3.823 \(\int \frac{(c (d \sin (e+f x))^p)^n}{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=189 \[ \frac{\cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n p}{2};\frac{1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt{\cos ^2(e+f x)}}-\frac{n p \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) \sqrt{\cos ^2(e+f x)}}-\frac{\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a \sin (e+f x)+a)} \]

[Out]

(Cos[e + f*x]*Hypergeometric2F1[1/2, (n*p)/2, (2 + n*p)/2, Sin[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*Sqrt
[Cos[e + f*x]^2]) - (n*p*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e +
f*x]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) - (Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)
/(f*(a + a*Sin[e + f*x]))

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Rubi [A]  time = 0.229935, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2826, 2769, 2748, 2643} \[ \frac{\cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n p}{2};\frac{1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt{\cos ^2(e+f x)}}-\frac{n p \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) \sqrt{\cos ^2(e+f x)}}-\frac{\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a \sin (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x]),x]

[Out]

(Cos[e + f*x]*Hypergeometric2F1[1/2, (n*p)/2, (2 + n*p)/2, Sin[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*Sqrt
[Cos[e + f*x]^2]) - (n*p*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e +
f*x]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) - (Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)
/(f*(a + a*Sin[e + f*x]))

Rule 2826

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

Rule 2769

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

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

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

Mathematica [A]  time = 0.299217, size = 157, normalized size = 0.83 \[ \frac{\sin (e+f x) \cos (e+f x) \sqrt{\cos ^2(e+f x)} \left ((n p+1) \sin (e+f x) \, _2F_1\left (\frac{3}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right )-(n p+2) \, _2F_1\left (\frac{3}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right )\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) (n p+2) (\sin (e+f x)-1) (\sin (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x]),x]

[Out]

(Cos[e + f*x]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(-((2 + n*p)*Hypergeometric2F1[3/2, (
1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]) + (1 + n*p)*Hypergeometric2F1[3/2, 1 + (n*p)/2, 2 + (n*p)/2, Sin[e +
f*x]^2]*Sin[e + f*x]))/(a*f*(1 + n*p)*(2 + n*p)*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x]))

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c \left ( d\sin \left ( fx+e \right ) \right ) ^{p} \right ) ^{n}}{a+a\sin \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e)),x)

[Out]

int((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate(((d*sin(f*x + e))^p*c)^n/(a*sin(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral(((d*sin(f*x + e))^p*c)^n/(a*sin(f*x + e) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\left (c \left (d \sin{\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin{\left (e + f x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sin(f*x+e))**p)**n/(a+a*sin(f*x+e)),x)

[Out]

Integral((c*(d*sin(e + f*x))**p)**n/(sin(e + f*x) + 1), x)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(((d*sin(f*x + e))^p*c)^n/(a*sin(f*x + e) + a), x)