3.3.46 \(\int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx\) [246]
Optimal. Leaf size=132 \[ \frac {a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac {2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac {2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5} \]
[Out]
1/11*a^2*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^8+1/33*a^2*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^7+2/231*a^2*cos(f*x+
e)^5/f/(c-c*sin(f*x+e))^6+2/1155*a^2*cos(f*x+e)^5/c/f/(c-c*sin(f*x+e))^5
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Rubi [A]
time = 0.17, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 5, 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^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}+\frac {2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac {a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^6,x]
[Out]
(a^2*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*c*Cos[e + f*x]^5)/(33*f*(c - c*Sin[e + f*x])^7)
+ (2*a^2*Cos[e + f*x]^5)/(231*f*(c - c*Sin[e + f*x])^6) + (2*a^2*Cos[e + f*x]^5)/(1155*c*f*(c - c*Sin[e + f*x]
)^5)
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))^2}{(c-c \sin (e+f x))^6} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {1}{11} \left (3 a^2 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac {1}{33} \left (2 a^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac {2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac {\left (2 a^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx}{231 c}\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac {2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac {2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 133, normalized size = 1.01 \begin {gather*} \frac {a^2 \left (2079 \cos \left (\frac {1}{2} (e+f x)\right )-825 \cos \left (\frac {3}{2} (e+f x)\right )-55 \cos \left (\frac {7}{2} (e+f x)\right )+\cos \left (\frac {11}{2} (e+f x)\right )+2541 \sin \left (\frac {1}{2} (e+f x)\right )+1155 \sin \left (\frac {3}{2} (e+f x)\right )-165 \sin \left (\frac {5}{2} (e+f x)\right )+11 \sin \left (\frac {9}{2} (e+f x)\right )\right )}{9240 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])^2/(c - c*Sin[e + f*x])^6,x]
[Out]
(a^2*(2079*Cos[(e + f*x)/2] - 825*Cos[(3*(e + f*x))/2] - 55*Cos[(7*(e + f*x))/2] + Cos[(11*(e + f*x))/2] + 254
1*Sin[(e + f*x)/2] + 1155*Sin[(3*(e + f*x))/2] - 165*Sin[(5*(e + f*x))/2] + 11*Sin[(9*(e + f*x))/2]))/(9240*c^
6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11)
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Maple [A]
time = 0.50, size = 178, normalized size = 1.35
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {4 i a^{2} \left (2079 i {\mathrm e}^{6 i \left (f x +e \right )}+1155 \,{\mathrm e}^{7 i \left (f x +e \right )}-825 i {\mathrm e}^{4 i \left (f x +e \right )}-2541 \,{\mathrm e}^{5 i \left (f x +e \right )}-55 i {\mathrm e}^{2 i \left (f x +e \right )}+165 \,{\mathrm e}^{3 i \left (f x +e \right )}+i-11 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{1155 f \,c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11}}\) |
\(110\) |
| derivativedivides |
\(\frac {2 a^{2} \left (-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {88}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {30}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {512}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {128}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {2376}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {288}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {932}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}\right )}{f \,c^{6}}\) |
\(178\) |
| default |
\(\frac {2 a^{2} \left (-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {88}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {30}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {512}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {128}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {2376}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {288}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {932}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}\right )}{f \,c^{6}}\) |
\(178\) |
| norman |
\(\frac {\frac {52 a^{2} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {304 a^{2}}{1155 c f}-\frac {2 a^{2} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {6 a^{2} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {28 a^{2} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {94 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{105 c f}-\frac {574 a^{2} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}+\frac {714 a^{2} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {17534 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}+\frac {2228 a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {3466 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{385 c f}-\frac {3524 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{55 c f}+\frac {6056 a^{2} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}-\frac {7016 a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}+\frac {10138 a^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) |
\(351\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)
[Out]
2/f*a^2/c^6*(-7/(tan(1/2*f*x+1/2*e)-1)^2-88/(tan(1/2*f*x+1/2*e)-1)^4-1/(tan(1/2*f*x+1/2*e)-1)-30/(tan(1/2*f*x+
1/2*e)-1)^3-512/3/(tan(1/2*f*x+1/2*e)-1)^9-292/(tan(1/2*f*x+1/2*e)-1)^6-128/11/(tan(1/2*f*x+1/2*e)-1)^11-2376/
7/(tan(1/2*f*x+1/2*e)-1)^7-288/(tan(1/2*f*x+1/2*e)-1)^8-932/5/(tan(1/2*f*x+1/2*e)-1)^5-64/(tan(1/2*f*x+1/2*e)-
1)^10)
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1452 vs.
\(2 (136) = 272\).
time = 0.35, size = 1452, normalized size = 11.00 \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))^2/(c-c*sin(f*x+e))^6,x, algorithm="maxima")
[Out]
-2/3465*(5*a^2*(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) - 6*a^2*(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) + 4*a^2*(25
3*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/(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))/f
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs.
