3.159 \(\int \frac{(b \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=69 \[ \frac{b^2 x \sqrt{b \sec (c+d x)}}{2 \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{b \sec (c+d x)}}{2 d \sec ^{\frac{3}{2}}(c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0154454, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {17, 2635, 8} \[ \frac{b^2 x \sqrt{b \sec (c+d x)}}{2 \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{b \sec (c+d x)}}{2 d \sec ^{\frac{3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]

[Out]

(b^2*x*Sqrt[b*Sec[c + d*x]])/(2*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*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{(b \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \int 1 \, dx}{2 \sqrt{\sec (c+d x)}}\\ &=\frac{b^2 x \sqrt{b \sec (c+d x)}}{2 \sqrt{\sec (c+d x)}}+\frac{b^2 \sqrt{b \sec (c+d x)} \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.1127, size = 45, normalized size = 0.65 \[ \frac{(2 (c+d x)+\sin (2 (c+d x))) (b \sec (c+d x))^{5/2}}{4 d \sec ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]

[Out]

((b*Sec[c + d*x])^(5/2)*(2*(c + d*x) + Sin[2*(c + d*x)]))/(4*d*Sec[c + d*x]^(5/2))

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Maple [A]  time = 0.105, size = 54, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +dx+c}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}} \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))^(5/2)/sec(d*x+c)^(9/2),x)

[Out]

1/2/d*(cos(d*x+c)*sin(d*x+c)+d*x+c)*(b/cos(d*x+c))^(5/2)/cos(d*x+c)^2/(1/cos(d*x+c))^(9/2)

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Maxima [A]  time = 2.12771, size = 43, normalized size = 0.62 \begin{align*} \frac{{\left (2 \,{\left (d x + c\right )} b^{2} + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt{b}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(5/2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/4*(2*(d*x + c)*b^2 + b^2*sin(2*d*x + 2*c))*sqrt(b)/d

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Fricas [A]  time = 2.04022, size = 444, normalized size = 6.43 \begin{align*} \left [\frac{2 \, b^{2} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) + \sqrt{-b} b^{2} \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 )}{4 \, d}, \frac{b^{2} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) + b^{\frac{5}{2}} \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 )}{2 \, d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(5/2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

[1/4*(2*b^2*sqrt(b/cos(d*x + c))*cos(d*x + c)^(3/2)*sin(d*x + c) + sqrt(-b)*b^2*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, 1/2*(b^2*sqrt(b/cos(d*x + c))*cos(d*x + c
)^(3/2)*sin(d*x + c) + b^(5/2)*arctan(sqrt(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))**(5/2)/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{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}}}{\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))^(5/2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c))^(5/2)/sec(d*x + c)^(9/2), x)