3.4.72 \(\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx\) [372]

Optimal. Leaf size=89 \[ \frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{10 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f} \]

[Out]

1/10*c^2*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(1/2)+1/5*c*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)*(c
-c*sin(f*x+e))^(1/2)/f

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Rubi [A]
time = 0.12, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817} \begin {gather*} \frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]

[Out]

(c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*(a + a*Sin[e +
f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f)

Rule 2817

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rule 2819

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Dist[a*((2*m - 1)/(
m + n)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m
]) &&  !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f}+\frac {1}{5} (2 c) \int (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{10 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 93, normalized size = 1.04 \begin {gather*} -\frac {a^3 c \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (20 \cos (2 (e+f x))+5 \cos (4 (e+f x))-70 \sin (e+f x)-5 \sin (3 (e+f x))+\sin (5 (e+f x)))}{80 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]

[Out]

-1/80*(a^3*c*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(20*Cos[2*(e + f*x)] + 5*Cos[4*(
e + f*x)] - 70*Sin[e + f*x] - 5*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/f

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Maple [A]
time = 16.94, size = 107, normalized size = 1.20

method	result	size
default	\(\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \left (2 \left (\cos ^{6}\left (f x +e \right )\right )-\sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+2 \left (\cos ^{4}\left (f x +e \right )\right )-3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-6 \sin \left (f x +e \right )+6\right )}{10 f \cos \left (f x +e \right )^{7}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/10/f*(-c*(sin(f*x+e)-1))^(3/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(7/2)*(2*cos(f*x+e)^6-sin(f*x+e)*cos(f*x+e)^4+2
*cos(f*x+e)^4-3*sin(f*x+e)*cos(f*x+e)^2-6*sin(f*x+e)+6)/cos(f*x+e)^7

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(3/2), x)

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Fricas [A]
time = 0.34, size = 108, normalized size = 1.21 \begin {gather*} -\frac {{\left (5 \, a^{3} c \cos \left (f x + e\right )^{4} - 5 \, a^{3} c + 2 \, {\left (a^{3} c \cos \left (f x + e\right )^{4} - 2 \, a^{3} c \cos \left (f x + e\right )^{2} - 4 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{10 \, f \cos \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/10*(5*a^3*c*cos(f*x + e)^4 - 5*a^3*c + 2*(a^3*c*cos(f*x + e)^4 - 2*a^3*c*cos(f*x + e)^2 - 4*a^3*c)*sin(f*x
+ e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [A]
time = 0.50, size = 110, normalized size = 1.24 \begin {gather*} \frac {8 \, {\left (4 \, a^{3} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 5 \, a^{3} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{5 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

8/5*(4*a^3*c*cos(-1/4*pi + 1/2*f*x + 1/2*e)^10*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x +
1/2*e)) - 5*a^3*c*cos(-1/4*pi + 1/2*f*x + 1/2*e)^8*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*
f*x + 1/2*e)))*sqrt(a)*sqrt(c)/f

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Mupad [B]
time = 9.19, size = 109, normalized size = 1.22 \begin {gather*} -\frac {a^3\,c\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (20\,\cos \left (e+f\,x\right )+25\,\cos \left (3\,e+3\,f\,x\right )+5\,\cos \left (5\,e+5\,f\,x\right )-75\,\sin \left (2\,e+2\,f\,x\right )-4\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )\right )}{80\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2),x)

[Out]

-(a^3*c*(a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(20*cos(e + f*x) + 25*cos(3*e + 3*f*x) + 5*
cos(5*e + 5*f*x) - 75*sin(2*e + 2*f*x) - 4*sin(4*e + 4*f*x) + sin(6*e + 6*f*x)))/(80*f*(cos(2*e + 2*f*x) + 1))

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