3.3.69 \(\int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx\) [269]

Optimal. Leaf size=148 \[ \frac {105 c^5 x}{2 a^2}+\frac {35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac {105 c^5 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2} \]

[Out]

105/2*c^5*x/a^2+35*c^5*cos(f*x+e)^3/a^2/f+105/2*c^5*cos(f*x+e)*sin(f*x+e)/a^2/f-2/3*a^4*c^5*cos(f*x+e)^9/f/(a+
a*sin(f*x+e))^6+6*a^2*c^5*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^4+42*c^5*cos(f*x+e)^5/f/(a+a*sin(f*x+e))^2

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Rubi [A]
time = 0.17, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2759, 2761, 2715, 8} \begin {gather*} -\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac {35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac {105 c^5 \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac {105 c^5 x}{2 a^2}+\frac {42 c^5 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]

[Out]

(105*c^5*x)/(2*a^2) + (35*c^5*Cos[e + f*x]^3)/(a^2*f) + (105*c^5*Cos[e + f*x]*Sin[e + f*x])/(2*a^2*f) - (2*a^4
*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^6) + (6*a^2*c^5*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) + (4
2*c^5*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

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 {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx &=\left (a^5 c^5\right ) \int \frac {\cos ^{10}(e+f x)}{(a+a \sin (e+f x))^7} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}-\left (3 a^3 c^5\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^5} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\left (21 a c^5\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}+\frac {\left (105 c^5\right ) \int \frac {\cos ^4(e+f x)}{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac {35 c^5 \cos ^3(e+f x)}{a^2 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}+\frac {\left (105 c^5\right ) \int \cos ^2(e+f x) \, dx}{a^2}\\ &=\frac {35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac {105 c^5 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}+\frac {\left (105 c^5\right ) \int 1 \, dx}{2 a^2}\\ &=\frac {105 c^5 x}{2 a^2}+\frac {35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac {105 c^5 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 276, normalized size = 1.86 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (256 \sin \left (\frac {1}{2} (e+f x)\right )-128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-1664 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+630 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+285 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-\cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-21 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (a+a \sin (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2]) - 1664*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 630*(e + f*x)*(Cos[(e + f*
x)/2] + Sin[(e + f*x)/2])^3 + 285*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - Cos[3*(e + f*x)]*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 21*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*f*(Cos
[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(a + a*Sin[e + f*x])^2)

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Maple [A]
time = 0.41, size = 138, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {2 c^{5} \left (-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {48}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+\frac {\frac {7 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+23 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+48 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+\frac {71}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {105 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{2}}\)	\(138\)
default	\(\frac {2 c^{5} \left (-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {48}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}+\frac {\frac {7 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+23 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+48 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+\frac {71}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {105 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{2}}\)	\(138\)
risch	\(\frac {105 c^{5} x}{2 a^{2}}+\frac {7 i c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 f \,a^{2}}+\frac {95 c^{5} {\mathrm e}^{i \left (f x +e \right )}}{8 a^{2} f}+\frac {95 c^{5} {\mathrm e}^{-i \left (f x +e \right )}}{8 a^{2} f}-\frac {7 i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 f \,a^{2}}+\frac {256 i c^{5} {\mathrm e}^{i \left (f x +e \right )}+160 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {416 c^{5}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}-\frac {c^{5} \cos \left (3 f x +3 e \right )}{12 a^{2} f}\)	\(170\)
norman	\(\frac {\frac {3675 c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {2625 c^{5} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {840 c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {420 c^{5} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {315 c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {105 c^{5} x}{2 a}+\frac {323 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {105 c^{5} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {103 c^{5} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2983 c^{5} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {5593 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {7385 c^{5} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {10694 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {9490 c^{5} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {10270 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {6596 c^{5} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {1655 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2311 c^{5} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {420 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {840 c^{5} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {2625 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {3675 c^{5} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {2100 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {2100 c^{5} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {391 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {494 c^{5}}{3 a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\)	\(574\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/f*c^5/a^2*(-64/3/(tan(1/2*f*x+1/2*e)+1)^3+32/(tan(1/2*f*x+1/2*e)+1)^2+48/(tan(1/2*f*x+1/2*e)+1)+(7/2*tan(1/2
*f*x+1/2*e)^5+23*tan(1/2*f*x+1/2*e)^4+48*tan(1/2*f*x+1/2*e)^2-7/2*tan(1/2*f*x+1/2*e)+71/3)/(1+tan(1/2*f*x+1/2*
e)^2)^3+105/2*arctan(tan(1/2*f*x+1/2*e)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (151) = 302\).
time = 0.57, size = 1418, normalized size = 9.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

