3.1 \(\int (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx\)

Optimal. Leaf size=373 \[ \frac{a^3 (A (4 n+11)+B (4 n+9)) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{d^2 f (n+2) (n+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 \left (A \left (4 n^2+21 n+20\right )+B \left (4 n^2+19 n+15\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1) (n+2) (n+4) \sqrt{\cos ^2(e+f x)}}-\frac{a^3 \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2) (n+3) (n+4)}-\frac{(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{d f (n+3) (n+4)}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)} \]

[Out]

-((a^3*(B*(27 + 14*n + 2*n^2) + A*(28 + 15*n + 2*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n)*(3
+ n)*(4 + n))) + (a^3*(B*(15 + 19*n + 4*n^2) + A*(20 + 21*n + 4*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 +
n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*(4 + n)*Sqrt[Cos[e + f*x]^2])
+ (a^3*(B*(9 + 4*n) + A*(11 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]
*(d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n)*(3 + n)*Sqrt[Cos[e + f*x]^2]) - (a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(
1 + n)*(a + a*Sin[e + f*x])^2)/(d*f*(4 + n)) - ((A*(4 + n) + B*(6 + n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*
(a^3 + a^3*Sin[e + f*x]))/(d*f*(3 + n)*(4 + n))

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Rubi [A]  time = 0.840783, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {2976, 2968, 3023, 2748, 2643} \[ \frac{a^3 (A (4 n+11)+B (4 n+9)) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{d^2 f (n+2) (n+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 \left (A \left (4 n^2+21 n+20\right )+B \left (4 n^2+19 n+15\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1) (n+2) (n+4) \sqrt{\cos ^2(e+f x)}}-\frac{a^3 \left (A \left (2 n^2+15 n+28\right )+B \left (2 n^2+14 n+27\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2) (n+3) (n+4)}-\frac{(A (n+4)+B (n+6)) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (d \sin (e+f x))^{n+1}}{d f (n+3) (n+4)}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (d \sin (e+f x))^{n+1}}{d f (n+4)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]),x]

[Out]

-((a^3*(B*(27 + 14*n + 2*n^2) + A*(28 + 15*n + 2*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n)*(3
+ n)*(4 + n))) + (a^3*(B*(15 + 19*n + 4*n^2) + A*(20 + 21*n + 4*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 +
n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*(4 + n)*Sqrt[Cos[e + f*x]^2])
+ (a^3*(B*(9 + 4*n) + A*(11 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]
*(d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n)*(3 + n)*Sqrt[Cos[e + f*x]^2]) - (a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(
1 + n)*(a + a*Sin[e + f*x])^2)/(d*f*(4 + n)) - ((A*(4 + n) + B*(6 + n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*
(a^3 + a^3*Sin[e + f*x]))/(d*f*(3 + n)*(4 + n))

Rule 2976

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

Rule 2968

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 3023

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]

