\(\int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [172]

Optimal result

Integrand size = 23, antiderivative size = 30 \[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 a \cos ^5(c+d x)}{5 d (a+a \sin (c+d x))^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2752} \[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 a \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}} \]

[In]

Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]

[Out]

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

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos ^5(c+d x)}{5 d (a+a \sin (c+d x))^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 \cos ^5(c+d x) \sqrt {a (1+\sin (c+d x))}}{5 a^2 d (1+\sin (c+d x))^3} \]

[In]

Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(-2*Cos[c + d*x]^5*Sqrt[a*(1 + Sin[c + d*x])])/(5*a^2*d*(1 + Sin[c + d*x])^3)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57

method	result	size
default	\(\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{3}}{5 a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(47\)

[In]

int(cos(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/5/a*(1+sin(d*x+c))*(sin(d*x+c)-1)^3/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.27 \[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]

[In]

integrate(cos(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

2/5*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 - (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) -
4)*sqrt(a*sin(d*x + c) + a)/(a^2*d*cos(d*x + c) + a^2*d*sin(d*x + c) + a^2*d)

Sympy [F]

\[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(cos(d*x+c)**4/(a+a*sin(d*x+c))**(3/2),x)

[Out]

Integral(cos(c + d*x)**4/(a*(sin(c + d*x) + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^4/(a*sin(d*x + c) + a)^(3/2), x)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {8 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}{5 \, a^{\frac {3}{2}} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]

[In]

integrate(cos(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

8/5*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5/(a^(3/2)*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]

[In]

int(cos(c + d*x)^4/(a + a*sin(c + d*x))^(3/2),x)

[Out]

int(cos(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x)