1/3*(5*c^5*((75*sin(f*x + e)/(cos(f*x + e) + 1) + 97*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 126*sin(f*x + e)^3/
(cos(f*x + e) + 1)^3 + 98*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 21*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 5*a^2*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 7*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5
*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos
(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 2*c^5*((57*sin(f*x + e)/(cos(f*x + e) +
1) + 99*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 155*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 153*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 + 135*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 85*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 45*sin(
f*x + e)^7/(cos(f*x + e) + 1)^7 + 15*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 24)/(a^2 + 3*a^2*sin(f*x + e)/(cos(
f*x + e) + 1) + 6*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12*a^
2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 12*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 10*a^2*sin(f*x + e)^6/(co
s(f*x + e) + 1)^6 + 6*a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*a^2*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^
2*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 40*c^5*((12*sin(f*x
+ e)/(cos(f*x + e) + 1) + 11*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 4*a^2*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 + 4*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 +
a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 20*c^5*((9*sin(f*x
+ e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e)
+ 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 3*arctan(sin(f*x
+ e)/(cos(f*x + e) + 1))/a^2) - 2*c^5*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1
)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*
x + e)^3/(cos(f*x + e) + 1)^3) + 10*c^5*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^2*sin(f*x + e)/(cos
(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3))/f

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Fricas [A]
time = 0.33, size = 252, normalized size = 1.70 \begin {gather*} -\frac {2 \, c^{5} \cos \left (f x + e\right )^{5} + 19 \, c^{5} \cos \left (f x + e\right )^{4} - 106 \, c^{5} \cos \left (f x + e\right )^{3} + 630 \, c^{5} f x - 64 \, c^{5} - 7 \, {\left (45 \, c^{5} f x - 77 \, c^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (315 \, c^{5} f x + 598 \, c^{5}\right )} \cos \left (f x + e\right ) - {\left (2 \, c^{5} \cos \left (f x + e\right )^{4} - 17 \, c^{5} \cos \left (f x + e\right )^{3} - 630 \, c^{5} f x - 123 \, c^{5} \cos \left (f x + e\right )^{2} - 64 \, c^{5} - {\left (315 \, c^{5} f x + 662 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/6*(2*c^5*cos(f*x + e)^5 + 19*c^5*cos(f*x + e)^4 - 106*c^5*cos(f*x + e)^3 + 630*c^5*f*x - 64*c^5 - 7*(45*c^5
*f*x - 77*c^5)*cos(f*x + e)^2 + (315*c^5*f*x + 598*c^5)*cos(f*x + e) - (2*c^5*cos(f*x + e)^4 - 17*c^5*cos(f*x
+ e)^3 - 630*c^5*f*x - 123*c^5*cos(f*x + e)^2 - 64*c^5 - (315*c^5*f*x + 662*c^5)*cos(f*x + e))*sin(f*x + e))/(
a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3641 vs. \(2 (144) = 288\).
time = 15.58, size = 3641, normalized size = 24.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))**5/(a+a*sin(f*x+e))**2,x)

[Out]

Piecewise((315*c**5*f*x*tan(e/2 + f*x/2)**9/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36
*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/
2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6
*a**2*f) + 945*c**5*f*x*tan(e/2 + f*x/2)**8/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36
*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/
2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6
*a**2*f) + 1890*c**5*f*x*tan(e/2 + f*x/2)**7/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 3
6*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e
/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) +
6*a**2*f) + 3150*c**5*f*x*tan(e/2 + f*x/2)**6/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 +
36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(
e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) +
6*a**2*f) + 3780*c**5*f*x*tan(e/2 + f*x/2)**5/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 +
36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan
(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2)
+ 6*a**2*f) + 3780*c**5*f*x*tan(e/2 + f*x/2)**4/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8
+ 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*ta
n(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2)
+ 6*a**2*f) + 3150*c**5*f*x*tan(e/2 + f*x/2)**3/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8
+ 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*t
an(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2
) + 6*a**2*f) + 1890*c**5*f*x*tan(e/2 + f*x/2)**2/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**
8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*
tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/
2) + 6*a**2*f) + 945*c**5*f*x*tan(e/2 + f*x/2)/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 +
36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan
(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2)
+ 6*a**2*f) + 315*c**5*f*x/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 +
f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 6
0*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 618*c*
*5*tan(e/2 + f*x/2)**8/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x
/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a*
*2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 1938*c**5*
tan(e/2 + f*x/2)**7/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)
**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*
f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 3386*c**5*tan
(e/2 + f*x/2)**6/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7
+ 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*t
an(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 6054*c**5*tan(e/
2 + f*x/2)**5/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 +
60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(
e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 5802*c**5*tan(e/2 +
f*x/2)**4/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*
a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2
+ f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 6494*c**5*tan(e/2 + f*
x/2)**3/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2...

