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

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

[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))

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Rubi [A]  time = 0.0704151, antiderivative size = 170, normalized size of antiderivative = 1., 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 = {2703} \[ -\frac{e (e \cos (c+d x))^{p-1} \left (-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac{1-p}{2}} \left (\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{1-p}{2}} F_1\left (8-p;\frac{1-p}{2},\frac{1-p}{2};9-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right )}{b d (8-p) (a+b \sin (c+d x))^7} \]

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 2703

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*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])])/(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)), 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*}

Mathematica [F]  time = 66.8614, size = 0, normalized size = 0. \[ \int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx \]

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 = 4.677, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e\cos \left ( dx+c \right ) \right ) ^{p}}{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{8}}}\, dx \end{align*}

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)]  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((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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (e \cos \left (d x + c\right )\right )^{p}}{b^{8} \cos \left (d x + c\right )^{8} + a^{8} + 28 \, a^{6} b^{2} + 70 \, a^{4} b^{4} + 28 \, a^{2} b^{6} + b^{8} - 4 \,{\left (7 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \,{\left (35 \, a^{4} b^{4} + 42 \, a^{2} b^{6} + 3 \, b^{8}\right )} \cos \left (d x + c\right )^{4} - 4 \,{\left (7 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 21 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} - 8 \,{\left (a b^{7} \cos \left (d x + c\right )^{6} - a^{7} b - 7 \, a^{5} b^{3} - 7 \, a^{3} b^{5} - a b^{7} -{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} +{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}, x\right ) \end{align*}

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((e*cos(d*x + c))^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)]  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((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., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{p}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{8}}\,{d x} \end{align*}

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