3.136 \(\int \sin ^n(e+f x) (a+a \sin (e+f x))^m \, dx\)
Optimal. Leaf size=87 \[ -\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f} \]
[Out]
-((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(1 + Sin[e
+ f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)
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Rubi [A] time = 0.0981528, antiderivative size = 87, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{2787, 2785, 133} \[ -\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m F_1\left (\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f} \]
Antiderivative was successfully verified.
[In]
Int[Sin[e + f*x]^n*(a + a*Sin[e + f*x])^m,x]
[Out]
-((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(1 + Sin[e
+ f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)
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 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 \sin ^n(e+f x) (a+a \sin (e+f x))^m \, dx &=\left ((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\\ &=-\frac{\left (\cos (e+f x) (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{1}{2}+m} 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) (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^m}{f}\\ \end{align*}
Mathematica [B] time = 6.30142, size = 2807, normalized size = 32.26 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sin[e + f*x]^n*(a + a*Sin[e + f*x])^m,x]
[Out]
(-3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Si
n[e + f*x]^(2*n)*(a + a*Sin[e + f*x])^m)/(f*(Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2
, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e
+ Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi
/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)*((-3*n*AppellF1[1/2, -n, 1 + m + n,
3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]^2*Sin[e + f*x]^(-1 + n))/((Sec[(-e
+ Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/
2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (
1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e
+ Pi/2 - f*x)/2]^2)) + (3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x
)/2]^2]*Sin[e + f*x]^(1 + n))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + P
i/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x
)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^
2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*m*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e
+ Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*Tan[(-e + Pi/2 - f*x)/2])/((Sec[
(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f
*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]
+ (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan
[(-e + Pi/2 - f*x)/2]^2)) + (3*Cos[e + f*x]*Sin[e + f*x]^n*(-(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e +
Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3 - ((1 +
m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + P
i/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m + n,
3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[
(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[(-e +
Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)) - (3*AppellF1[1/2, -n, 1 + m + n
, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Cos[e + f*x]*Sin[e + f*x]^n*(-2*(n*AppellF1[3/
2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2,
-n, 2 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[
(-e + Pi/2 - f*x)/2] + 3*(-(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2
- f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/3 - ((1 + m + n)*AppellF1[3/2, -n, 2 + m +
n, 5/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f
*x)/2])/3) - 2*Tan[(-e + Pi/2 - f*x)/2]^2*(n*((-3*(1 + m + n)*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e + P
i/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 + (3*(1 -
n)*AppellF1[5/2, 2 - n, 1 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi
/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5) + (1 + m + n)*((-3*n*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Tan[(-e
+ Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e + Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5 - (3*(
2 + m + n)*AppellF1[5/2, -n, 3 + m + n, 7/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2]*Sec[(-e
+ Pi/2 - f*x)/2]^2*Tan[(-e + Pi/2 - f*x)/2])/5))))/((Sec[(-e + Pi/2 - f*x)/2]^2)^m*(3*AppellF1[1/2, -n, 1 + m
+ n, 3/2, Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2,
Tan[(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Tan[
(-e + Pi/2 - f*x)/2]^2, -Tan[(-e + Pi/2 - f*x)/2]^2])*Tan[(-e + Pi/2 - f*x)/2]^2)^2)))
________________________________________________________________________________________
Maple [F] time = 0.636, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(f*x+e)^n*(a+a*sin(f*x+e))^m,x)
[Out]
int(sin(f*x+e)^n*(a+a*sin(f*x+e))^m,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^m,x, algorithm="maxima")
[Out]
integrate((a*sin(f*x + e) + a)^m*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 (a \sin \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^m,x, algorithm="fricas")
[Out]
integral((a*sin(f*x + e) + a)^m*sin(f*x + e)^n, x)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{m} \sin ^{n}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)**n*(a+a*sin(f*x+e))**m,x)
[Out]
Integral((a*(sin(e + f*x) + 1))**m*sin(e + f*x)**n, x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^m,x, algorithm="giac")
[Out]
integrate((a*sin(f*x + e) + a)^m*sin(f*x + e)^n, x)