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Giac [A]
time = 0.46, size = 203, normalized size = 1.37 \begin {gather*} \frac {\frac {315 \, {\left (f x + e\right )} c^{5}}{a^{2}} + \frac {2 \, {\left (309 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 969 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 1693 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 3027 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2901 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 3247 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1995 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1173 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 494 \, c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3} a^{2}}}{6 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

1/6*(315*(f*x + e)*c^5/a^2 + 2*(309*c^5*tan(1/2*f*x + 1/2*e)^8 + 969*c^5*tan(1/2*f*x + 1/2*e)^7 + 1693*c^5*tan
(1/2*f*x + 1/2*e)^6 + 3027*c^5*tan(1/2*f*x + 1/2*e)^5 + 2901*c^5*tan(1/2*f*x + 1/2*e)^4 + 3247*c^5*tan(1/2*f*x
+ 1/2*e)^3 + 1995*c^5*tan(1/2*f*x + 1/2*e)^2 + 1173*c^5*tan(1/2*f*x + 1/2*e) + 494*c^5)/((tan(1/2*f*x + 1/2*e
)^3 + tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e) + 1)^3*a^2))/f

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Mupad [B]
time = 10.86, size = 372, normalized size = 2.51 \begin {gather*} \frac {105\,c^5\,x}{2\,a^2}-\frac {\frac {105\,c^5\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (945\,e+945\,f\,x+2346\right )}{6}\right )-\frac {c^5\,\left (315\,e+315\,f\,x+988\right )}{6}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (945\,e+945\,f\,x+618\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (315\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (1890\,e+1890\,f\,x+1938\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (315\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (1890\,e+1890\,f\,x+3990\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (525\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3150\,e+3150\,f\,x+3386\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (525\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3150\,e+3150\,f\,x+6494\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+5802\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+6054\right )}{6}\right )}{a^2\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c*sin(e + f*x))^5/(a + a*sin(e + f*x))^2,x)

[Out]

(105*c^5*x)/(2*a^2) - ((105*c^5*(e + f*x))/2 + tan(e/2 + (f*x)/2)*((315*c^5*(e + f*x))/2 - (c^5*(945*e + 945*f
*x + 2346))/6) - (c^5*(315*e + 315*f*x + 988))/6 + tan(e/2 + (f*x)/2)^8*((315*c^5*(e + f*x))/2 - (c^5*(945*e +
945*f*x + 618))/6) + tan(e/2 + (f*x)/2)^7*(315*c^5*(e + f*x) - (c^5*(1890*e + 1890*f*x + 1938))/6) + tan(e/2
+ (f*x)/2)^2*(315*c^5*(e + f*x) - (c^5*(1890*e + 1890*f*x + 3990))/6) + tan(e/2 + (f*x)/2)^6*(525*c^5*(e + f*x
) - (c^5*(3150*e + 3150*f*x + 3386))/6) + tan(e/2 + (f*x)/2)^3*(525*c^5*(e + f*x) - (c^5*(3150*e + 3150*f*x +
6494))/6) + tan(e/2 + (f*x)/2)^4*(630*c^5*(e + f*x) - (c^5*(3780*e + 3780*f*x + 5802))/6) + tan(e/2 + (f*x)/2)
^5*(630*c^5*(e + f*x) - (c^5*(3780*e + 3780*f*x + 6054))/6))/(a^2*f*(tan(e/2 + (f*x)/2) + tan(e/2 + (f*x)/2)^2
+ tan(e/2 + (f*x)/2)^3 + 1)^3)

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