∫ (c−c sin(e+fx))5∕2 ___________________________

Optimal. Leaf size=98 \[ -\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f} \]

16/3*c*sec(f*x+e)*(c-c*sin(f*x+e))^(3/2)/a/f+2/3*sec(f*x+e)*(c-c*sin(f*x+e))^(5/2)/a/f-64/3*c^2*sec(f*x+e)*(c-

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Rubi [A]
time = 0.18, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753, 2752} \begin {gather*} -\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f} \end {gather*}

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

(-64*c^2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a*f) + (16*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*a*

f) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f)

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 - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq

Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(

g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[a*((2*m + p - 1)/(m + p)), 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] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

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,

\begin {align*} \int \frac {(c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx &=\frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {8 \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{3 a}\\ &=\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {(32 c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{3 a}\\ &=-\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 102, normalized size = 1.04 \begin {gather*} -\frac {c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (45+\cos (2 (e+f x))+20 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{3 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \end {gather*}

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

-1/3*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(45 + Cos[2*(e + f*x)] + 20*Sin[e + f*x])*Sqrt[c - c*Sin[e + f

*x]])/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))

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method	result	size

default	\(-\frac {2 c^{3} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{2}\left (f x +e \right )-10 \sin \left (f x +e \right )-23\right )}{3 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(59\)

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

-2/3*c^3/a*(sin(f*x+e)-1)*(sin(f*x+e)^2-10*sin(f*x+e)-23)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (92) = 184\).
time = 0.60, size = 208, normalized size = 2.12 \begin {gather*} \frac {2 \, {\left (23 \, c^{\frac {5}{2}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {40 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {23 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )}}{3 \, {\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \end {gather*}

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

2/3*(23*c^(5/2) + 20*c^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 65*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2

+ 40*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 65*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 20*c^(5/2)

*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 23*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)/((a + a*sin(f*x + e)/(c

os(f*x + e) + 1))*f*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2))

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Fricas [A]
time = 0.36, size = 62, normalized size = 0.63 \begin {gather*} -\frac {2 \, {\left (c^{2} \cos \left (f x + e\right )^{2} + 10 \, c^{2} \sin \left (f x + e\right ) + 22 \, c^{2}\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\right )} \end {gather*}

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

-2/3*(c^2*cos(f*x + e)^2 + 10*c^2*sin(f*x + e) + 22*c^2)*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \end {gather*}

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

(Integral(c**2*sqrt(-c*sin(e + f*x) + c)/(sin(e + f*x) + 1), x) + Integral(-2*c**2*sqrt(-c*sin(e + f*x) + c)*s

in(e + f*x)/(sin(e + f*x) + 1), x) + Integral(c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2/(sin(e + f*x) + 1

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (92) = 184\).
time = 0.57, size = 237, normalized size = 2.42 \begin {gather*} \frac {8 \, \sqrt {2} \sqrt {c} {\left (\frac {3 \, c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}} - \frac {5 \, c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {12 \, c^{2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {3 \, c^{2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}}{a {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{3}}\right )}}{3 \, f} \end {gather*}

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

8/3*sqrt(2)*sqrt(c)*(3*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(a*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-

1/4*pi + 1/2*f*x + 1/2*e) + 1) + 1)) - (5*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 12*c^2*(cos(-1/4*pi + 1/2*

f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 3*c^2*(cos(-1/4*p

i + 1/2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2)/(a*((c

os(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 1)^3))/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}

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

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

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3.4.18 \(\int \frac {(c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx\) [318]