3.4.7 \(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx\) [307]

Optimal. Leaf size=145 \[ \frac {256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \]

[Out]

256/3003*a^3*c^7*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+64/429*a^3*c^6*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(5/2)+24
/143*a^3*c^5*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(3/2)+2/13*a^3*c^4*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.23, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(256*a^3*c^7*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*c^6*Cos[e + f*x]^7)/(429*f*(c - c*S
in[e + f*x])^(5/2)) + (24*a^3*c^5*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^3*c^4*Cos[e + f*x]
^7)/(13*f*Sqrt[c - c*Sin[e + f*x]])

Rule 2752

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]

Rule 2753

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]

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 (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{13} \left (12 a^3 c^4\right ) \int \frac {\cos ^6(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{143} \left (96 a^3 c^5\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{429} \left (128 a^3 c^6\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac {256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 5.40, size = 112, normalized size = 0.77 \begin {gather*} \frac {a^3 c^3 \cos ^6(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (5230-1890 \cos (2 (e+f x))-6377 \sin (e+f x)+231 \sin (3 (e+f x)))}{6006 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c^3*Cos[e + f*x]^6*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(5230 - 1890*Cos[2*(e +
f*x)] - 6377*Sin[e + f*x] + 231*Sin[3*(e + f*x)]))/(6006*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)

________________________________________________________________________________________

Maple [A]
time = 2.32, size = 81, normalized size = 0.56

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (231 \left (\sin ^{3}\left (f x +e \right )\right )-945 \left (\sin ^{2}\left (f x +e \right )\right )+1421 \sin \left (f x +e \right )-835\right )}{3003 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(81\)

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))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/3003*(sin(f*x+e)-1)*c^4*(1+sin(f*x+e))^4*a^3*(231*sin(f*x+e)^3-945*sin(f*x+e)^2+1421*sin(f*x+e)-835)/cos(f*x
+e)/(c-c*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (137) = 274\).
time = 0.34, size = 282, normalized size = 1.94 \begin {gather*} \frac {2 \, {\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{7} - 21 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 28 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} - 40 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} - 128 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3} + {\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 252 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} + 280 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 320 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 384 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3003 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\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))^(7/2),x, algorithm="fricas")

[Out]

2/3003*(231*a^3*c^3*cos(f*x + e)^7 - 21*a^3*c^3*cos(f*x + e)^6 + 28*a^3*c^3*cos(f*x + e)^5 - 40*a^3*c^3*cos(f*
x + e)^4 + 64*a^3*c^3*cos(f*x + e)^3 - 128*a^3*c^3*cos(f*x + e)^2 + 512*a^3*c^3*cos(f*x + e) + 1024*a^3*c^3 +
(231*a^3*c^3*cos(f*x + e)^6 + 252*a^3*c^3*cos(f*x + e)^5 + 280*a^3*c^3*cos(f*x + e)^4 + 320*a^3*c^3*cos(f*x +
e)^3 + 384*a^3*c^3*cos(f*x + e)^2 + 512*a^3*c^3*cos(f*x + e) + 1024*a^3*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e
) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5985 deep

________________________________________________________________________________________

Giac [A]
time = 0.60, size = 257, normalized size = 1.77 \begin {gather*} -\frac {\sqrt {2} {\left (60060 \, a^{3} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15015 \, a^{3} c^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 9009 \, a^{3} c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2574 \, a^{3} c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2002 \, a^{3} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 273 \, a^{3} c^{3} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 231 \, a^{3} c^{3} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{96096 \, f} \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))^(7/2),x, algorithm="giac")

[Out]

-1/96096*sqrt(2)*(60060*a^3*c^3*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 15015*a^3
*c^3*cos(-3/4*pi + 3/2*f*x + 3/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 9009*a^3*c^3*cos(-5/4*pi + 5/2*f*x +
5/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 2574*a^3*c^3*cos(-7/4*pi + 7/2*f*x + 7/2*e)*sgn(sin(-1/4*pi + 1/
2*f*x + 1/2*e)) + 2002*a^3*c^3*cos(-9/4*pi + 9/2*f*x + 9/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 273*a^3*c^
3*cos(-11/4*pi + 11/2*f*x + 11/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 231*a^3*c^3*cos(-13/4*pi + 13/2*f*x
+ 13/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(c)/f

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \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))^(7/2),x)

[Out]

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

________________________________________________________________________________________