3.79 \(\int \cos ^5(c+d x) \sqrt {b \sec (c+d x)} \, dx\)

Optimal. Leaf size=98 \[ \frac {2 b^4 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b^2 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \]

[Out]

2/9*b^4*sin(d*x+c)/d/(b*sec(d*x+c))^(7/2)+14/45*b^2*sin(d*x+c)/d/(b*sec(d*x+c))^(3/2)+14/15*b*(cos(1/2*d*x+1/2
*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)/(b*sec(d*x+c))^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3769, 3771, 2639} \[ \frac {2 b^4 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b^2 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sqrt[b*Sec[c + d*x]],x]

[Out]

(14*b*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^4*Sin[c + d*x])/(9*d*(b
*Sec[c + d*x])^(7/2)) + (14*b^2*Sin[c + d*x])/(45*d*(b*Sec[c + d*x])^(3/2))

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

\begin {align*} \int \cos ^5(c+d x) \sqrt {b \sec (c+d x)} \, dx &=b^5 \int \frac {1}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac {2 b^4 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {1}{9} \left (7 b^3\right ) \int \frac {1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac {2 b^4 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b^2 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {1}{15} (7 b) \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx\\ &=\frac {2 b^4 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b^2 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {(7 b) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=\frac {14 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^4 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b^2 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 71, normalized size = 0.72 \[ \frac {\sqrt {b \sec (c+d x)} \left ((33 \sin (c+d x)+5 \sin (3 (c+d x))) \cos ^2(c+d x)+84 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{90 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sqrt[b*Sec[c + d*x]],x]

[Out]

(Sqrt[b*Sec[c + d*x]]*(84*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^2*(33*Sin[c + d*x] + 5*S
in[3*(c + d*x)])))/(90*d)

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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (d x + c\right )} \cos \left (d x + c\right )^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c))*cos(d*x + c)^5, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (d x + c\right )} \cos \left (d x + c\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c))*cos(d*x + c)^5, x)

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maple [C]  time = 0.94, size = 325, normalized size = 3.32 \[ \frac {2 \sqrt {\frac {b}{\cos \left (d x +c \right )}}\, \left (21 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-21 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-5 \left (\cos ^{6}\left (d x +c \right )\right )+21 i \sin \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-21 i \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-2 \left (\cos ^{4}\left (d x +c \right )\right )-14 \left (\cos ^{2}\left (d x +c \right )\right )+21 \cos \left (d x +c \right )\right )}{45 d \sin \left (d x +c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45/d*(b/cos(d*x+c))^(1/2)*(21*I*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(-1+cos(
d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)-21*I*cos(d*x+c)*sin(d*x+c)*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+
c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-5*cos(d*x+c)^6+21*I*sin(d*x+c)*EllipticF(I*(-
1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-21*I*sin(d*x+c)*Ellipti
cE(I*(-1+cos(d*x+c))/sin(d*x+c),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-2*cos(d*x+c)^4-1
4*cos(d*x+c)^2+21*cos(d*x+c))/sin(d*x+c)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (d x + c\right )} \cos \left (d x + c\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c))*cos(d*x + c)^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^5\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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