3.211 \(\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx\)

Optimal. Leaf size=267 \[ \frac{2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c^2 f (2 m+7) \left (4 m^2+16 m+15\right )}+\frac{2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^3 f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right )}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4}}{f (2 m+7)}+\frac{(3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{c f (2 m+5) (2 m+7)} \]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m))/(f*(7 + 2*m)) + ((3*A - 2*B*(2 + m
))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(c*f*(5 + 2*m)*(7 + 2*m)) + (2*(3*A - 2*
B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c^2*f*(7 + 2*m)*(15 + 16*m + 4*
m^2)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^3*f*(5 +
2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2))

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Rubi [A]  time = 0.42687, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2972, 2743, 2742} \[ \frac{2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c^2 f (2 m+7) \left (4 m^2+16 m+15\right )}+\frac{2 (3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^3 f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right )}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4}}{f (2 m+7)}+\frac{(3 A-2 B (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{c f (2 m+5) (2 m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - m),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m))/(f*(7 + 2*m)) + ((3*A - 2*B*(2 + m
))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(c*f*(5 + 2*m)*(7 + 2*m)) + (2*(3*A - 2*
B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c^2*f*(7 + 2*m)*(15 + 16*m + 4*
m^2)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^3*f*(5 +
2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2))

Rule 2972

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] + Dist[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)), 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, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2
- b^2, 0] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] &&  !SumSimplerQ[n, 1])) && NeQ[2*m + 1, 0]

Rule 2743

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*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m
+ 1)), 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, m, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m
, 1] ||  !SumSimplerQ[n, 1])

Rule 2742

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*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] /; FreeQ[{a, b, c, d, e, f
, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -2^(-1)]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-m} \, dx &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac{(3 A-2 B (2+m)) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx}{c (7+2 m)}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac{(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac{(2 (3 A-2 B (2+m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx}{c^2 (5+2 m) (7+2 m)}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac{(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac{2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c^2 f (3+2 m) (5+2 m) (7+2 m)}+\frac{(2 (3 A-2 B (2+m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx}{c^3 (3+2 m) (5+2 m) (7+2 m)}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m}}{f (7+2 m)}+\frac{(3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{c f (5+2 m) (7+2 m)}+\frac{2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c^2 f (3+2 m) (5+2 m) (7+2 m)}+\frac{2 (3 A-2 B (2+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^3 f (1+2 m) (3+2 m) (5+2 m) (7+2 m)}\\ \end{align*}

Mathematica [A]  time = 12.3615, size = 353, normalized size = 1.32 \[ -\frac{2^{-m-18} \cos \left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right ) \csc ^{21}\left (\frac{1}{8} \left (-e-f x+\frac{\pi }{2}\right )\right ) \sec ^7\left (\frac{1}{8} \left (-e-f x+\frac{\pi }{2}\right )\right ) \sin ^{-2 m}\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right ) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{-2 (-m-4)} \left (\left (8 m^2+32 m+29\right ) (3 A-2 B (m+2)) \sin (e+f x)+4 (m+2) (2 B (m+2)-3 A) \cos \left (2 \left (-e-f x+\frac{\pi }{2}\right )\right )+3 A \cos \left (3 \left (-e-f x+\frac{\pi }{2}\right )\right )-16 A m^3-96 A m^2-176 A m-96 A-2 B m \cos \left (3 \left (-e-f x+\frac{\pi }{2}\right )\right )-4 B \cos \left (3 \left (-e-f x+\frac{\pi }{2}\right )\right )+16 B m^2+64 B m+58 B\right )}{f (2 m+1) (2 m+3) (2 m+5) (2 m+7) \left (\cot ^2\left (\frac{1}{8} \left (-e-f x+\frac{\pi }{2}\right )\right )-1\right )^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - m),x]

[Out]

-((2^(-18 - m)*Cos[(-e + Pi/2 - f*x)/2]*Csc[(-e + Pi/2 - f*x)/8]^21*Sec[(-e + Pi/2 - f*x)/8]^7*(a + a*Sin[e +
f*x])^m*(c - c*Sin[e + f*x])^(-4 - m)*(-96*A + 58*B - 176*A*m + 64*B*m - 96*A*m^2 + 16*B*m^2 - 16*A*m^3 + 4*(2
+ m)*(-3*A + 2*B*(2 + m))*Cos[2*(-e + Pi/2 - f*x)] + 3*A*Cos[3*(-e + Pi/2 - f*x)] - 4*B*Cos[3*(-e + Pi/2 - f*
x)] - 2*B*m*Cos[3*(-e + Pi/2 - f*x)] + (29 + 32*m + 8*m^2)*(3*A - 2*B*(2 + m))*Sin[e + f*x]))/(f*(1 + 2*m)*(3
+ 2*m)*(5 + 2*m)*(7 + 2*m)*(-1 + Cot[(-e + Pi/2 - f*x)/8]^2)^7*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2
] - Sin[(e + f*x)/2])^(2*(-4 - m))))

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Maple [F]  time = 0.56, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c-c\sin \left ( fx+e \right ) \right ) ^{-4-m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x)

[Out]

int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.18018, size = 506, normalized size = 1.9 \begin{align*} \frac{{\left (4 \,{\left (2 \, B m^{2} -{\left (3 \, A - 8 \, B\right )} m - 6 \, A + 8 \, B\right )} \cos \left (f x + e\right )^{3} +{\left (8 \, A m^{3} + 12 \,{\left (4 \, A - B\right )} m^{2} + 2 \,{\left (47 \, A - 24 \, B\right )} m + 60 \, A - 45 \, B\right )} \cos \left (f x + e\right ) -{\left (2 \,{\left (2 \, B m - 3 \, A + 4 \, B\right )} \cos \left (f x + e\right )^{3} -{\left (8 \, B m^{3} - 12 \,{\left (A - 4 \, B\right )} m^{2} - 2 \,{\left (24 \, A - 47 \, B\right )} m - 45 \, A + 60 \, B\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 4}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x, algorithm="fricas")

[Out]

(4*(2*B*m^2 - (3*A - 8*B)*m - 6*A + 8*B)*cos(f*x + e)^3 + (8*A*m^3 + 12*(4*A - B)*m^2 + 2*(47*A - 24*B)*m + 60
*A - 45*B)*cos(f*x + e) - (2*(2*B*m - 3*A + 4*B)*cos(f*x + e)^3 - (8*B*m^3 - 12*(A - 4*B)*m^2 - 2*(24*A - 47*B
)*m - 45*A + 60*B)*cos(f*x + e))*sin(f*x + e))*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 4)/(16*f*m^4
+ 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(-4-m),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-m),x, algorithm="giac")

[Out]

Exception raised: AttributeError