3.143 \(\int \cos ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=49 \[ \frac{4 (a \sin (c+d x)+a)^{11/2}}{11 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0661054, antiderivative size = 49, 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} \[ \frac{4 (a \sin (c+d x)+a)^{11/2}}{11 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.133037, size = 44, normalized size = 0.9 \[ \frac{2 \left (26 a (a \sin (c+d x)+a)^{11/2}-11 (a \sin (c+d x)+a)^{13/2}\right )}{143 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

(2*(26*a*(a + a*Sin[c + d*x])^(11/2) - 11*(a + a*Sin[c + d*x])^(13/2)))/(143*a^3*d)

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Maple [A]  time = 0.082, size = 31, normalized size = 0.6 \begin{align*} -{\frac{22\,\sin \left ( dx+c \right ) -30}{143\,{a}^{2}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)^3*(a+a*sin(d*x+c))^(7/2),x)

[Out]

-2/143/a^2*(a+a*sin(d*x+c))^(11/2)*(11*sin(d*x+c)-15)/d

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Maxima [A]  time = 0.9574, size = 51, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (11 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 26 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a\right )}}{143 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

-2/143*(11*(a*sin(d*x + c) + a)^(13/2) - 26*(a*sin(d*x + c) + a)^(11/2)*a)/(a^3*d)

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Fricas [B]  time = 1.65446, size = 250, normalized size = 5.1 \begin{align*} \frac{2 \,{\left (11 \, a^{3} \cos \left (d x + c\right )^{6} - 68 \, a^{3} \cos \left (d x + c\right )^{4} + 8 \, a^{3} \cos \left (d x + c\right )^{2} + 64 \, a^{3} - 8 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{143 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

2/143*(11*a^3*cos(d*x + c)^6 - 68*a^3*cos(d*x + c)^4 + 8*a^3*cos(d*x + c)^2 + 64*a^3 - 8*(5*a^3*cos(d*x + c)^4
- 5*a^3*cos(d*x + c)^2 - 8*a^3)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/d

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^(7/2)*cos(d*x + c)^3, x)