∫ cos3 (c+dx)______∘ a + b sin(c + dx) dx [507]

Optimal. Leaf size=81 \[ -\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}{b^3 d}+\frac {4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac {2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d} \]

4/3*a*(a+b*sin(d*x+c))^(3/2)/b^3/d-2/5*(a+b*sin(d*x+c))^(5/2)/b^3/d-2*(a^2-b^2)*(a+b*sin(d*x+c))^(1/2)/b^3/d

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Rubi [A]
time = 0.05, antiderivative size = 81, 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 = {2747, 711} \begin {gather*} -\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}{b^3 d}-\frac {2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d} \end {gather*}

Int[Cos[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]],x]

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

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*

x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer

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

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Mathematica [A]
time = 0.65, size = 108, normalized size = 1.33 \begin {gather*} \frac {b (3 b \cos (2 (c+d x))+8 a \sin (c+d x)) (a+b \sin (c+d x))-a \left (16 a^2-27 b^2\right ) \sqrt {1+\frac {b \sin (c+d x)}{a}} \left (-1+\sqrt {1+\frac {b \sin (c+d x)}{a}}\right )}{15 b^3 d \sqrt {a+b \sin (c+d x)}} \end {gather*}

Integrate[Cos[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]],x]

(b*(3*b*Cos[2*(c + d*x)] + 8*a*Sin[c + d*x])*(a + b*Sin[c + d*x]) - a*(16*a^2 - 27*b^2)*Sqrt[1 + (b*Sin[c + d*

x])/a]*(-1 + Sqrt[1 + (b*Sin[c + d*x])/a]))/(15*b^3*d*Sqrt[a + b*Sin[c + d*x]])

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method	result	size

default	\(-\frac {2 \sqrt {a +b \sin \left (d x +c \right )}\, \left (-3 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )-4 a b \sin \left (d x +c \right )+8 a^{2}-12 b^{2}\right )}{15 b^{3} d}\)	\(55\)

int(cos(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

-2/15/b^3*(a+b*sin(d*x+c))^(1/2)*(-3*b^2*cos(d*x+c)^2-4*a*b*sin(d*x+c)+8*a^2-12*b^2)/d

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Maxima [A]
time = 0.27, size = 75, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (15 \, \sqrt {b \sin \left (d x + c\right ) + a} - \frac {3 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2}}{b^{2}}\right )}}{15 \, b d} \end {gather*}

integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

2/15*(15*sqrt(b*sin(d*x + c) + a) - (3*(b*sin(d*x + c) + a)^(5/2) - 10*(b*sin(d*x + c) + a)^(3/2)*a + 15*sqrt(

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Fricas [A]
time = 0.35, size = 54, normalized size = 0.67 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} \cos \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) - 8 \, a^{2} + 12 \, b^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{15 \, b^{3} d} \end {gather*}

integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

2/15*(3*b^2*cos(d*x + c)^2 + 4*a*b*sin(d*x + c) - 8*a^2 + 12*b^2)*sqrt(b*sin(d*x + c) + a)/(b^3*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)**3/(a+b*sin(d*x+c))**(1/2),x)

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Giac [A]
time = 5.00, size = 72, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2} - 15 \, \sqrt {b \sin \left (d x + c\right ) + a} b^{2}\right )}}{15 \, b^{3} d} \end {gather*}

integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

-2/15*(3*(b*sin(d*x + c) + a)^(5/2) - 10*(b*sin(d*x + c) + a)^(3/2)*a + 15*sqrt(b*sin(d*x + c) + a)*a^2 - 15*s

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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 )}^3}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}

int(cos(c + d*x)^3/(a + b*sin(c + d*x))^(1/2),x)

int(cos(c + d*x)^3/(a + b*sin(c + d*x))^(1/2), x)

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3.6.7 \(\int \frac {\cos ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [507]