3.141 \(\int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=73 \[ \frac {2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]

[Out]

8/13*(a+a*sin(d*x+c))^(13/2)/a^3/d-8/15*(a+a*sin(d*x+c))^(15/2)/a^4/d+2/17*(a+a*sin(d*x+c))^(17/2)/a^5/d

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Rubi [A]  time = 0.07, 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 = {2667, 43} \[ \frac {2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

(8*(a + a*Sin[c + d*x])^(13/2))/(13*a^3*d) - (8*(a + a*Sin[c + d*x])^(15/2))/(15*a^4*d) + (2*(a + a*Sin[c + d*
x])^(17/2))/(17*a^5*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 \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{11/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{11/2}-4 a (a+x)^{13/2}+(a+x)^{15/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{13/2}}{13 a^3 d}-\frac {8 (a+a \sin (c+d x))^{15/2}}{15 a^4 d}+\frac {2 (a+a \sin (c+d x))^{17/2}}{17 a^5 d}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 54, normalized size = 0.74 \[ \frac {2 a^3 (\sin (c+d x)+1)^6 \left (195 \sin ^2(c+d x)-494 \sin (c+d x)+331\right ) \sqrt {a (\sin (c+d x)+1)}}{3315 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

(2*a^3*(1 + Sin[c + d*x])^6*Sqrt[a*(1 + Sin[c + d*x])]*(331 - 494*Sin[c + d*x] + 195*Sin[c + d*x]^2))/(3315*d)

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fricas [B]  time = 0.84, size = 128, normalized size = 1.75 \[ \frac {2 \, {\left (195 \, a^{3} \cos \left (d x + c\right )^{8} - 1072 \, a^{3} \cos \left (d x + c\right )^{6} + 56 \, a^{3} \cos \left (d x + c\right )^{4} + 128 \, a^{3} \cos \left (d x + c\right )^{2} + 1024 \, a^{3} - 4 \, {\left (169 \, a^{3} \cos \left (d x + c\right )^{6} - 126 \, a^{3} \cos \left (d x + c\right )^{4} - 160 \, a^{3} \cos \left (d x + c\right )^{2} - 256 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

2/3315*(195*a^3*cos(d*x + c)^8 - 1072*a^3*cos(d*x + c)^6 + 56*a^3*cos(d*x + c)^4 + 128*a^3*cos(d*x + c)^2 + 10
24*a^3 - 4*(169*a^3*cos(d*x + c)^6 - 126*a^3*cos(d*x + c)^4 - 160*a^3*cos(d*x + c)^2 - 256*a^3)*sin(d*x + c))*
sqrt(a*sin(d*x + c) + a)/d

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giac [B]  time = 3.13, size = 537, normalized size = 7.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

1/98017920*sqrt(2)*(51051*a^3*cos(1/4*pi + 15/2*d*x + 15/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d - 696150*a
^3*cos(1/4*pi + 11/2*d*x + 11/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d - 5469750*a^3*cos(1/4*pi + 7/2*d*x +
7/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d - 16846830*a^3*cos(1/4*pi + 3/2*d*x + 3/2*c)*sgn(cos(-1/4*pi + 1/
2*d*x + 1/2*c))/d + 45045*a^3*cos(-1/4*pi + 17/2*d*x + 17/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d - 589050*
a^3*cos(-1/4*pi + 13/2*d*x + 13/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d - 4254250*a^3*cos(-1/4*pi + 9/2*d*x
+ 9/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d - 10108098*a^3*cos(-1/4*pi + 5/2*d*x + 5/2*c)*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))/d - 353430*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4*pi + 13/2*d*x + 13/2*c)/d - 850
850*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4*pi + 9/2*d*x + 9/2*c)/d + 5207202*a^3*sgn(cos(-1/4*pi + 1/
2*d*x + 1/2*c))*sin(1/4*pi + 5/2*d*x + 5/2*c)/d + 84234150*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4*pi
+ 1/2*d*x + 1/2*c)/d - 306306*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 15/2*d*x + 15/2*c)/d - 696
150*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 11/2*d*x + 11/2*c)/d + 3719430*a^3*sgn(cos(-1/4*pi +
1/2*d*x + 1/2*c))*sin(-1/4*pi + 7/2*d*x + 7/2*c)/d + 28078050*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/
4*pi + 3/2*d*x + 3/2*c)/d)*sqrt(a)

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maple [A]  time = 0.17, size = 41, normalized size = 0.56 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {13}{2}} \left (195 \left (\cos ^{2}\left (d x +c \right )\right )+494 \sin \left (d x +c \right )-526\right )}{3315 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x)

[Out]

-2/3315/a^3*(a+a*sin(d*x+c))^(13/2)*(195*cos(d*x+c)^2+494*sin(d*x+c)-526)/d

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maxima [A]  time = 0.32, size = 55, normalized size = 0.75 \[ \frac {2 \, {\left (195 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {17}{2}} - 884 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a + 1020 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a^{2}\right )}}{3315 \, a^{5} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

2/3315*(195*(a*sin(d*x + c) + a)^(17/2) - 884*(a*sin(d*x + c) + a)^(15/2)*a + 1020*(a*sin(d*x + c) + a)^(13/2)
*a^2)/(a^5*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*(a + a*sin(c + d*x))^(7/2),x)

[Out]

int(cos(c + d*x)^5*(a + a*sin(c + d*x))^(7/2), 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)**5*(a+a*sin(d*x+c))**(7/2),x)

[Out]

Timed out

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