3.4.30 \(\int (e \cos (c+d x))^p (a+a \sin (c+d x)) \, dx\) [330]

Optimal. Leaf size=93 \[ -\frac {2^{\frac {3}{2}+\frac {p}{2}} a (e \cos (c+d x))^{1+p} \, _2F_1\left (\frac {1}{2} (-1-p),\frac {1+p}{2};\frac {3+p}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-p)}}{d e (1+p)} \]

[Out]

-2^(3/2+1/2*p)*a*(e*cos(d*x+c))^(1+p)*hypergeom([-1/2-1/2*p, 1/2+1/2*p],[3/2+1/2*p],1/2-1/2*sin(d*x+c))*(1+sin
(d*x+c))^(-1/2-1/2*p)/d/e/(1+p)

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Rubi [A]
time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2767, 71} \begin {gather*} -\frac {a 2^{\frac {p}{2}+\frac {3}{2}} (\sin (c+d x)+1)^{\frac {1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac {1}{2} (-p-1),\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d e (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(3/2 + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-1 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d
*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*e*(1 + p)))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 2767

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

Rubi steps

\begin {align*} \int (e \cos (c+d x))^p (a+a \sin (c+d x)) \, dx &=\frac {\left (a (e \cos (c+d x))^{1+p} (1-\sin (c+d x))^{\frac {1}{2} (-1-p)} (1+\sin (c+d x))^{\frac {1}{2} (-1-p)}\right ) \text {Subst}\left (\int (1-x)^{\frac {1}{2} (-1+p)} (1+x)^{1+\frac {1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=-\frac {2^{\frac {3}{2}+\frac {p}{2}} a (e \cos (c+d x))^{1+p} \, _2F_1\left (\frac {1}{2} (-1-p),\frac {1+p}{2};\frac {3+p}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {1}{2} (-1-p)}}{d e (1+p)}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.54, size = 266, normalized size = 2.86 \begin {gather*} \frac {2^{-1-p} a \left (1+e^{2 i (c+d x)}\right )^{-1-p} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{1+p} \cos ^{-p}(c+d x) (e \cos (c+d x))^p \left (-\left ((-1+p) p \, _2F_1\left (\frac {1}{2} (-1-p),-p;\frac {1-p}{2};-e^{2 i (c+d x)}\right )\right )+e^{i (c+d x)} (1+p) \left (e^{i (c+d x)} p \, _2F_1\left (\frac {1-p}{2},-p;\frac {3-p}{2};-e^{2 i (c+d x)}\right )+2 i (-1+p) \, _2F_1\left (-p,-\frac {p}{2};1-\frac {p}{2};-e^{2 i (c+d x)}\right )\right )\right ) (1+\sin (c+d x))}{d (-1+p) p (1+p) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-1 - p)*a*(1 + E^((2*I)*(c + d*x)))^(-1 - p)*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + p)*(e*Cos[c
+ d*x])^p*(-((-1 + p)*p*Hypergeometric2F1[(-1 - p)/2, -p, (1 - p)/2, -E^((2*I)*(c + d*x))]) + E^(I*(c + d*x))*
(1 + p)*(E^(I*(c + d*x))*p*Hypergeometric2F1[(1 - p)/2, -p, (3 - p)/2, -E^((2*I)*(c + d*x))] + (2*I)*(-1 + p)*
Hypergeometric2F1[-p, -1/2*p, 1 - p/2, -E^((2*I)*(c + d*x))]))*(1 + Sin[c + d*x]))/(d*(-1 + p)*p*(1 + p)*Cos[c
+ d*x]^p*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)

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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{p} \left (a +a \sin \left (d x +c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*cos(d*x+c))^p*(a+a*sin(d*x+c)),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))^p*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)*(cos(d*x + c)*e)^p, 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))^p*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral((a*sin(d*x + c) + a)*(cos(d*x + c)*e)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**p*(a+a*sin(d*x+c)),x)

[Out]

a*(Integral((e*cos(c + d*x))**p, x) + Integral((e*cos(c + d*x))**p*sin(c + d*x), 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))^p*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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