3.834 \(\int (c (d \sin (e+f x))^p)^n (a+b \sin (e+f x)) \, dx\)

Optimal. Leaf size=163 \[ \frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}} \]

[Out]

(a*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)/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2
, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])

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Rubi [A]  time = 0.114439, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2826, 2748, 2643} \[ \frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*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)/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2
, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])

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 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 \left (c (d \sin (e+f x))^p\right )^n (a+b \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 (d \sin (e+f x))^{n p} (a+b \sin (e+f x)) \, dx\\ &=\left (a (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+\frac{\left (b (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}{d}\\ &=\frac{a \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}{f (1+n p) \sqrt{\cos ^2(e+f x)}}+\frac{b \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (2+n p);\frac{1}{2} (4+n p);\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.248195, size = 129, normalized size = 0.79 \[ \frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a (n p+2) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right )+b (n p+1) \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n p}{2}+1;\frac{n p}{2}+2;\sin ^2(e+f x)\right )\right )}{f (n p+1) (n p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.165, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\sin \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+b\sin \left ( fx+e \right ) \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+b*sin(f*x+e)),x)

[Out]

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

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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