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 (d \sin (e+f x))^n (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) \, dx &=-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}+\frac{\int (d \sin (e+f x))^n (a+a \sin (e+f x))^2 (a d (B (1+n)+A (4+n))+a d (A (4+n)+B (6+n)) \sin (e+f x)) \, dx}{d (4+n)}\\ &=-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}-\frac{(A (4+n)+B (6+n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^3+a^3 \sin (e+f x)\right )}{d f (3+n) (4+n)}+\frac{\int (d \sin (e+f x))^n (a+a \sin (e+f x)) \left (a^2 d^2 \left (2 A \left (8+6 n+n^2\right )+B \left (9+11 n+2 n^2\right )\right )+a^2 d^2 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right ) \sin (e+f x)\right ) \, dx}{d^2 (3+n) (4+n)}\\ &=-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}-\frac{(A (4+n)+B (6+n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^3+a^3 \sin (e+f x)\right )}{d f (3+n) (4+n)}+\frac{\int (d \sin (e+f x))^n \left (a^3 d^2 \left (2 A \left (8+6 n+n^2\right )+B \left (9+11 n+2 n^2\right )\right )+\left (a^3 d^2 \left (2 A \left (8+6 n+n^2\right )+B \left (9+11 n+2 n^2\right )\right )+a^3 d^2 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right )\right ) \sin (e+f x)+a^3 d^2 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right ) \sin ^2(e+f x)\right ) \, dx}{d^2 (3+n) (4+n)}\\ &=-\frac{a^3 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n) (4+n)}-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}-\frac{(A (4+n)+B (6+n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^3+a^3 \sin (e+f x)\right )}{d f (3+n) (4+n)}+\frac{\int (d \sin (e+f x))^n \left (a^3 d^3 (3+n) \left (B \left (15+19 n+4 n^2\right )+A \left (20+21 n+4 n^2\right )\right )+a^3 d^3 (2+n) (4+n) (B (9+4 n)+A (11+4 n)) \sin (e+f x)\right ) \, dx}{d^3 (2+n) (3+n) (4+n)}\\ &=-\frac{a^3 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n) (4+n)}-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}-\frac{(A (4+n)+B (6+n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^3+a^3 \sin (e+f x)\right )}{d f (3+n) (4+n)}+\frac{\left (a^3 (B (9+4 n)+A (11+4 n))\right ) \int (d \sin (e+f x))^{1+n} \, dx}{d (3+n)}+\frac{\left (a^3 \left (B \left (15+19 n+4 n^2\right )+A \left (20+21 n+4 n^2\right )\right )\right ) \int (d \sin (e+f x))^n \, dx}{(2+n) (4+n)}\\ &=-\frac{a^3 \left (B \left (27+14 n+2 n^2\right )+A \left (28+15 n+2 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n) (3+n) (4+n)}+\frac{a^3 \left (B \left (15+19 n+4 n^2\right )+A \left (20+21 n+4 n^2\right )\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) (4+n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 (B (9+4 n)+A (11+4 n)) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{2+n}{2};\frac{4+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) (3+n) \sqrt{\cos ^2(e+f x)}}-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^2}{d f (4+n)}-\frac{(A (4+n)+B (6+n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \left (a^3+a^3 \sin (e+f x)\right )}{d f (3+n) (4+n)}\\ \end{align*}

Mathematica [A]  time = 2.26348, size = 248, normalized size = 0.66 \[ \frac{a^3 \sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left (\sin (e+f x) \left (\frac{(3 A+B) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{n+2}+\sin (e+f x) \left (\frac{3 (A+B) \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(e+f x)\right )}{n+3}+\sin (e+f x) \left (\frac{(A+3 B) \, _2F_1\left (\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(e+f x)\right )}{n+4}+\frac{B \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n+5}{2};\frac{n+7}{2};\sin ^2(e+f x)\right )}{n+5}\right )\right )\right )+\frac{A \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{n+1}\right )}{f \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]),x]

[Out]

(a^3*Cos[e + f*x]*Sin[e + f*x]*(d*Sin[e + f*x])^n*((A*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x
]^2])/(1 + n) + Sin[e + f*x]*(((3*A + B)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2])/(2 + n)
+ Sin[e + f*x]*((3*(A + B)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Sin[e + f*x]^2])/(3 + n) + Sin[e + f*
x]*(((A + 3*B)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Sin[e + f*x]^2])/(4 + n) + (B*Hypergeometric2F1[1/
2, (5 + n)/2, (7 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x])/(5 + n))))))/(f*Sqrt[Cos[e + f*x]^2])

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Maple [F]  time = 3.333, 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 ) ^{3} \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))^3*(A+B*sin(f*x+e)),x)

[Out]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^3*(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 )}^{3} \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))^3*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(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 a^{3} \cos \left (f x + e\right )^{4} -{\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \,{\left (A + B\right )} a^{3} -{\left ({\left (A + 3 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \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))^3*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*a^3*cos(f*x + e)^4 - (3*A + 5*B)*a^3*cos(f*x + e)^2 + 4*(A + B)*a^3 - ((A + 3*B)*a^3*cos(f*x + e)^
2 - 4*(A + B)*a^3)*sin(f*x + e))*(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))**3*(A+B*sin(f*x+e)),x)

[Out]

Timed out

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Giac [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 )}^{3} \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))^3*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(d*sin(f*x + e))^n, x)