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

Optimal. Leaf size=73 \[ \frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \]

[Out]

(8*(a + a*Sin[c + d*x])^(11/2))/(11*a^3*d) - (8*(a + a*Sin[c + d*x])^(13/2))/(13*a^4*d) + (2*(a + a*Sin[c + d*
x])^(15/2))/(15*a^5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0735895, antiderivative size = 73, normalized size of antiderivative = 1., 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} \[ \frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(a + a*Sin[c + d*x])^(11/2))/(11*a^3*d) - (8*(a + a*Sin[c + d*x])^(13/2))/(13*a^4*d) + (2*(a + a*Sin[c + d*
x])^(15/2))/(15*a^5*d)

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])

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])

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^{9/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{9/2}-4 a (a+x)^{11/2}+(a+x)^{13/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 (a+a \sin (c+d x))^{11/2}}{11 a^3 d}-\frac{8 (a+a \sin (c+d x))^{13/2}}{13 a^4 d}+\frac{2 (a+a \sin (c+d x))^{15/2}}{15 a^5 d}\\ \end{align*}

Mathematica [A]  time = 0.201794, size = 51, normalized size = 0.7 \[ \frac{2 (\sin (c+d x)+1)^3 \left (143 \sin ^2(c+d x)-374 \sin (c+d x)+263\right ) (a (\sin (c+d x)+1))^{5/2}}{2145 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + Sin[c + d*x])^3*(a*(1 + Sin[c + d*x]))^(5/2)*(263 - 374*Sin[c + d*x] + 143*Sin[c + d*x]^2))/(2145*d)

________________________________________________________________________________________

Maple [A]  time = 0.082, size = 41, normalized size = 0.6 \begin{align*} -{\frac{286\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+748\,\sin \left ( dx+c \right ) -812}{2145\,{a}^{3}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2145/a^3*(a+a*sin(d*x+c))^(11/2)*(143*cos(d*x+c)^2+374*sin(d*x+c)-406)/d

________________________________________________________________________________________

Maxima [A]  time = 0.955465, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (143 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{15}{2}} - 660 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a + 780 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a^{2}\right )}}{2145 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2145*(143*(a*sin(d*x + c) + a)^(15/2) - 660*(a*sin(d*x + c) + a)^(13/2)*a + 780*(a*sin(d*x + c) + a)^(11/2)*
a^2)/(a^5*d)

________________________________________________________________________________________

Fricas [A]  time = 1.68519, size = 296, normalized size = 4.05 \begin{align*} -\frac{2 \,{\left (341 \, a^{2} \cos \left (d x + c\right )^{6} - 28 \, a^{2} \cos \left (d x + c\right )^{4} - 64 \, a^{2} \cos \left (d x + c\right )^{2} - 512 \, a^{2} +{\left (143 \, a^{2} \cos \left (d x + c\right )^{6} - 252 \, a^{2} \cos \left (d x + c\right )^{4} - 320 \, a^{2} \cos \left (d x + c\right )^{2} - 512 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{2145 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2145*(341*a^2*cos(d*x + c)^6 - 28*a^2*cos(d*x + c)^4 - 64*a^2*cos(d*x + c)^2 - 512*a^2 + (143*a^2*cos(d*x +
c)^6 - 252*a^2*cos(d*x + c)^4 - 320*a^2*cos(d*x + c)^2 - 512*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/d

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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