3.174 \(\int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+m} \, dx\)
Optimal. Leaf size=57 \[ -\frac{g \log (1-\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{c f} \]
[Out]
-((g*Log[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m)/(c*f*(g*Cos[e + f*x])^(2*m)))
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Rubi [A] time = 0.227553, antiderivative size = 57, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used =
{2843, 12, 2667, 31} \[ -\frac{g \log (1-\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{c f} \]
Antiderivative was successfully verified.
[In]
Int[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + m),x]
[Out]
-((g*Log[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m)/(c*f*(g*Cos[e + f*x])^(2*m)))
Rule 2843
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*(c + d*Sin[e
+ f*x])^FracPart[m])/(g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m])), Int[(g*Cos[e + f*x])^(2*m + p)/(c +
d*Sin[e + f*x]), x], 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] && EqQ[m - n - 1, 0]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 2667
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/2])
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+m} \, dx &=\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \frac{g \cos (e+f x)}{c-c \sin (e+f x)} \, dx\\ &=\left (g (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx\\ &=-\frac{\left (g (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c f}\\ &=-\frac{g (g \cos (e+f x))^{-2 m} \log (1-\sin (e+f x)) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m}{c f}\\ \end{align*}
Mathematica [B] time = 80.9138, size = 132, normalized size = 2.32 \[ \frac{g 2^{m+1} \cos ^{2 m}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{-2 m} \left (\log \left (\csc ^2\left (\frac{1}{8} (2 e+2 f x+3 \pi )\right )\right )-\log \left (\tan \left (\frac{1}{8} (-2 e-2 f x+\pi )\right )\right )\right )}{c f} \]
Antiderivative was successfully verified.
[In]
Integrate[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + m),x]
[Out]
(2^(1 + m)*g*Cos[(2*e + Pi + 2*f*x)/4]^(2*m)*(Log[Csc[(2*e + 3*Pi + 2*f*x)/8]^2] - Log[Tan[(-2*e + Pi - 2*f*x)
/8]])*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^m)/(c*f*(g*Cos[e + f*x])^(2*m)*(Cos[(e + f*x)/2] - Sin[(e
+ f*x)/2])^(2*m))
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Maple [C] time = 5.367, size = 9871, normalized size = 173.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-1+m),x)
[Out]
result too large to display
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \cos \left (f x + e\right )\right )^{-2 \, m + 1}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{m - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-1+m),x, algorithm="maxima")
[Out]
integrate((g*cos(f*x + e))^(-2*m + 1)*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(m - 1), x)
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Fricas [A] time = 1.72801, size = 69, normalized size = 1.21 \begin{align*} -\frac{a \left (\frac{a c}{g^{2}}\right )^{m - 1} \log \left (-\sin \left (f x + e\right ) + 1\right )}{f g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-1+m),x, algorithm="fricas")
[Out]
-a*(a*c/g^2)^(m - 1)*log(-sin(f*x + e) + 1)/(f*g)
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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))**(1-2*m)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(-1+m),x)
[Out]
Timed out
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Giac [B] time = 27.2214, size = 1314, normalized size = 23.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(1-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-1+m),x, algorithm="giac")
[Out]
1/2*(4*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*floor(1/4*(pi + 2*f*
x - 4*pi*floor(1/2*(pi + f*x + e)/pi) + 2*e)/pi)*tan(1/4*pi + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*
sgn(c) + 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4
*pi*sgn(c) + 1/4*pi*sgn(g))^2 + 4*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(ab
s(g)))*floor(1/2*(pi + f*x + e)/pi)*tan(1/4*pi + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*sgn(c) + 1) +
1/4*pi*m*sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4*pi*sgn(c) +
1/4*pi*sgn(g))^2 + 2*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*sgn(ta
n(1/2*f*x + 1/2*e)^2 - 1)*tan(1/4*pi + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*sgn(c) + 1) + 1/4*pi*m*
sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4*pi*sgn(c) + 1/4*pi*sgn
(g))^2 + 3*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*tan(1/4*pi + pi*
m*floor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*sgn(c) + 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn(g)
+ 1/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4*pi*sgn(c) + 1/4*pi*sgn(g))^2 - 4*pi*e^(m*log(abs(a)) + m*log(abs
(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*floor(1/4*(pi + 2*f*x - 4*pi*floor(1/2*(pi + f*x + e)/pi)
+ 2*e)/pi) - 4*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*floor(1/2*(p
i + f*x + e)/pi) - 2*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*sgn(ta
n(1/2*f*x + 1/2*e)^2 - 1) - 4*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)))*
log(2*(tan(1/2*f*x + 1/2*e)^2 - 2*tan(1/2*f*x + 1/2*e) + 1)/(tan(1/2*f*x + 1/2*e)^2 + 1))*tan(1/4*pi + pi*m*fl
oor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*sgn(c) + 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn(g) + 1
/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4*pi*sgn(c) + 1/4*pi*sgn(g)) - 2*e^(m*log(abs(a)) + m*log(abs(c)) - 2*
m*log(abs(g)) - log(abs(c)) + log(abs(g)) + 1)*tan(1/4*pi + pi*m*floor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*sg
n(c) + 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn(g) + 1/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4*p
i*sgn(c) + 1/4*pi*sgn(g))^2 - 3*pi*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(
g))) + 2*e^(m*log(abs(a)) + m*log(abs(c)) - 2*m*log(abs(g)) - log(abs(c)) + log(abs(g)) + 1))/(f*tan(1/4*pi +
pi*m*floor(-1/4*sgn(a) + 1/2) + pi*m*floor(-1/4*sgn(c) + 1) + 1/4*pi*m*sgn(a) + 1/4*pi*m*sgn(c) - 1/2*pi*m*sgn
(g) + 1/4*pi*m - pi*floor(-1/4*sgn(c) + 1) - 1/4*pi*sgn(c) + 1/4*pi*sgn(g))^2 + f)