\(2 (136) = 272\).
time = 0.32, size = 356, normalized size = 2.70 \begin {gather*} -\frac {2 \, a^{2} \cos \left (f x + e\right )^{6} + 12 \, a^{2} \cos \left (f x + e\right )^{5} - 25 \, a^{2} \cos \left (f x + e\right )^{4} - 70 \, a^{2} \cos \left (f x + e\right )^{3} - 245 \, a^{2} \cos \left (f x + e\right )^{2} + 210 \, a^{2} \cos \left (f x + e\right ) + 420 \, a^{2} - {\left (2 \, a^{2} \cos \left (f x + e\right )^{5} - 10 \, a^{2} \cos \left (f x + e\right )^{4} - 35 \, a^{2} \cos \left (f x + e\right )^{3} + 35 \, a^{2} \cos \left (f x + e\right )^{2} - 210 \, a^{2} \cos \left (f x + e\right ) - 420 \, a^{2}\right )} \sin \left (f x + e\right )}{1155 \, {\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))^2/(c-c*sin(f*x+e))^6,x, algorithm="fricas")
[Out]
-1/1155*(2*a^2*cos(f*x + e)^6 + 12*a^2*cos(f*x + e)^5 - 25*a^2*cos(f*x + e)^4 - 70*a^2*cos(f*x + e)^3 - 245*a^
2*cos(f*x + e)^2 + 210*a^2*cos(f*x + e) + 420*a^2 - (2*a^2*cos(f*x + e)^5 - 10*a^2*cos(f*x + e)^4 - 35*a^2*cos
(f*x + e)^3 + 35*a^2*cos(f*x + e)^2 - 210*a^2*cos(f*x + e) - 420*a^2)*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.
\(2 (117) = 234\).
time = 39.37, size = 2509, normalized size = 19.01 \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))**2/(c-c*sin(f*x+e))**6,x)
[Out]
Piecewise((-2310*a**2*tan(e/2 + f*x/2)**10/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**
10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7
- 533610*c**6*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 +
190575*c**6*f*tan(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c*
*6*f) + 6930*a**2*tan(e/2 + f*x/2)**9/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 +
63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533
610*c**6*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 19057
5*c**6*f*tan(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f)
- 27720*a**2*tan(e/2 + f*x/2)**8/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 6352
5*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*
c**6*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c*
*6*f*tan(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) + 4
6200*a**2*tan(e/2 + f*x/2)**7/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c*
*6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6
*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f
*tan(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) - 74844
*a**2*tan(e/2 + f*x/2)**6/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f
*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*t
an(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan
(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) + 65604*a**
2*tan(e/2 + f*x/2)**5/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan
(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e
/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2
+ f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) - 54120*a**2*ta
n(e/2 + f*x/2)**4/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2
+ f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e/2 +
f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f
*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) + 22440*a**2*tan(e/
2 + f*x/2)**3/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f
*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e/2 + f*x
/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2
)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) - 9790*a**2*tan(e/2 + f
*x/2)**2/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)
**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e/2 + f*x/2)**
6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2)**3
- 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) + 1034*a**2*tan(e/2 + f*x/2)
/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 19
0575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e/2 + f*x/2)**6 + 5336
10*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2)**3 - 63525*
c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) - 304*a**2/(1155*c**6*f*tan(e/2 + f*
x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/
2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)
**5 - 381150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2
+ 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f), Ne(f, 0)), (x*(a*sin(e) + a)**2/(-c*sin(e) + c)**6, True))
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Giac [A]
time = 0.49, size = 196, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (1155 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 3465 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 13860 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 23100 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 37422 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 32802 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 27060 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 11220 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4895 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 517 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 152 \, a^{2}\right )}}{1155 \, 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))^2/(c-c*sin(f*x+e))^6,x, algorithm="giac")
[Out]
-2/1155*(1155*a^2*tan(1/2*f*x + 1/2*e)^10 - 3465*a^2*tan(1/2*f*x + 1/2*e)^9 + 13860*a^2*tan(1/2*f*x + 1/2*e)^8
- 23100*a^2*tan(1/2*f*x + 1/2*e)^7 + 37422*a^2*tan(1/2*f*x + 1/2*e)^6 - 32802*a^2*tan(1/2*f*x + 1/2*e)^5 + 27
060*a^2*tan(1/2*f*x + 1/2*e)^4 - 11220*a^2*tan(1/2*f*x + 1/2*e)^3 + 4895*a^2*tan(1/2*f*x + 1/2*e)^2 - 517*a^2*
tan(1/2*f*x + 1/2*e) + 152*a^2)/(c^6*f*(tan(1/2*f*x + 1/2*e) - 1)^11)
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Mupad [B]
time = 9.36, size = 143, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {2}\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (697\,\cos \left (e+f\,x\right )+\frac {7623\,\sin \left (e+f\,x\right )}{4}+\frac {3977\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {3203\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {461\,\cos \left (4\,e+4\,f\,x\right )}{8}+\frac {75\,\cos \left (5\,e+5\,f\,x\right )}{16}-462\,\sin \left (2\,e+2\,f\,x\right )-\frac {4983\,\sin \left (3\,e+3\,f\,x\right )}{16}+\frac {187\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {77\,\sin \left (5\,e+5\,f\,x\right )}{16}-\frac {12721}{8}\right )}{36960\,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))^2/(c - c*sin(e + f*x))^6,x)
[Out]
-(2^(1/2)*a^2*cos(e/2 + (f*x)/2)*(697*cos(e + f*x) + (7623*sin(e + f*x))/4 + (3977*cos(2*e + 2*f*x))/4 - (3203
*cos(3*e + 3*f*x))/16 - (461*cos(4*e + 4*f*x))/8 + (75*cos(5*e + 5*f*x))/16 - 462*sin(2*e + 2*f*x) - (4983*sin
(3*e + 3*f*x))/16 + (187*sin(4*e + 4*f*x))/4 + (77*sin(5*e + 5*f*x))/16 - 12721/8))/(36960*c^6*f*cos(e/2 + pi/
4 + (f*x)/2)^11)
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