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]

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

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Rubi [A]  time = 0.08, antiderivative size = 97, 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 = {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 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])

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])

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*}

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Mathematica [A]  time = 0.30, 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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fricas [A]  time = 0.69, size = 82, normalized size = 0.85 \[ \frac {2 \, {\left (231 \, \cos \left (d x + c\right )^{6} + 28 \, \cos \left (d x + c\right )^{4} + 64 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 512\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3003 \, a d} \]

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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giac [B]  time = 2.70, size = 430, normalized size = 4.43 \[ \frac {2 \, {\left (\frac {835 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3003 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3926 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {6006 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15301 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {21021 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15444 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15444 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {21021 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15301 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {6006 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {3926 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {835 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {3003 \, a^{6}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{3003 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {13}{2}} d} \]

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*(835*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (3003*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (3926*a^6/sgn(tan(1/
2*d*x + 1/2*c) + 1) + (6006*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (15301*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (21
021*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (15444*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (15444*a^6/sgn(tan(1/2*d*x
+ 1/2*c) + 1) + (21021*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (15301*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (6006*a^
6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (3926*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1) + (835*a^6*tan(1/2*d*x + 1/2*c)/sgn(
tan(1/2*d*x + 1/2*c) + 1) + 3003*a^6/sgn(tan(1/2*d*x + 1/2*c) + 1))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c)
)*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c)
)*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c)
)/((a*tan(1/2*d*x + 1/2*c)^2 + a)^(13/2)*d)

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maple [A]  time = 0.16, size = 57, normalized size = 0.59 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (231 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-945 \left (\cos ^{2}\left (d x +c \right )\right )-1652 \sin \left (d x +c \right )+1780\right )}{3003 a^{4} d} \]

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 = 0.41, size = 281, normalized size = 2.90 \[ \frac {2 \, {\left (15015 \, \sqrt {a \sin \left (d x + c\right ) + a} - \frac {3003 \, {\left (3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac {143 \, {\left (35 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{4}\right )}}{a^{4}} - \frac {5 \, {\left (231 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 1638 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{6}\right )}}{a^{6}}\right )}}{15015 \, a d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^7}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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