3.7.22 \(\int \frac {(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx\) [622]

Optimal. Leaf size=170 \[ -\frac {e F_1\left (8-p;\frac {1-p}{2},\frac {1-p}{2};9-p;\frac {a+b}{a+b \sin (c+d x)},\frac {a-b}{a+b \sin (c+d x)}\right ) (e \cos (c+d x))^{-1+p} \left (-\frac {b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1-p}{2}} \left (\frac {b (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1-p}{2}}}{b d (8-p) (a+b \sin (c+d x))^7} \]

[Out]

-e*AppellF1(8-p,1/2-1/2*p,1/2-1/2*p,9-p,(a-b)/(a+b*sin(d*x+c)),(a+b)/(a+b*sin(d*x+c)))*(e*cos(d*x+c))^(-1+p)*(
-b*(1-sin(d*x+c))/(a+b*sin(d*x+c)))^(1/2-1/2*p)*(b*(1+sin(d*x+c))/(a+b*sin(d*x+c)))^(1/2-1/2*p)/b/d/(8-p)/(a+b
*sin(d*x+c))^7

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Rubi [A]
time = 0.05, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2782} \begin {gather*} -\frac {e (e \cos (c+d x))^{p-1} \left (-\frac {b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1-p}{2}} \left (\frac {b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac {1-p}{2}} F_1\left (8-p;\frac {1-p}{2},\frac {1-p}{2};9-p;\frac {a+b}{a+b \sin (c+d x)},\frac {a-b}{a+b \sin (c+d x)}\right )}{b d (8-p) (a+b \sin (c+d x))^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8,x]

[Out]

-((e*AppellF1[8 - p, (1 - p)/2, (1 - p)/2, 9 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*
(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))
/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(8 - p)*(a + b*Sin[c + d*x])^7))

Rule 2782

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)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p)*((-b)*((1 - Sin[e + f*x])/(a + b*Sin[e + f*x])
))^((p - 1)/2)*(b*((1 + Sin[e + f*x])/(a + b*Sin[e + f*x])))^((p - 1)/2)))*AppellF1[-p - m, (1 - p)/2, (1 - p)
/2, 1 - p - m, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& NeQ[a^2 - b^2, 0] && ILtQ[m, 0] &&  !IGtQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx &=-\frac {e F_1\left (8-p;\frac {1-p}{2},\frac {1-p}{2};9-p;\frac {a+b}{a+b \sin (c+d x)},\frac {a-b}{a+b \sin (c+d x)}\right ) (e \cos (c+d x))^{-1+p} \left (-\frac {b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1-p}{2}} \left (\frac {b (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac {1-p}{2}}}{b d (8-p) (a+b \sin (c+d x))^7}\\ \end {align*}

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Mathematica [F]
time = 46.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8,x]

[Out]

Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8, x]

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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\left (e \cos \left (d x +c \right )\right )^{p}}{\left (a +b \sin \left (d x +c \right )\right )^{8}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^p/(a+b*sin(d*x+c))^8,x)

[Out]

int((e*cos(d*x+c))^p/(a+b*sin(d*x+c))^8,x)

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Maxima [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((e*cos(d*x+c))^p/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

Timed out

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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))^p/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

integral((cos(d*x + c)*e)^p/(b^8*cos(d*x + c)^8 + a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8 - 4*(7*a^2*
b^6 + b^8)*cos(d*x + c)^6 + 2*(35*a^4*b^4 + 42*a^2*b^6 + 3*b^8)*cos(d*x + c)^4 - 4*(7*a^6*b^2 + 35*a^4*b^4 + 2
1*a^2*b^6 + b^8)*cos(d*x + c)^2 - 8*(a*b^7*cos(d*x + c)^6 - a^7*b - 7*a^5*b^3 - 7*a^3*b^5 - a*b^7 - (7*a^3*b^5
+ 3*a*b^7)*cos(d*x + c)^4 + (7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2)*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((e*cos(d*x+c))**p/(a+b*sin(d*x+c))**8,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((e*cos(d*x+c))^p/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

integrate((cos(d*x + c)*e)^p/(b*sin(d*x + c) + a)^8, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^p}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(c + d*x))^p/(a + b*sin(c + d*x))^8,x)

[Out]

int((e*cos(c + d*x))^p/(a + b*sin(c + d*x))^8, x)

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