3.7.50 \(\int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx\) [650]

Optimal. Leaf size=132 \[ \frac {e (e \cos (c+d x))^{-2-m} \, _2F_1\left (1+m,\frac {2+m}{2};2+m;\frac {2 (a+b \sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right ) (1-\sin (c+d x)) \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{(a+b) d (1+m)} \]

[Out]

e*(e*cos(d*x+c))^(-2-m)*hypergeom([1+m, 1+1/2*m],[2+m],2*(a+b*sin(d*x+c))/(a+b)/(1+sin(d*x+c)))*(1-sin(d*x+c))
*(-(a-b)*(1-sin(d*x+c))/(a+b)/(1+sin(d*x+c)))^(1/2*m)*(a+b*sin(d*x+c))^(1+m)/(a+b)/d/(1+m)

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Rubi [A]
time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2777} \begin {gather*} \frac {e (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right )^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (m+1,\frac {m+2}{2};m+2;\frac {2 (a+b \sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right )}{d (m+1) (a+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (2*(a + b*Sin[c + d*x]))/((a + b)*(1 +
Sin[c + d*x]))]*(1 - Sin[c + d*x])*(-(((a - b)*(1 - Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))))^(m/2)*(a +
b*Sin[c + d*x])^(1 + m))/((a + b)*d*(1 + m))

Rule 2777

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*(1 - Sin[e + f*x])*(a + b*Sin[e + f*x])^(m + 1)*(((-(a - b))*((1 - Sin[e + f*x])/((a + b)
*(1 + Sin[e + f*x]))))^(m/2)/(f*(a + b)*(m + 1)))*Hypergeometric2F1[m + 1, m/2 + 1, m + 2, 2*((a + b*Sin[e + f
*x])/((a + b)*(1 + Sin[e + f*x])))], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p +
1, 0]

Rubi steps

\begin {align*} \int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx &=\frac {e (e \cos (c+d x))^{-2-m} \, _2F_1\left (1+m,\frac {2+m}{2};2+m;\frac {2 (a+b \sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right ) (1-\sin (c+d x)) \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{(a+b) d (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 132, normalized size = 1.00 \begin {gather*} -\frac {e (e \cos (c+d x))^{-2-m} \, _2F_1\left (1+m,\frac {2+m}{2};2+m;-\frac {2 (a+b \sin (c+d x))}{(a-b) (-1+\sin (c+d x))}\right ) (1+\sin (c+d x)) \left (\frac {(a+b) (1+\sin (c+d x))}{(a-b) (-1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{(a-b) d (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (-2*(a + b*Sin[c + d*x]))/((a - b)*(
-1 + Sin[c + d*x]))]*(1 + Sin[c + d*x])*(((a + b)*(1 + Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x])))^(m/2)*(a
+ b*Sin[c + d*x])^(1 + m))/((a - b)*d*(1 + m)))

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{-1-m} \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x)

[Out]

int((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^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((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

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

[Out]

integral((cos(d*x + c)*e)^(-m - 1)*(b*sin(d*x + c) + a)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**(-1-m)*(a+b*sin(d*x+c))**m,x)

[Out]

Integral((e*cos(c + d*x))**(-m - 1)*(a + b*sin(c + d*x))**m, 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((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x, algorithm="giac")

[Out]

integrate((cos(d*x + c)*e)^(-m - 1)*(b*sin(d*x + c) + a)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 1),x)

[Out]

int((a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 1), x)

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