.48, size = 196, normalized size = 1.94 \begin {gather*} -\frac {2 \, {\left (693 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 1386 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 8085 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 10626 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 21252 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 15246 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15444 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4950 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2959 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 176 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 79 \, a^{3}\right )}}{693 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}} \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))^6,x, algorithm="giac")
[Out]
-2/693*(693*a^3*tan(1/2*f*x + 1/2*e)^10 - 1386*a^3*tan(1/2*f*x + 1/2*e)^9 + 8085*a^3*tan(1/2*f*x + 1/2*e)^8 -
10626*a^3*tan(1/2*f*x + 1/2*e)^7 + 21252*a^3*tan(1/2*f*x + 1/2*e)^6 - 15246*a^3*tan(1/2*f*x + 1/2*e)^5 + 15444
*a^3*tan(1/2*f*x + 1/2*e)^4 - 4950*a^3*tan(1/2*f*x + 1/2*e)^3 + 2959*a^3*tan(1/2*f*x + 1/2*e)^2 - 176*a^3*tan(
1/2*f*x + 1/2*e) + 79*a^3)/(c^6*f*(tan(1/2*f*x + 1/2*e) - 1)^11)
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Mupad [B]
time = 9.34, size = 143, normalized size = 1.42 \begin {gather*} -\frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {6635\,\cos \left (e+f\,x\right )}{16}+\frac {13629\,\sin \left (e+f\,x\right )}{16}+565\,\cos \left (2\,e+2\,f\,x\right )-\frac {3527\,\cos \left (3\,e+3\,f\,x\right )}{32}-29\,\cos \left (4\,e+4\,f\,x\right )+\frac {81\,\cos \left (5\,e+5\,f\,x\right )}{32}-\frac {1617\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {5049\,\sin \left (3\,e+3\,f\,x\right )}{32}+\frac {407\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {77\,\sin \left (5\,e+5\,f\,x\right )}{32}-922\right )}{22176\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^{11}} \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))^6,x)
[Out]
-(2^(1/2)*a^3*cos(e/2 + (f*x)/2)*((6635*cos(e + f*x))/16 + (13629*sin(e + f*x))/16 + 565*cos(2*e + 2*f*x) - (3
527*cos(3*e + 3*f*x))/32 - 29*cos(4*e + 4*f*x) + (81*cos(5*e + 5*f*x))/32 - (1617*sin(2*e + 2*f*x))/8 - (5049*
sin(3*e + 3*f*x))/32 + (407*sin(4*e + 4*f*x))/16 + (77*sin(5*e + 5*f*x))/32 - 922))/(22176*c^6*f*cos(e/2 + pi/
4 + (f*x)/2)^11)
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