3.2.38 \(\int (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx\) [138]

Optimal. Leaf size=107 \[ -\frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2};-n,\frac {1}{2}-m;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f} \]

[Out]

-2^(1/2+m)*AppellF1(1/2,-n,1/2-m,3/2,1-sin(f*x+e),1/2-1/2*sin(f*x+e))*cos(f*x+e)*(d*sin(f*x+e))^n*(1+sin(f*x+e
))^(-1/2-m)*(a+a*sin(f*x+e))^m/f/(sin(f*x+e)^n)

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Rubi [A]
time = 0.11, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2866, 2865, 2864, 138} \begin {gather*} -\frac {2^{m+\frac {1}{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,\frac {1}{2}-m;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*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]*(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))

Rule 138

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

Rule 2864

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] &&  !
IntegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]

Rule 2865

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 2866

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[a^Int
Part[m]*((a + b*Sin[e + f*x])^FracPart[m]/(1 + (b/a)*Sin[e + f*x])^FracPart[m]), Int[(1 + (b/a)*Sin[e + f*x])^
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]

Rubi steps

\begin {align*} \int (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx &=\left ((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 (\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\\ &=-\frac {\left (\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 ) \text {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) \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(995\) vs. \(2(107)=214\).
time = 2.69, size = 995, normalized size = 9.30 \begin {gather*} \frac {15 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \cos (e+f x) (d \sin (e+f x))^n (a (1+\sin (e+f x)))^m}{f \left (15 n F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \cos (e+f x) \cot (e+f x)+10 \left (n F_1\left (\frac {3}{2};1-n,1+m+n;\frac {5}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )+(1+m+n) F_1\left (\frac {3}{2};-n,2+m+n;\frac {5}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )\right ) \cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )+\frac {9 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \cos (e+f x) \cot \left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \left (-5 n F_1\left (\frac {3}{2};1-n,1+m+n;\frac {5}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )-5 (1+m+n) F_1\left (\frac {3}{2};-n,2+m+n;\frac {5}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )+2 \left (2 n (1+m+n) F_1\left (\frac {5}{2};1-n,2+m+n;\frac {7}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )+(-1+n) n F_1\left (\frac {5}{2};2-n,1+m+n;\frac {7}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )+\left (2+m^2+3 n+n^2+m (3+2 n)\right ) F_1\left (\frac {5}{2};-n,3+m+n;\frac {7}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )\right ) \cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right ) \csc ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{3 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )-2 \left (n F_1\left (\frac {3}{2};1-n,1+m+n;\frac {5}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )+(1+m+n) F_1\left (\frac {3}{2};-n,2+m+n;\frac {5}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )\right ) \cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}-15 F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \sin (e+f x)+30 m F_1\left (\frac {1}{2};-n,1+m+n;\frac {3}{2};\cot ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right ),-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(15*AppellF1[1/2, -n, 1 + m + n, 3/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*
(d*Sin[e + f*x])^n*(a*(1 + Sin[e + f*x]))^m)/(f*(15*n*AppellF1[1/2, -n, 1 + m + n, 3/2, Cot[(2*e + Pi + 2*f*x)
/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*Cot[e + f*x] + 10*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Co
t[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Cot[(
2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Cot[(2*e + Pi + 2*f*x)/4]^2 + (9*AppellF1[1/2, -n, 1 +
m + n, 3/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Cos[e + f*x]*Cot[(2*e + Pi + 2*f*x)/4]*
(-5*n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] - 5*(1 +
m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + 2*(2*n*
(1 + m + n)*AppellF1[5/2, 1 - n, 2 + m + n, 7/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] +
(-1 + n)*n*AppellF1[5/2, 2 - n, 1 + m + n, 7/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (
2 + m^2 + 3*n + n^2 + m*(3 + 2*n))*AppellF1[5/2, -n, 3 + m + n, 7/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e -
Pi + 2*f*x)/4]^2])*Cot[(2*e + Pi + 2*f*x)/4]^2)*Csc[(2*e + Pi + 2*f*x)/4]^2)/(3*AppellF1[1/2, -n, 1 + m + n, 3
/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] - 2*(n*AppellF1[3/2, 1 - n, 1 + m + n, 5/2, Cot
[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n)*AppellF1[3/2, -n, 2 + m + n, 5/2, Cot[(2
*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Cot[(2*e + Pi + 2*f*x)/4]^2) - 15*AppellF1[1/2, -n, 1 +
m + n, 3/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Sin[e + f*x] + 30*m*AppellF1[1/2, -n, 1
+ m + n, 3/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*Sin[(2*e - Pi + 2*f*x)/4]^2))

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]

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,x)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (d \sin {\left (e + f x \right )}\right )^{n}\, dx \end {gather*}

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,x)

[Out]

Integral((a*(sin(e + f*x) + 1))**m*(d*sin(e + f*x))**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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