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

Optimal. Leaf size=24 \[ \frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a d} \]

[Out]

2/9*(a+a*sin(d*x+c))^(9/2)/a/d

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 32} \[ \frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + a*Sin[c + d*x])^(9/2))/(9*a*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 (c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 24, normalized size = 1.00 \[ \frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + a*Sin[c + d*x])^(9/2))/(9*a*d)

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 74, normalized size = 3.08 \[ \frac {2 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} - 4 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{9 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(a^3*cos(d*x + c)^4 - 8*a^3*cos(d*x + c)^2 + 8*a^3 - 4*(a^3*cos(d*x + c)^2 - 2*a^3)*sin(d*x + c))*sqrt(a*s
in(d*x + c) + a)/d

________________________________________________________________________________________

giac [B]  time = 0.97, size = 273, normalized size = 11.38 \[ \frac {1}{2520} \, \sqrt {2} {\left (\frac {45 \, a^{3} \cos \left (\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {1470 \, a^{3} \cos \left (\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {35 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {882 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {378 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {4410 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} - \frac {270 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {1470 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*sqrt(2)*(45*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 - 1470*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 + 35*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 - 882*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 - 3
78*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 + 4410*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 - 270*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 + 1470*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)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 21, normalized size = 0.88 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(a+a*sin(d*x+c))^(9/2)/d/a

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}}}{9 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(a*sin(d*x + c) + a)^(9/2)/(a*d)

________________________________________________________________________________________

mupad [B]  time = 4.84, size = 20, normalized size = 0.83 \[ \frac {2\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^{9/2}}{9\,a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a*(sin(c + d*x) + 1))^(9/2))/(9*a*d)

________________________________________________________________________________________

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)*(a+a*sin(d*x+c))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________