∫ cos5 (c+dx)___ (a+a sin(c+dx))3∕2 dx [171]

Optimal. Leaf size=73 \[ \frac {8 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {8 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}+\frac {2 (a+a \sin (c+d x))^{7/2}}{7 a^5 d} \]

8/3*(a+a*sin(d*x+c))^(3/2)/a^3/d-8/5*(a+a*sin(d*x+c))^(5/2)/a^4/d+2/7*(a+a*sin(d*x+c))^(7/2)/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)^{7/2}}{7 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \end {gather*}

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

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

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])

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])

\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int (a-x)^2 \sqrt {a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^2 \sqrt {a+x}-4 a (a+x)^{3/2}+(a+x)^{5/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {8 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}+\frac {2 (a+a \sin (c+d x))^{7/2}}{7 a^5 d}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 44, normalized size = 0.60 \begin {gather*} \frac {2 (a (1+\sin (c+d x)))^{3/2} \left (71-54 \sin (c+d x)+15 \sin ^2(c+d x)\right )}{105 a^3 d} \end {gather*}

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

(2*(a*(1 + Sin[c + d*x]))^(3/2)*(71 - 54*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(105*a^3*d)

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method	result	size

default	\(-\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (15 \left (\cos ^{2}\left (d x +c \right )\right )+54 \sin \left (d x +c \right )-86\right )}{105 a^{3} d}\)	\(41\)

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

-2/105/a^3*(a+a*sin(d*x+c))^(3/2)*(15*cos(d*x+c)^2+54*sin(d*x+c)-86)/d

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Maxima [A]
time = 0.29, size = 55, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 84 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 140 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, a^{5} d} \end {gather*}

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

2/105*(15*(a*sin(d*x + c) + a)^(7/2) - 84*(a*sin(d*x + c) + a)^(5/2)*a + 140*(a*sin(d*x + c) + a)^(3/2)*a^2)/(

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Fricas [A]
time = 0.33, size = 52, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (39 \, \cos \left (d x + c\right )^{2} - {\left (15 \, \cos \left (d x + c\right )^{2} - 32\right )} \sin \left (d x + c\right ) + 32\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{105 \, a^{2} d} \end {gather*}

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

2/105*(39*cos(d*x + c)^2 - (15*cos(d*x + c)^2 - 32)*sin(d*x + c) + 32)*sqrt(a*sin(d*x + c) + a)/(a^2*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*}

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

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Giac [A]
time = 4.05, size = 90, normalized size = 1.23 \begin {gather*} \frac {16 \, {\left (15 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 42 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )}}{105 \, a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \end {gather*}

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

16/105*(15*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^7 - 42*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c

)^5 + 35*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^3)/(a^2*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^5}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}

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

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

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3.2.71 \(\int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [171]