3.156 \(\int \frac{\cos ^7(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d} \]

[Out]

(16*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(3*a^5*d) + (12*(a + a*Sin[c + d*x]
)^(11/2))/(11*a^6*d) - (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^7*d)

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Rubi [A]  time = 0.075736, antiderivative size = 97, normalized size of antiderivative = 1., 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 = {2667, 43} \[ -\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(16*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(3*a^5*d) + (12*(a + a*Sin[c + d*x]
)^(11/2))/(11*a^6*d) - (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^7*d)

Rule 2667

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])

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^{5/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{5/2}-12 a^2 (a+x)^{7/2}+6 a (a+x)^{9/2}-(a+x)^{11/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{16 (a+a \sin (c+d x))^{7/2}}{7 a^4 d}-\frac{8 (a+a \sin (c+d x))^{9/2}}{3 a^5 d}+\frac{12 (a+a \sin (c+d x))^{11/2}}{11 a^6 d}-\frac{2 (a+a \sin (c+d x))^{13/2}}{13 a^7 d}\\ \end{align*}

Mathematica [A]  time = 0.288566, size = 61, normalized size = 0.63 \[ -\frac{2 (\sin (c+d x)+1)^4 \left (231 \sin ^3(c+d x)-945 \sin ^2(c+d x)+1421 \sin (c+d x)-835\right )}{3003 d \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*(1 + Sin[c + d*x])^4*(-835 + 1421*Sin[c + d*x] - 945*Sin[c + d*x]^2 + 231*Sin[c + d*x]^3))/(3003*d*Sqrt[a*
(1 + Sin[c + d*x])])

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Maple [A]  time = 0.089, size = 57, normalized size = 0.6 \begin{align*}{\frac{462\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -1890\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-3304\,\sin \left ( dx+c \right ) +3560}{3003\,{a}^{4}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a+a*sin(d*x+c))^(1/2),x)

[Out]

2/3003/a^4*(a+a*sin(d*x+c))^(7/2)*(231*cos(d*x+c)^2*sin(d*x+c)-945*cos(d*x+c)^2-1652*sin(d*x+c)+1780)/d

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Maxima [B]  time = 0.969764, size = 379, normalized size = 3.91 \begin{align*} \frac{2 \,{\left (15015 \, \sqrt{a \sin \left (d x + c\right ) + a} - \frac{3003 \,{\left (3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac{143 \,{\left (35 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{4}\right )}}{a^{4}} - \frac{5 \,{\left (231 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 1638 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 5005 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2} - 8580 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{4} - 6006 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{5} + 3003 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{6}\right )}}{a^{6}}\right )}}{15015 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2/15015*(15015*sqrt(a*sin(d*x + c) + a) - 3003*(3*(a*sin(d*x + c) + a)^(5/2) - 10*(a*sin(d*x + c) + a)^(3/2)*a
+ 15*sqrt(a*sin(d*x + c) + a)*a^2)/a^2 + 143*(35*(a*sin(d*x + c) + a)^(9/2) - 180*(a*sin(d*x + c) + a)^(7/2)*
a + 378*(a*sin(d*x + c) + a)^(5/2)*a^2 - 420*(a*sin(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(a*sin(d*x + c) + a)*a^4
)/a^4 - 5*(231*(a*sin(d*x + c) + a)^(13/2) - 1638*(a*sin(d*x + c) + a)^(11/2)*a + 5005*(a*sin(d*x + c) + a)^(9
/2)*a^2 - 8580*(a*sin(d*x + c) + a)^(7/2)*a^3 + 9009*(a*sin(d*x + c) + a)^(5/2)*a^4 - 6006*(a*sin(d*x + c) + a
)^(3/2)*a^5 + 3003*sqrt(a*sin(d*x + c) + a)*a^6)/a^6)/(a*d)

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Fricas [A]  time = 1.91648, size = 228, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (231 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{2} + 4 \,{\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 ) + 512\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3003 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/3003*(231*cos(d*x + c)^6 + 28*cos(d*x + c)^4 + 64*cos(d*x + c)^2 + 4*(63*cos(d*x + c)^4 + 80*cos(d*x + c)^2
+ 128)*sin(d*x + c) + 512)*sqrt(a*sin(d*x + c) + a)/(a*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)**7/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.30596, size = 97, normalized size = 1. \begin{align*} -\frac{2 \,{\left (231 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 1638 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 4004 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2} - 3432 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3}\right )}}{3003 \, a^{7} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-2/3003*(231*(a*sin(d*x + c) + a)^(13/2) - 1638*(a*sin(d*x + c) + a)^(11/2)*a + 4004*(a*sin(d*x + c) + a)^(9/2
)*a^2 - 3432*(a*sin(d*x + c) + a)^(7/2)*a^3)/(a^7*d)