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

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

[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]) + (a*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 = 0.118872, 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} \[ \frac{a \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*(d*Sin[e + f*x])^p)^n*(a + a*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]) + (a*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+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 (d \sin (e+f x))^{n p} (a+a \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 (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))^{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{a \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 [C]  time = 1.53437, size = 270, normalized size = 1.66 \[ \frac{a 2^{-n p-1} (\sin (e+f x)+1) \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{n p+1} \left (2 \left (n^2 p^2-1\right ) e^{i (e+f x)} \, _2F_1\left (1,\frac{n p}{2}+1;1-\frac{n p}{2};e^{2 i (e+f x)}\right )+i n p \left ((n p-1) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (1-n p);e^{2 i (e+f x)}\right )-(n p+1) e^{2 i (e+f x)} \, _2F_1\left (1,\frac{1}{2} (n p+3);\frac{1}{2} (3-n p);e^{2 i (e+f x)}\right )\right )\right ) \sin ^{-n p}(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f n p (n p-1) (n p+1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\sin \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+a\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+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{\left (a \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+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((a*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 (a \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+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((a*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*} a \left (\int \left (c \left (d \sin{\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int \left (c \left (d \sin{\left (e + f x \right )}\right )^{p}\right )^{n} \sin{\left (e + f x \right )}\, dx\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)

[Out]

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

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

[Out]

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