3.5.7 \(\int (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx\) [407]

Optimal. Leaf size=84 \[ -\frac {2^{\frac {1}{2}+m} a^2 c \cos ^3(e+f x) \, _2F_1\left (\frac {3}{2},\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-2+m}}{3 f} \]

[Out]

-1/3*2^(1/2+m)*a^2*c*cos(f*x+e)^3*hypergeom([3/2, 1/2-m],[5/2],1/2-1/2*sin(f*x+e))*(1+sin(f*x+e))^(1/2-m)*(a+a
*sin(f*x+e))^(-2+m)/f

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Rubi [A]
time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2815, 2768, 72, 71} \begin {gather*} -\frac {a^2 c 2^{m+\frac {1}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (\frac {3}{2},\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(2^(1/2 + m)*a^2*c*Cos[e + f*x]^3*Hypergeometric2F1[3/2, 1/2 - m, 5/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e +
f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-2 + m))/f

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 2768

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

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))^m (c-c \sin (e+f x)) \, dx &=(a c) \int \cos ^2(e+f x) (a+a \sin (e+f x))^{-1+m} \, dx\\ &=\frac {\left (a^3 c \cos ^3(e+f x)\right ) \text {Subst}\left (\int \sqrt {a-a x} (a+a x)^{-\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}\\ &=\frac {\left (2^{-\frac {1}{2}+m} a^3 c \cos ^3(e+f x) (a+a \sin (e+f x))^{-2+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m} \sqrt {a-a x} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2}}\\ &=-\frac {2^{\frac {1}{2}+m} a^2 c \cos ^3(e+f x) \, _2F_1\left (\frac {3}{2},\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-2+m}}{3 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.08, size = 285, normalized size = 3.39 \begin {gather*} -\frac {i 2^{-1-2 m} c e^{-i (e+f x)} \left (1+i e^{-i (e+f x)}\right )^{-2 m} \left (-(-1)^{3/4} e^{-\frac {1}{2} i (e+f x)} \left (i+e^{i (e+f x)}\right )\right )^{2 m} \left (e^{2 i (e+f x)} (-1+m) m \, _2F_1\left (-1-m,-2 m;-m;-i e^{-i (e+f x)}\right )+(1+m) \left (m \, _2F_1\left (1-m,-2 m;2-m;-i e^{-i (e+f x)}\right )-2 e^{i (e+f x)} (-1+m) \, _2F_1\left (-2 m,-m;1-m;-i e^{-i (e+f x)}\right )\right )\right ) (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^m \sin ^{-2 m}\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{f (-1+m) m (1+m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*2^(-1 - 2*m)*c*(-(((-1)^(3/4)*(I + E^(I*(e + f*x))))/E^((I/2)*(e + f*x))))^(2*m)*(E^((2*I)*(e + f*x))*(-
1 + m)*m*Hypergeometric2F1[-1 - m, -2*m, -m, (-I)/E^(I*(e + f*x))] + (1 + m)*(m*Hypergeometric2F1[1 - m, -2*m,
2 - m, (-I)/E^(I*(e + f*x))] - 2*E^(I*(e + f*x))*(-1 + m)*Hypergeometric2F1[-2*m, -m, 1 - m, (-I)/E^(I*(e + f
*x))]))*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^m)/(E^(I*(e + f*x))*(1 + I/E^(I*(e + f*x)))^(2*m)*f*(-1 + m
)*m*(1 + m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(2*e + Pi + 2*f*x)/4]^(2*m))

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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e)),x)

[Out]

int((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e)),x)

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

[Out]

-integrate((c*sin(f*x + e) - c)*(a*sin(f*x + e) + a)^m, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(c*sin(f*x + e) - c)*(a*sin(f*x + e) + a)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**m*(c-c*sin(f*x+e)),x)

[Out]

-c*(Integral((a*sin(e + f*x) + a)**m*sin(e + f*x), x) + Integral(-(a*sin(e + f*x) + a)**m, x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(c*sin(f*x + e) - c)*(a*sin(f*x + e) + a)^m, x)

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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 )}^m\,\left (c-c\,\sin \left (e+f\,x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^m*(c - c*sin(e + f*x)),x)

[Out]

int((a + a*sin(e + f*x))^m*(c - c*sin(e + f*x)), x)

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