3.176 \(\int (g \cos (e+f x))^{3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\)

Optimal. Leaf size=127 \[ -\frac{2 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+2) (-m+n+3)}-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+3)} \]

[Out]

(-2*a^2*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(2 - m + n)*(3 -
m + n)) - (a*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n
))

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Rubi [A]  time = 0.407658, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2846, 2844} \[ -\frac{2 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+2) (-m+n+3)}-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(g*Cos[e + f*x])^(3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]

[Out]

(-2*a^2*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(2 - m + n)*(3 -
m + n)) - (a*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n
))

Rule 2846

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e +
f*x])^n)/(f*g*(m + n + p)), x] + Dist[(a*(2*m + p - 1))/(m + n + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && Eq
Q[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] &&  !LtQ[n, -1] &&  !(IGtQ[Simplify[n + p/2 - 1/2], 0] &&
GtQ[m - n, 0]) &&  !(ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + (3*p)/2 + 1], 0])

Rule 2844

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

Rubi steps

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

Mathematica [A]  time = 1.32339, size = 143, normalized size = 1.13 \[ -\frac{g^3 \cos ^{2 n}(e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4 (g \cos (e+f x))^{-2 m} ((-m+n+2) \sin (e+f x)-m+n+4) (a (\sin (e+f x)+1))^{m-n} \exp (n (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x))-2 \log (\cos (e+f x))))}{f (-m+n+2) (-m+n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(g*Cos[e + f*x])^(3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]

[Out]

-((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*g^3*Cos[e + f*x]^(2*n)*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(a*(1 + Sin[e + f*x]))^(m - n)*(4 - m + n + (2 - m + n)*Sin[e + f*x]))/
(f*(2 - m + n)*(3 - m + n)*(g*Cos[e + f*x])^(2*m)))

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Maple [F]  time = 11.848, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{3-2\,m} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)

[Out]

int((g*cos(f*x+e))^(3-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)

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Maxima [B]  time = 2.21669, size = 655, normalized size = 5.16 \begin{align*} \frac{{\left (a^{m} c^{n} g^{3}{\left (m - n - 4\right )} - \frac{2 \, a^{m} c^{n} g^{3}{\left (m - n - 6\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{a^{m} c^{n} g^{3}{\left (m - n + 12\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{4 \, a^{m} c^{n} g^{3}{\left (m - n + 2\right )} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac{a^{m} c^{n} g^{3}{\left (m - n + 12\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{2 \, a^{m} c^{n} g^{3}{\left (m - n - 6\right )} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac{a^{m} c^{n} g^{3}{\left (m - n - 4\right )} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} e^{\left (2 \, n \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right ) - 2 \, m \log \left (-\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) + m \log \left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right ) - n \log \left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left ({\left (m^{2} - m{\left (2 \, n + 5\right )} + n^{2} + 5 \, n + 6\right )} g^{2 \, m} + \frac{3 \,{\left (m^{2} - m{\left (2 \, n + 5\right )} + n^{2} + 5 \, n + 6\right )} g^{2 \, m} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{3 \,{\left (m^{2} - m{\left (2 \, n + 5\right )} + n^{2} + 5 \, n + 6\right )} g^{2 \, m} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{{\left (m^{2} - m{\left (2 \, n + 5\right )} + n^{2} + 5 \, n + 6\right )} g^{2 \, m} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="maxima")

[Out]

(a^m*c^n*g^3*(m - n - 4) - 2*a^m*c^n*g^3*(m - n - 6)*sin(f*x + e)/(cos(f*x + e) + 1) - a^m*c^n*g^3*(m - n + 12
)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4*a^m*c^n*g^3*(m - n + 2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - a^m*c^
n*g^3*(m - n + 12)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 2*a^m*c^n*g^3*(m - n - 6)*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 + a^m*c^n*g^3*(m - n - 4)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)*e^(2*n*log(sin(f*x + e)/(cos(f*x + e)
+ 1) - 1) - 2*m*log(-sin(f*x + e)/(cos(f*x + e) + 1) + 1) + m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1) - n
*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/(((m^2 - m*(2*n + 5) + n^2 + 5*n + 6)*g^(2*m) + 3*(m^2 - m*(2*n
+ 5) + n^2 + 5*n + 6)*g^(2*m)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*(m^2 - m*(2*n + 5) + n^2 + 5*n + 6)*g^(
2*m)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (m^2 - m*(2*n + 5) + n^2 + 5*n + 6)*g^(2*m)*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6)*f)

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Fricas [B]  time = 1.88668, size = 741, normalized size = 5.83 \begin{align*} \frac{{\left ({\left (m - n - 2\right )} \cos \left (f x + e\right )^{2} +{\left (m - n - 4\right )} \cos \left (f x + e\right ) +{\left ({\left (m - n - 2\right )} \cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - 2\right )} \left (g \cos \left (f x + e\right )\right )^{-2 \, m + 3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} e^{\left (2 \, n \log \left (g \cos \left (f x + e\right )\right ) - n \log \left (a \sin \left (f x + e\right ) + a\right ) + n \log \left (\frac{a c}{g^{2}}\right )\right )}}{2 \, f m^{2} + 2 \, f n^{2} -{\left (f m^{2} + f n^{2} - 5 \, f m -{\left (2 \, f m - 5 \, f\right )} n + 6 \, f\right )} \cos \left (f x + e\right )^{2} - 10 \, f m - 2 \,{\left (2 \, f m - 5 \, f\right )} n +{\left (f m^{2} + f n^{2} - 5 \, f m -{\left (2 \, f m - 5 \, f\right )} n + 6 \, f\right )} \cos \left (f x + e\right ) +{\left (2 \, f m^{2} + 2 \, f n^{2} - 10 \, f m - 2 \,{\left (2 \, f m - 5 \, f\right )} n +{\left (f m^{2} + f n^{2} - 5 \, f m -{\left (2 \, f m - 5 \, f\right )} n + 6 \, f\right )} \cos \left (f x + e\right ) + 12 \, f\right )} \sin \left (f x + e\right ) + 12 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

((m - n - 2)*cos(f*x + e)^2 + (m - n - 4)*cos(f*x + e) + ((m - n - 2)*cos(f*x + e) + 2)*sin(f*x + e) - 2)*(g*c
os(f*x + e))^(-2*m + 3)*(a*sin(f*x + e) + a)^m*e^(2*n*log(g*cos(f*x + e)) - n*log(a*sin(f*x + e) + a) + n*log(
a*c/g^2))/(2*f*m^2 + 2*f*n^2 - (f*m^2 + f*n^2 - 5*f*m - (2*f*m - 5*f)*n + 6*f)*cos(f*x + e)^2 - 10*f*m - 2*(2*
f*m - 5*f)*n + (f*m^2 + f*n^2 - 5*f*m - (2*f*m - 5*f)*n + 6*f)*cos(f*x + e) + (2*f*m^2 + 2*f*n^2 - 10*f*m - 2*
(2*f*m - 5*f)*n + (f*m^2 + f*n^2 - 5*f*m - (2*f*m - 5*f)*n + 6*f)*cos(f*x + e) + 12*f)*sin(f*x + e) + 12*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((g*cos(f*x+e))**(3-2*m)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError