3.6.6 \(\int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [506]

Optimal. Leaf size=152 \[ \frac {2 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}{b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d} \]

[Out]

-8/3*a*(a^2-b^2)*(a+b*sin(d*x+c))^(3/2)/b^5/d+4/5*(3*a^2-b^2)*(a+b*sin(d*x+c))^(5/2)/b^5/d-8/7*a*(a+b*sin(d*x+
c))^(7/2)/b^5/d+2/9*(a+b*sin(d*x+c))^(9/2)/b^5/d+2*(a^2-b^2)^2*(a+b*sin(d*x+c))^(1/2)/b^5/d

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Rubi [A]
time = 0.07, antiderivative size = 152, 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 {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {2 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}{b^5 d}+\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(2*(a^2 - b^2)^2*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) +
(4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (2*(a +
b*Sin[c + d*x])^(9/2))/(9*b^5*d)

Rule 711

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]

Rule 2747

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
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{\sqrt {a+x}} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\left (a^2-b^2\right )^2}{\sqrt {a+x}}-4 \left (a^3-a b^2\right ) \sqrt {a+x}+2 \left (3 a^2-b^2\right ) (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {2 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}{b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}\\ \end {align*}

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Mathematica [A]
time = 1.37, size = 161, normalized size = 1.06 \begin {gather*} \frac {a \left (1024 a^4-2496 a^2 b^2+2121 b^4\right ) \sqrt {1+\frac {b \sin (c+d x)}{a}} \left (-1+\sqrt {1+\frac {b \sin (c+d x)}{a}}\right )-b (a+b \sin (c+d x)) \left (-35 b^3 \cos (4 (c+d x))+32 a \left (16 a^2-37 b^2\right ) \sin (c+d x)-4 b \cos (2 (c+d x)) \left (-48 a^2+91 b^2+40 a b \sin (c+d x)\right )\right )}{1260 b^5 d \sqrt {a+b \sin (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(a*(1024*a^4 - 2496*a^2*b^2 + 2121*b^4)*Sqrt[1 + (b*Sin[c + d*x])/a]*(-1 + Sqrt[1 + (b*Sin[c + d*x])/a]) - b*(
a + b*Sin[c + d*x])*(-35*b^3*Cos[4*(c + d*x)] + 32*a*(16*a^2 - 37*b^2)*Sin[c + d*x] - 4*b*Cos[2*(c + d*x)]*(-4
8*a^2 + 91*b^2 + 40*a*b*Sin[c + d*x])))/(1260*b^5*d*Sqrt[a + b*Sin[c + d*x]])

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Maple [A]
time = 2.63, size = 126, normalized size = 0.83

method	result	size
default	\(\frac {2 \sqrt {a +b \sin \left (d x +c \right )}\, \left (35 b^{4} \left (\cos ^{4}\left (d x +c \right )\right )+40 a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-48 a^{2} b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+56 b^{4} \left (\cos ^{2}\left (d x +c \right )\right )-64 a^{3} b \sin \left (d x +c \right )+128 a \,b^{3} \sin \left (d x +c \right )+128 a^{4}-288 a^{2} b^{2}+224 b^{4}\right )}{315 b^{5} d}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/315/b^5*(a+b*sin(d*x+c))^(1/2)*(35*b^4*cos(d*x+c)^4+40*a*b^3*cos(d*x+c)^2*sin(d*x+c)-48*a^2*b^2*cos(d*x+c)^2
+56*b^4*cos(d*x+c)^2-64*a^3*b*sin(d*x+c)+128*a*b^3*sin(d*x+c)+128*a^4-288*a^2*b^2+224*b^4)/d

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Maxima [A]
time = 0.30, size = 160, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (315 \, \sqrt {b \sin \left (d x + c\right ) + a} - \frac {42 \, {\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}\right )}}{b^{2}} + \frac {35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{4}}{b^{4}}\right )}}{315 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(b*sin(d*x + c) + a) - 42*(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)/b^2 + (35*(b*sin(d*x + c) + a)^(9/2) - 180*(b*sin(d*x + c) + a)^(7/2)*a + 378*(b
*sin(d*x + c) + a)^(5/2)*a^2 - 420*(b*sin(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(b*sin(d*x + c) + a)*a^4)/b^4)/(b*
d)

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Fricas [A]
time = 0.35, size = 111, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} \cos \left (d x + c\right )^{4} + 128 \, a^{4} - 288 \, a^{2} b^{2} + 224 \, b^{4} - 8 \, {\left (6 \, a^{2} b^{2} - 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (5 \, a b^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} b + 16 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{315 \, b^{5} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*cos(d*x + c)^4 + 128*a^4 - 288*a^2*b^2 + 224*b^4 - 8*(6*a^2*b^2 - 7*b^4)*cos(d*x + c)^2 + 8*(5*a
*b^3*cos(d*x + c)^2 - 8*a^3*b + 16*a*b^3)*sin(d*x + c))*sqrt(b*sin(d*x + c) + a)/(b^5*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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5/(a+b*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 6.25, size = 161, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{4} - 126 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} b^{2} + 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a b^{2} - 630 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2} b^{2} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} b^{4}\right )}}{315 \, b^{5} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

2/315*(35*(b*sin(d*x + c) + a)^(9/2) - 180*(b*sin(d*x + c) + a)^(7/2)*a + 378*(b*sin(d*x + c) + a)^(5/2)*a^2 -
420*(b*sin(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(b*sin(d*x + c) + a)*a^4 - 126*(b*sin(d*x + c) + a)^(5/2)*b^2 +
420*(b*sin(d*x + c) + a)^(3/2)*a*b^2 - 630*sqrt(b*sin(d*x + c) + a)*a^2*b^2 + 315*sqrt(b*sin(d*x + c) + a)*b^4
)/(b^5*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a + b*sin(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^5/(a + b*sin(c + d*x))^(1/2), x)

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