3.2.16 \(\int \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [116]

Optimal. Leaf size=73 \[ \frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^3 d}-\frac {8 (a+a \sin (c+d x))^{11/2}}{11 a^4 d}+\frac {2 (a+a \sin (c+d x))^{13/2}}{13 a^5 d} \]

[Out]

8/9*(a+a*sin(d*x+c))^(9/2)/a^3/d-8/11*(a+a*sin(d*x+c))^(11/2)/a^4/d+2/13*(a+a*sin(d*x+c))^(13/2)/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)^{13/2}}{13 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(8*(a + a*Sin[c + d*x])^(9/2))/(9*a^3*d) - (8*(a + a*Sin[c + d*x])^(11/2))/(11*a^4*d) + (2*(a + a*Sin[c + d*x]
)^(13/2))/(13*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) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\text {Subst}\left (\int (a-x)^2 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^2 (a+x)^{7/2}-4 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^3 d}-\frac {8 (a+a \sin (c+d x))^{11/2}}{11 a^4 d}+\frac {2 (a+a \sin (c+d x))^{13/2}}{13 a^5 d}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 51, normalized size = 0.70 \begin {gather*} \frac {2 (1+\sin (c+d x))^3 (a (1+\sin (c+d x)))^{3/2} \left (203-270 \sin (c+d x)+99 \sin ^2(c+d x)\right )}{1287 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(2*(1 + Sin[c + d*x])^3*(a*(1 + Sin[c + d*x]))^(3/2)*(203 - 270*Sin[c + d*x] + 99*Sin[c + d*x]^2))/(1287*d)

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

method	result	size
default	\(-\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (99 \left (\cos ^{2}\left (d x +c \right )\right )+270 \sin \left (d x +c \right )-302\right )}{1287 a^{3} d}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/1287/a^3*(a+a*sin(d*x+c))^(9/2)*(99*cos(d*x+c)^2+270*sin(d*x+c)-302)/d

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Maxima [A]
time = 0.30, size = 55, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (99 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 468 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 572 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2}\right )}}{1287 \, 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))^(3/2),x, algorithm="maxima")

[Out]

2/1287*(99*(a*sin(d*x + c) + a)^(13/2) - 468*(a*sin(d*x + c) + a)^(11/2)*a + 572*(a*sin(d*x + c) + a)^(9/2)*a^
2)/(a^5*d)

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Fricas [A]
time = 0.35, size = 88, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left (99 \, a \cos \left (d x + c\right )^{6} - 14 \, a \cos \left (d x + c\right )^{4} - 32 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (63 \, a \cos \left (d x + c\right )^{4} + 80 \, a \cos \left (d x + c\right )^{2} + 128 \, a\right )} \sin \left (d x + c\right ) - 256 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1287 \, 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))^(3/2),x, algorithm="fricas")

[Out]

-2/1287*(99*a*cos(d*x + c)^6 - 14*a*cos(d*x + c)^4 - 32*a*cos(d*x + c)^2 - 2*(63*a*cos(d*x + c)^4 + 80*a*cos(d
*x + c)^2 + 128*a)*sin(d*x + c) - 256*a)*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))**(3/2),x)

[Out]

Timed out

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Giac [A]
time = 5.02, size = 102, normalized size = 1.40 \begin {gather*} \frac {128 \, \sqrt {2} {\left (99 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 234 \, a \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 ) + 143 \, a \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 )\right )} \sqrt {a}}{1287 \, 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))^(3/2),x, algorithm="giac")

[Out]

128/1287*sqrt(2)*(99*a*cos(-1/4*pi + 1/2*d*x + 1/2*c)^13*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) - 234*a*cos(-1/4*
pi + 1/2*d*x + 1/2*c)^11*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 143*a*cos(-1/4*pi + 1/2*d*x + 1/2*c)^9*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\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,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))^(3/2),x)

[Out]

int(cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2), x)

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