3.12 \(\int (d \sin (e+f x))^n (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\)

Optimal. Leaf size=221 \[ -\frac{2^{m+\frac{1}{2}} (A-B) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f}-\frac{B 2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left (\frac{1}{2};-n,-m-\frac{1}{2};\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f} \]

[Out]

-((2^(3/2 + m)*B*AppellF1[1/2, -n, -1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[
e + f*x])^n*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^n)) - (2^(1/2 + m)*(A - B)*A
ppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(1 + Si
n[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^n)

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Rubi [A]  time = 0.453302, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {2987, 2787, 2786, 2785, 133} \[ -\frac{2^{m+\frac{1}{2}} (A-B) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f}-\frac{B 2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left (\frac{1}{2};-n,-m-\frac{1}{2};\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

-((2^(3/2 + m)*B*AppellF1[1/2, -n, -1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[
e + f*x])^n*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^n)) - (2^(1/2 + m)*(A - B)*A
ppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(1 + Si
n[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^n)

Rule 2987

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

Rule 2787

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a^In
tPart[m]*(a + b*Sin[e + f*x])^FracPart[m])/(1 + (b*Sin[e + f*x])/a)^FracPart[m], Int[(1 + (b*Sin[e + f*x])/a)^
m*(d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] &&  !GtQ
[a, 0]

Rule 2786

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[((d/b
)^IntPart[n]*(d*Sin[e + f*x])^FracPart[n])/(b*Sin[e + f*x])^FracPart[n], Int[(a + b*Sin[e + f*x])^m*(b*Sin[e +
f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] && GtQ[a, 0] &&  !Gt
Q[d/b, 0]

Rule 2785

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Dist[(b*(d
/b)^n*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[((a - x)^n*(2*a - x)^(m -
1/2))/Sqrt[x], x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !In
tegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int (d \sin (e+f x))^n (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=(A-B) \int (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx+\frac{B \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{1+m} \, dx}{a}\\ &=\left ((A-B) (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx+\left (B (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (d \sin (e+f x))^n (1+\sin (e+f x))^{1+m} \, dx\\ &=\left ((A-B) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int \sin ^n(e+f x) (1+\sin (e+f x))^m \, dx+\left (B \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int \sin ^n(e+f x) (1+\sin (e+f x))^{1+m} \, dx\\ &=-\frac{\left ((A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{(1-x)^n (2-x)^{-\frac{1}{2}+m}}{\sqrt{x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)}}-\frac{\left (B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{(1-x)^n (2-x)^{\frac{1}{2}+m}}{\sqrt{x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)}}\\ &=-\frac{2^{\frac{3}{2}+m} B F_1\left (\frac{1}{2};-n,-\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^m}{f}-\frac{2^{\frac{1}{2}+m} (A-B) F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^m}{f}\\ \end{align*}

Mathematica [B]  time = 22.213, size = 5918, normalized size = 26.78 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

Result too large to show

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Maple [F]  time = 4.251, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

[Out]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^m*(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 (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*(d*sin(f*x + e))^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 (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*(d*sin(f*x + e))^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((d*sin(f*x+e))**n*(a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

Exception raised: AttributeError