3.2.41 \(\int \sin ^3(c+d x) (a+a \sin (c+d x))^n \, dx\) [141]

Optimal. Leaf size=215 \[ -\frac {(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac {2^{\frac {1}{2}+n} n \left (5+3 n+n^2\right ) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac {n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )} \]

[Out]

-(4+n)*cos(d*x+c)*(a+a*sin(d*x+c))^n/d/(n^3+6*n^2+11*n+6)-cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^n/d/(3+n)-2
^(1/2+n)*n*(n^2+3*n+5)*cos(d*x+c)*hypergeom([1/2, 1/2-n],[3/2],1/2-1/2*sin(d*x+c))*(1+sin(d*x+c))^(-1/2-n)*(a+
a*sin(d*x+c))^n/d/(n^3+6*n^2+11*n+6)-n*cos(d*x+c)*(a+a*sin(d*x+c))^(1+n)/a/d/(n^2+5*n+6)

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Rubi [A]
time = 0.21, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2862, 3047, 3102, 2830, 2731, 2730} \begin {gather*} -\frac {2^{n+\frac {1}{2}} n \left (n^2+3 n+5\right ) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d (n+1) (n+2) (n+3)}-\frac {n \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d \left (n^2+5 n+6\right )}-\frac {\sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3)}-\frac {(n+4) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n,x]

[Out]

-(((4 + n)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n))) - (Cos[c + d*x]*Sin[c + d*x]^2*(a
+ a*Sin[c + d*x])^n)/(d*(3 + n)) - (2^(1/2 + n)*n*(5 + 3*n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n
, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n))
- (n*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(6 + 5*n + n^2))

Rule 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 2731

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[a^IntPart[n]*((a + b*Sin[c + d*x])^FracPart
[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n]), Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rule 2862

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(f*(m + n))), x] + Dist[1/(b*(m + n))
, Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 2)*Simp[d*(a*c*m + b*d*(n - 1)) + b*c^2*(m + n) + d*(a*
d*m + b*c*(m + 2*n - 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && E
qQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[n]

Rule 3047

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

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \sin ^3(c+d x) (a+a \sin (c+d x))^n \, dx &=-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}+\frac {\int \sin (c+d x) (a+a \sin (c+d x))^n (2 a+a n \sin (c+d x)) \, dx}{a (3+n)}\\ &=-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}+\frac {\int (a+a \sin (c+d x))^n \left (2 a \sin (c+d x)+a n \sin ^2(c+d x)\right ) \, dx}{a (3+n)}\\ &=-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac {n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}+\frac {\int (a+a \sin (c+d x))^n \left (a^2 n (1+n)+a^2 (4+n) \sin (c+d x)\right ) \, dx}{a^2 (2+n) (3+n)}\\ &=-\frac {(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac {n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}+\frac {\left (n \left (5+3 n+n^2\right )\right ) \int (a+a \sin (c+d x))^n \, dx}{(1+n) (2+n) (3+n)}\\ &=-\frac {(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac {n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}+\frac {\left (n \left (5+3 n+n^2\right ) (1+\sin (c+d x))^{-n} (a+a \sin (c+d x))^n\right ) \int (1+\sin (c+d x))^n \, dx}{(1+n) (2+n) (3+n)}\\ &=-\frac {(4+n) \cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac {\cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^n}{d (3+n)}-\frac {2^{\frac {1}{2}+n} n \left (5+3 n+n^2\right ) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n) (2+n) (3+n)}-\frac {n \cos (c+d x) (a+a \sin (c+d x))^{1+n}}{a d \left (6+5 n+n^2\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 120.76, size = 59941, normalized size = 278.80 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n,x]

[Out]

Result too large to show

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Maple [F]
time = 0.86, size = 0, normalized size = 0.00 \[\int \left (\sin ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3*(a+a*sin(d*x+c))^n,x)

[Out]

int(sin(d*x+c)^3*(a+a*sin(d*x+c))^n,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(sin(d*x+c)^3*(a+a*sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^n*sin(d*x + c)^3, 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(sin(d*x+c)^3*(a+a*sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^n*sin(d*x + c), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**3*(a+a*sin(d*x+c))**n,x)

[Out]

Timed out

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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(sin(d*x+c)^3*(a+a*sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^n*sin(d*x + c)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^3*(a + a*sin(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^3*(a + a*sin(c + d*x))^n, x)

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