3.2.3 \(\int \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [103]

Optimal. Leaf size=73 \[ \frac {8 (a+a \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}+\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^5 d} \]

[Out]

8/7*(a+a*sin(d*x+c))^(7/2)/a^3/d-8/9*(a+a*sin(d*x+c))^(9/2)/a^4/d+2/11*(a+a*sin(d*x+c))^(11/2)/a^5/d

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Rubi [A]
time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2746, 45} \begin {gather*} \frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(8*(a + a*Sin[c + d*x])^(7/2))/(7*a^3*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) + (2*(a + a*Sin[c + d*x])^
(11/2))/(11*a^5*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

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

Rubi steps

\begin {align*} \int \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int (a-x)^2 (a+x)^{5/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^2 (a+x)^{5/2}-4 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}+\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 64, normalized size = 0.88 \begin {gather*} -\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \sqrt {a (1+\sin (c+d x))} (-365+63 \cos (2 (c+d x))+364 \sin (c+d x))}{693 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-1/693*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*Sqrt[a*(1 + Sin[c + d*x])]*(-365 + 63*Cos[2*(c + d*x)] + 364*S
in[c + d*x]))/d

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Maple [A]
time = 0.27, size = 41, normalized size = 0.56

method	result	size
default	\(-\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (63 \left (\cos ^{2}\left (d x +c \right )\right )+182 \sin \left (d x +c \right )-214\right )}{693 a^{3} d}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/693/a^3*(a+a*sin(d*x+c))^(7/2)*(63*cos(d*x+c)^2+182*sin(d*x+c)-214)/d

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Maxima [A]
time = 0.31, size = 55, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (63 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 308 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 396 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}\right )}}{693 \, a^{5} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*(a*sin(d*x + c) + a)^(11/2) - 308*(a*sin(d*x + c) + a)^(9/2)*a + 396*(a*sin(d*x + c) + a)^(7/2)*a^2)
/(a^5*d)

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Fricas [A]
time = 0.35, size = 68, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (7 \, \cos \left (d x + c\right )^{4} + 16 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/693*(7*cos(d*x + c)^4 + 16*cos(d*x + c)^2 + (63*cos(d*x + c)^4 + 80*cos(d*x + c)^2 + 128)*sin(d*x + c) + 128
)*sqrt(a*sin(d*x + c) + a)/d

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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(cos(d*x+c)**5*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 7.52, size = 99, normalized size = 1.36 \begin {gather*} \frac {64 \, \sqrt {2} {\left (63 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 154 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 99 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{693 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

64/693*sqrt(2)*(63*cos(-1/4*pi + 1/2*d*x + 1/2*c)^11*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) - 154*cos(-1/4*pi + 1
/2*d*x + 1/2*c)^9*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 99*cos(-1/4*pi + 1/2*d*x + 1/2*c)^7*sgn(cos(-1/4*pi +
1/2*d*x + 1/2*c)))*sqrt(a)/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^5\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*(a + a*sin(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^5*(a + a*sin(c + d*x))^(1/2), x)

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