3.150 \(\int \frac {(b \sec (c+d x))^{3/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=72 \[ \frac {b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}-\frac {b \sin ^3(c+d x) \sqrt {b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}} \]

[Out]

b*sin(d*x+c)*(b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(1/2)-1/3*b*sin(d*x+c)^3*(b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(1/2
)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 72, 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 = {17, 2633} \[ \frac {b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}-\frac {b \sin ^3(c+d x) \sqrt {b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(9/2),x]

[Out]

(b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) - (b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sq
rt[Sec[c + d*x]])

Rule 17

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v])/Sqrt[a*v]
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \frac {(b \sec (c+d x))^{3/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=-\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\sec (c+d x)}}\\ &=\frac {b \sqrt {b \sec (c+d x)} \sin (c+d x)}{d \sqrt {\sec (c+d x)}}-\frac {b \sqrt {b \sec (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\sec (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 45, normalized size = 0.62 \[ \frac {\sin (c+d x) (\cos (2 (c+d x))+5) (b \sec (c+d x))^{3/2}}{6 d \sec ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(9/2),x]

[Out]

((5 + Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 51, normalized size = 0.71 \[ \frac {{\left (b \cos \left (d x + c\right )^{3} + 2 \, b \cos \left (d x + c\right )\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, d \sqrt {\cos \left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

1/3*(b*cos(d*x + c)^3 + 2*b*cos(d*x + c))*sqrt(b/cos(d*x + c))*sin(d*x + c)/(d*sqrt(cos(d*x + c)))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c))^(3/2)/sec(d*x + c)^(9/2), x)

________________________________________________________________________________________

maple [A]  time = 1.10, size = 52, normalized size = 0.72 \[ \frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{3 d \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \cos \left (d x +c \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(9/2),x)

[Out]

1/3/d*(2+cos(d*x+c)^2)*(b/cos(d*x+c))^(3/2)*sin(d*x+c)/(1/cos(d*x+c))^(9/2)/cos(d*x+c)^3

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 45, normalized size = 0.62 \[ \frac {{\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} \sqrt {b}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/12*(b*sin(3*d*x + 3*c) + 9*b*sin(1/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))))*sqrt(b)/d

________________________________________________________________________________________

mupad [B]  time = 0.44, size = 46, normalized size = 0.64 \[ \frac {b\,\left (9\,\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{12\,d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/cos(c + d*x))^(3/2)/(1/cos(c + d*x))^(9/2),x)

[Out]

(b*(9*sin(c + d*x) + sin(3*c + 3*d*x))*(b/cos(c + d*x))^(1/2))/(12*d*(1/cos(c + d*x))^(1/2))

________________________________________________________________________________________

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((b*sec(d*x+c))**(3/2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________