3.925 \(\int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=59 \[ \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2833, 68} \[ \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x])^2,x]

[Out]

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

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 59, normalized size = 1.00 \[ \frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x])^2,x]

[Out]

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

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(-(a*sin(f*x + e) + a)^m*cos(f*x + e)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c)^2, x)

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maple [F]  time = 4.58, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x)

[Out]

int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(e + f*x)*(a + a*sin(e + f*x))^m)/(c + d*sin(e + f*x))^2,x)

[Out]

int((cos(e + f*x)*(a + a*sin(e + f*x))^m)/(c + d*sin(e + f*x))^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))**m/(c+d*sin(f*x+e))**2,x)

[Out]

Timed out

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