3.15 \(\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx\)

Optimal. Leaf size=35 \[ -\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a \sin (c+d x)+a)^m}{d} \]

[Out]

-((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a + a*Sin[c + d*x])^m)/d)

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Rubi [A]  time = 0.0944697, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {2974} \[ -\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a \sin (c+d x)+a)^m}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^(-2 - m)*(a + a*Sin[c + d*x])^m*(1 + m - m*Sin[c + d*x]),x]

[Out]

-((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a + a*Sin[c + d*x])^m)/d)

Rule 2974

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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^(n + 1))/(f*(n + 1)*(c^2 - d^2)), 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] && EqQ[m + n + 2, 0] && EqQ[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n +
1)), 0]

Rubi steps

\begin{align*} \int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx &=-\frac{\cos (c+d x) \sin ^{-1-m}(c+d x) (a+a \sin (c+d x))^m}{d}\\ \end{align*}

Mathematica [A]  time = 0.362636, size = 35, normalized size = 1. \[ -\frac{\cos (c+d x) \sin ^{-m-1}(c+d x) (a (\sin (c+d x)+1))^m}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^(-2 - m)*(a + a*Sin[c + d*x])^m*(1 + m - m*Sin[c + d*x]),x]

[Out]

-((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a*(1 + Sin[c + d*x]))^m)/d)

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Maple [F]  time = 0.454, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( dx+c \right ) \right ) ^{-2-m} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m} \left ( 1+m-m\sin \left ( dx+c \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x)

[Out]

int(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (m \sin \left (d x + c\right ) - m - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x, algorithm="maxima")

[Out]

-integrate((m*sin(d*x + c) - m - 1)*(a*sin(d*x + c) + a)^m*sin(d*x + c)^(-m - 2), x)

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Fricas [A]  time = 1.46566, size = 101, normalized size = 2.89 \begin{align*} -\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x, algorithm="fricas")

[Out]

-(a*sin(d*x + c) + a)^m*sin(d*x + c)^(-m - 2)*cos(d*x + c)*sin(d*x + c)/d

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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(sin(d*x+c)**(-2-m)*(a+a*sin(d*x+c))**m*(1+m-m*sin(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (m \sin \left (d x + c\right ) - m - 1\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(-(m*sin(d*x + c) - m - 1)*(a*sin(d*x + c) + a)^m*sin(d*x + c)^(-m - 2), x)