3.3.98 \(\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx\) [298]

Optimal. Leaf size=145 \[ \frac {256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \]

[Out]

256/1155*a^2*c^6*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(5/2)+64/231*a^2*c^5*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(3/2)+8/
33*a^2*c^4*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(1/2)+2/11*a^2*c^3*cos(f*x+e)^5*(c-c*sin(f*x+e))^(1/2)/f

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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^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(256*a^2*c^6*Cos[e + f*x]^5)/(1155*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*c^5*Cos[e + f*x]^5)/(231*f*(c - c*S
in[e + f*x])^(3/2)) + (8*a^2*c^4*Cos[e + f*x]^5)/(33*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c^3*Cos[e + f*x]^5*S
qrt[c - c*Sin[e + f*x]])/(11*f)

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))^2 (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{11} \left (12 a^2 c^3\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{33} \left (32 a^2 c^4\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{231} \left (128 a^2 c^5\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1105\) vs. \(2(145)=290\).
time = 6.27, size = 1105, normalized size = 7.62 \begin {gather*} \frac {7 \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {\cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {11 \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {\cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {\cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {\cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {7 \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {3}{2} (e+f x)\right )}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {11 (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {5}{2} (e+f x)\right )}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {7}{2} (e+f x)\right )}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {9}{2} (e+f x)\right )}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {11}{2} (e+f x)\right )}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/
2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - (Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f
*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (11*Cos[(5
*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^
7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])
^(7/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (Cos[(9*(e +
f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(48*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/
2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (7*Sin[(e + f*x)
/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2])^4) + ((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(3*(e + f*x))/2])/(8*f
*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (11*(a + a*Sin[e + f*x])^2
*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)
/2] + Sin[(e + f*x)/2])^4) - ((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(7*(e + f*x))/2])/(112*f*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((a + a*Sin[e + f*x])^2*(c -
c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(48*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])^4) - ((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(176*f*(Cos[
(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

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Maple [A]
time = 1.77, size = 81, normalized size = 0.56

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (105 \left (\sin ^{3}\left (f x +e \right )\right )-455 \left (\sin ^{2}\left (f x +e \right )\right )+755 \sin \left (f x +e \right )-533\right )}{1155 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(sin(f*x+e)-1)*c^4*(1+sin(f*x+e))^3*a^2*(105*sin(f*x+e)^3-455*sin(f*x+e)^2+755*sin(f*x+e)-533)/cos(f*x+
e)/(c-c*sin(f*x+e))^(1/2)/f

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

[Out]

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

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Fricas [A]
time = 0.34, size = 249, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left (105 \, a^{2} c^{3} \cos \left (f x + e\right )^{6} + 245 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} - 20 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} + 32 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} - 64 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \, a^{2} c^{3} \cos \left (f x + e\right ) + 512 \, a^{2} c^{3} - {\left (105 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} - 140 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} - 160 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} - 192 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} - 256 \, a^{2} c^{3} \cos \left (f x + e\right ) - 512 \, a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{1155 \, {\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))^2*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/1155*(105*a^2*c^3*cos(f*x + e)^6 + 245*a^2*c^3*cos(f*x + e)^5 - 20*a^2*c^3*cos(f*x + e)^4 + 32*a^2*c^3*cos(f
*x + e)^3 - 64*a^2*c^3*cos(f*x + e)^2 + 256*a^2*c^3*cos(f*x + e) + 512*a^2*c^3 - (105*a^2*c^3*cos(f*x + e)^5 -
140*a^2*c^3*cos(f*x + e)^4 - 160*a^2*c^3*cos(f*x + e)^3 - 192*a^2*c^3*cos(f*x + e)^2 - 256*a^2*c^3*cos(f*x +
e) - 512*a^2*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4845 deep

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Giac [A]
time = 0.62, size = 222, normalized size = 1.53 \begin {gather*} -\frac {\sqrt {2} {\left (16170 \, a^{2} 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 ) + 2310 \, a^{2} 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 ) - 2541 \, a^{2} 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 ) + 165 \, a^{2} 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 ) + 385 \, a^{2} 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 ) - 105 \, a^{2} 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 )\right )} \sqrt {c}}{18480 \, f} \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))^(7/2),x, algorithm="giac")

[Out]

-1/18480*sqrt(2)*(16170*a^2*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)) + 2310*a^2*
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)) - 2541*a^2*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)) + 165*a^2*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)) + 385*a^2*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)) - 105*a^2*c^3*c
os(-11/4*pi + 11/2*f*x + 11/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(c)/f

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

[Out]

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

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