3.141 \(\int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=98 \[ \frac{3 x \sqrt{b \sec (c+d x)}}{8 \sqrt{\sec (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{b \sec (c+d x)}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{4 d \sec ^{\frac{7}{2}}(c+d x)} \]

[Out]

(3*x*Sqrt[b*Sec[c + d*x]])/(8*Sqrt[Sec[c + d*x]]) + (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(7/2
)) + (3*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2))

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Rubi [A]  time = 0.0251071, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {17, 2635, 8} \[ \frac{3 x \sqrt{b \sec (c+d x)}}{8 \sqrt{\sec (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{b \sec (c+d x)}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{4 d \sec ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*Sqrt[b*Sec[c + d*x]])/(8*Sqrt[Sec[c + d*x]]) + (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(7/2
)) + (3*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2))

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 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \sec (c+d x)} \int \cos ^4(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (3 \sqrt{b \sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (3 \sqrt{b \sec (c+d x)}\right ) \int 1 \, dx}{8 \sqrt{\sec (c+d x)}}\\ &=\frac{3 x \sqrt{b \sec (c+d x)}}{8 \sqrt{\sec (c+d x)}}+\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.116334, size = 55, normalized size = 0.56 \[ \frac{(12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x))) \sqrt{b \sec (c+d x)}}{32 d \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*Sec[c + d*x]]*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(32*d*Sqrt[Sec[c + d*x]])

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Maple [A]  time = 0.157, size = 74, normalized size = 0.8 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +3\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +3\,dx+3\,c}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d*(2*cos(d*x+c)^3*sin(d*x+c)+3*cos(d*x+c)*sin(d*x+c)+3*d*x+3*c)*(b/cos(d*x+c))^(1/2)/cos(d*x+c)^4/(1/cos(d
*x+c))^(9/2)

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Maxima [A]  time = 2.13244, size = 66, normalized size = 0.67 \begin{align*} \frac{{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} \sqrt{b}}{32 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/32*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))))*sqrt(b)/d

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Fricas [A]  time = 1.78951, size = 537, normalized size = 5.48 \begin{align*} \left [\frac{\frac{2 \,{\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 3 \, \sqrt{-b} \log \left (-2 \, \sqrt{-b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{16 \, d}, \frac{\frac{{\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 3 \, \sqrt{b} \arctan \left (\frac{\sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{b} \sqrt{\cos \left (d x + c\right )}}\right )}{8 \, d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

[1/16*(2*(2*cos(d*x + c)^4 + 3*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)) + 3*sqrt(-
b)*log(-2*sqrt(-b)*sqrt(b/cos(d*x + c))*cos(d*x + c)^(3/2)*sin(d*x + c) + 2*b*cos(d*x + c)^2 - b))/d, 1/8*((2*
cos(d*x + c)^4 + 3*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)) + 3*sqrt(b)*arctan(sqr
t(b/cos(d*x + c))*sin(d*x + c)/(sqrt(b)*sqrt(cos(d*x + c)))))/d]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))**(1/2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (d x + c\right )}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

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