3.102 \(\int \cos ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=127 \[ -\frac{24 a^2 \cos ^7(c+d x)}{143 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a \sin (c+d x)+a)^{5/2}}-\frac{256 a^4 \cos ^7(c+d x)}{3003 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-256*a^4*Cos[c + d*x]^7)/(3003*d*(a + a*Sin[c + d*x])^(7/2)) - (64*a^3*Cos[c + d*x]^7)/(429*d*(a + a*Sin[c +
d*x])^(5/2)) - (24*a^2*Cos[c + d*x]^7)/(143*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^7)/(13*d*Sqrt[a
+ a*Sin[c + d*x]])

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Rubi [A]  time = 0.258362, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ -\frac{24 a^2 \cos ^7(c+d x)}{143 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a \sin (c+d x)+a)^{5/2}}-\frac{256 a^4 \cos ^7(c+d x)}{3003 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-256*a^4*Cos[c + d*x]^7)/(3003*d*(a + a*Sin[c + d*x])^(7/2)) - (64*a^3*Cos[c + d*x]^7)/(429*d*(a + a*Sin[c +
d*x])^(5/2)) - (24*a^2*Cos[c + d*x]^7)/(143*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^7)/(13*d*Sqrt[a
+ a*Sin[c + d*x]])

Rule 2674

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

Rule 2673

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

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{13} (12 a) \int \frac{\cos ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{143} \left (96 a^2\right ) \int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a+a \sin (c+d x))^{5/2}}-\frac{24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{429} \left (128 a^3\right ) \int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\\ &=-\frac{256 a^4 \cos ^7(c+d x)}{3003 d (a+a \sin (c+d x))^{7/2}}-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a+a \sin (c+d x))^{5/2}}-\frac{24 a^2 \cos ^7(c+d x)}{143 d (a+a \sin (c+d x))^{3/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 3.90918, size = 99, normalized size = 0.78 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^7 (-6377 \sin (c+d x)+231 \sin (3 (c+d x))+1890 \cos (2 (c+d x))-5230)}{6006 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*Sqrt[a*(1 + Sin[c + d*x])]*(-5230 + 1890*Cos[2*(c + d*x)] - 6377*Sin[
c + d*x] + 231*Sin[3*(c + d*x)]))/(6006*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 0.122, size = 75, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) a \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4} \left ( 231\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+945\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+1421\,\sin \left ( dx+c \right ) +835 \right ) }{3003\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x)

[Out]

-2/3003*(1+sin(d*x+c))*a*(sin(d*x+c)-1)^4*(231*sin(d*x+c)^3+945*sin(d*x+c)^2+1421*sin(d*x+c)+835)/cos(d*x+c)/(
a+a*sin(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)^6, x)

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Fricas [A]  time = 1.68498, size = 494, normalized size = 3.89 \begin{align*} -\frac{2 \,{\left (231 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{3} - 128 \, \cos \left (d x + c\right )^{2} -{\left (231 \, \cos \left (d x + c\right )^{6} + 252 \, \cos \left (d x + c\right )^{5} + 280 \, \cos \left (d x + c\right )^{4} + 320 \, \cos \left (d x + c\right )^{3} + 384 \, \cos \left (d x + c\right )^{2} + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sin \left (d x + c\right ) + 512 \, \cos \left (d x + c\right ) + 1024\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3003 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-2/3003*(231*cos(d*x + c)^7 - 21*cos(d*x + c)^6 + 28*cos(d*x + c)^5 - 40*cos(d*x + c)^4 + 64*cos(d*x + c)^3 -
128*cos(d*x + c)^2 - (231*cos(d*x + c)^6 + 252*cos(d*x + c)^5 + 280*cos(d*x + c)^4 + 320*cos(d*x + c)^3 + 384*
cos(d*x + c)^2 + 512*cos(d*x + c) + 1024)*sin(d*x + c) + 512*cos(d*x + c) + 1024)*sqrt(a*sin(d*x + c) + a)/(d*
cos(d*x + c) + d*sin(d*x + c) + d)

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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(cos(d*x+c)**6*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)^6, x)