3.189 \(\int (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\)

Optimal. Leaf size=138 \[ \frac{c 2^{n-\frac{p}{2}+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac{1}{2} (-2 n+p+1)} \, _2F_1\left (\frac{1}{2} (2 m-p+1),\frac{1}{2} (-2 n+p+1);\frac{1}{2} (2 m-p+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m-p+1)} \]

[Out]

(2^(1/2 + n - p/2)*c*Cos[e + f*x]*Hypergeometric2F1[(1 + 2*m - p)/2, (1 - 2*n + p)/2, (3 + 2*m - p)/2, (1 + Si
n[e + f*x])/2]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 - 2*n + p)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f
*x])^(-1 + n))/(f*(1 + 2*m - p))

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Rubi [A]  time = 0.460227, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {2926, 2853, 2689, 70, 69} \[ \frac{c 2^{n-\frac{p}{2}+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac{1}{2} (-2 n+p+1)} \, _2F_1\left (\frac{1}{2} (2 m-p+1),\frac{1}{2} (-2 n+p+1);\frac{1}{2} (2 m-p+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m-p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(1/2 + n - p/2)*c*Cos[e + f*x]*Hypergeometric2F1[(1 + 2*m - p)/2, (1 - 2*n + p)/2, (3 + 2*m - p)/2, (1 + Si
n[e + f*x])/2]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 - 2*n + p)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f
*x])^(-1 + n))/(f*(1 + 2*m - p))

Rule 2926

Int[((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^FracP
art[p], Int[((a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, c, d, e
, f, g, m, n, p}, x] &&  !IntegerQ[p]

Rule 2853

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])^(n - m), 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] && (FractionQ[m] ||  !FractionQ[n])

Rule 2689

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 70

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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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]))

Rubi steps

\begin{align*} \int (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=\left ((g \cos (e+f x))^p (g \sec (e+f x))^p\right ) \int (g \cos (e+f x))^{-p} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\\ &=\left ((g \cos (e+f x))^{-2 m+p} (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int (g \cos (e+f x))^{2 m-p} (c-c \sin (e+f x))^{-m+n} \, dx\\ &=\frac{\left (c^2 \cos (e+f x) (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{m+\frac{1}{2} (-1-2 m+p)} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m+p)}\right ) \operatorname{Subst}\left (\int (c-c x)^{-m+n+\frac{1}{2} (-1+2 m-p)} (c+c x)^{\frac{1}{2} (-1+2 m-p)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (2^{-\frac{1}{2}+n-\frac{p}{2}} c^2 \cos (e+f x) (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-\frac{1}{2}+m+n-\frac{p}{2}+\frac{1}{2} (-1-2 m+p)} \left (\frac{c-c \sin (e+f x)}{c}\right )^{\frac{1}{2}-n+\frac{p}{2}} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m+p)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{-m+n+\frac{1}{2} (-1+2 m-p)} (c+c x)^{\frac{1}{2} (-1+2 m-p)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{2^{\frac{1}{2}+n-\frac{p}{2}} c \cos (e+f x) \, _2F_1\left (\frac{1}{2} (1+2 m-p),\frac{1}{2} (1-2 n+p);\frac{1}{2} (3+2 m-p);\frac{1}{2} (1+\sin (e+f x))\right ) (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac{1}{2} (1-2 n+p)} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{f (1+2 m-p)}\\ \end{align*}

Mathematica [A]  time = 40.1278, size = 139, normalized size = 1.01 \[ \frac{2 \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \sec (e+f x))^p \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )^{m+n-p} \, _2F_1\left (m+n-p+1,n-\frac{p}{2}+\frac{1}{2};n-\frac{p}{2}+\frac{3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f (2 n-p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Hypergeometric2F1[1 + m + n - p, 1/2 + n - p/2, 3/2 + n - p/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(g*Sec[e + f*x
])^p*(Sec[(2*e - Pi + 2*f*x)/4]^2)^(m + n - p)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n*Tan[(2*e - Pi +
2*f*x)/4])/(f*(1 + 2*n - p))

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Maple [F]  time = 5.358, size = 0, normalized size = 0. \begin{align*} \int \left ( g\sec \left ( fx+e \right ) \right ) ^{p} \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*sec(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)

[Out]

int((g*sec(f*x+e))^p*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \sec \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*sec(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (g \sec \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((g*sec(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)

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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*sec(f*x+e))**p*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \sec \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*sec(f*x + e))^p*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)