3.112 \(\int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=203 \[ \frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {710 a^3 \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {284 a^3 \sin (c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}-\frac {568 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{693 d}+\frac {284 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d} \]

[Out]

284/231*a*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+284/99*a^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+710/693*a^3*cos(d
*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+46/99*a^3*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-568/693
*a^2*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/11*a^2*cos(d*x+c)^4*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.36, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2763, 2981, 2770, 2759, 2751, 2646} \[ \frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {710 a^3 \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}-\frac {568 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{693 d}+\frac {284 a^3 \sin (c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {284 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(284*a^3*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (710*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a +
a*Cos[c + d*x]]) + (46*a^3*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (568*a^2*Sqrt[a + a
*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a^2*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (
284*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d)

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2759

Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*
Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 2763

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*
(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2770

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(2*n*(b*c + a*d)
)/(b*(2*n + 1)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rubi steps

\begin {align*} \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx &=\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {19 a^2}{2}+\frac {23}{2} a^2 \cos (c+d x)\right ) \, dx\\ &=\frac {46 a^3 \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{99} \left (355 a^2\right ) \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {710 a^3 \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {46 a^3 \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{231} \left (710 a^2\right ) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {710 a^3 \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {46 a^3 \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {284 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac {1}{231} (284 a) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {710 a^3 \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {46 a^3 \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {568 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {284 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac {1}{99} \left (142 a^2\right ) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {284 a^3 \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {710 a^3 \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {46 a^3 \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {568 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {284 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}\\ \end {align*}

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Mathematica [A]  time = 0.50, size = 107, normalized size = 0.53 \[ \frac {a^2 \left (31878 \sin \left (\frac {1}{2} (c+d x)\right )+8778 \sin \left (\frac {3}{2} (c+d x)\right )+3465 \sin \left (\frac {5}{2} (c+d x)\right )+1287 \sin \left (\frac {7}{2} (c+d x)\right )+385 \sin \left (\frac {9}{2} (c+d x)\right )+63 \sin \left (\frac {11}{2} (c+d x)\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{11088 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(31878*Sin[(c + d*x)/2] + 8778*Sin[(3*(c + d*x))/2] + 3465*Si
n[(5*(c + d*x))/2] + 1287*Sin[(7*(c + d*x))/2] + 385*Sin[(9*(c + d*x))/2] + 63*Sin[(11*(c + d*x))/2]))/(11088*
d)

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fricas [A]  time = 0.98, size = 101, normalized size = 0.50 \[ \frac {2 \, {\left (63 \, a^{2} \cos \left (d x + c\right )^{5} + 224 \, a^{2} \cos \left (d x + c\right )^{4} + 355 \, a^{2} \cos \left (d x + c\right )^{3} + 426 \, a^{2} \cos \left (d x + c\right )^{2} + 568 \, a^{2} \cos \left (d x + c\right ) + 1136 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{693 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*a^2*cos(d*x + c)^5 + 224*a^2*cos(d*x + c)^4 + 355*a^2*cos(d*x + c)^3 + 426*a^2*cos(d*x + c)^2 + 568*
a^2*cos(d*x + c) + 1136*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)

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giac [A]  time = 2.03, size = 171, normalized size = 0.84 \[ \frac {1}{11088} \, \sqrt {2} {\left (\frac {63 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {385 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {1287 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {3465 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {8778 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {31878 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/11088*sqrt(2)*(63*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(11/2*d*x + 11/2*c)/d + 385*a^2*sgn(cos(1/2*d*x + 1/2*c))
*sin(9/2*d*x + 9/2*c)/d + 1287*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(7/2*d*x + 7/2*c)/d + 3465*a^2*sgn(cos(1/2*d*x
+ 1/2*c))*sin(5/2*d*x + 5/2*c)/d + 8778*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(3/2*d*x + 3/2*c)/d + 31878*a^2*sgn(
cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c)/d)*sqrt(a)

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maple [A]  time = 0.18, size = 112, normalized size = 0.55 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (504 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-364 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+178 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+100 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200\right ) \sqrt {2}}{693 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/693*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(504*cos(1/2*d*x+1/2*c)^10-364*cos(1/2*d*x+1/2*c)^8+178*cos(1/
2*d*x+1/2*c)^6+75*cos(1/2*d*x+1/2*c)^4+100*cos(1/2*d*x+1/2*c)^2+200)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

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maxima [A]  time = 1.22, size = 111, normalized size = 0.55 \[ \frac {{\left (63 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{11088 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

1/11088*(63*sqrt(2)*a^2*sin(11/2*d*x + 11/2*c) + 385*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 1287*sqrt(2)*a^2*sin(7
/2*d*x + 7/2*c) + 3465*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 8778*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 31878*sqrt(2
)*a^2*sin(1/2*d*x + 1/2*c))*sqrt(a)/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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