3.140 \(\int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=70 \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}}-\frac{\sin ^3(c+d x) \sqrt{b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0163579, antiderivative size = 70, normalized size of antiderivative = 1., 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} \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}}-\frac{\sin ^3(c+d x) \sqrt{b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(7/2),x]

[Out]

(Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) - (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[S
ec[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{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \sec (c+d x)} \int \cos ^3(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=-\frac{\sqrt{b \sec (c+d x)} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{d \sqrt{\sec (c+d x)}}-\frac{\sqrt{b \sec (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.108713, size = 45, normalized size = 0.64 \[ \frac{\sin (c+d x) (\cos (2 (c+d x))+5) \sqrt{b \sec (c+d x)}}{6 d \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(7/2),x]

[Out]

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

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Maple [A]  time = 0.135, size = 52, normalized size = 0.7 \begin{align*}{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) }{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{7}{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)^(7/2),x)

[Out]

1/3/d*(cos(d*x+c)^2+2)*(b/cos(d*x+c))^(1/2)*sin(d*x+c)/(1/cos(d*x+c))^(7/2)/cos(d*x+c)^3

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Maxima [A]  time = 2.07164, size = 57, normalized size = 0.81 \begin{align*} \frac{\sqrt{b}{\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \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 )}}{12 \, 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)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.38295, size = 130, normalized size = 1.86 \begin{align*} \frac{{\left (\cos \left (d x + c\right )^{3} + 2 \, \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 )}} \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)^(7/2),x, algorithm="fricas")

[Out]

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

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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)**(7/2),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{7}{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)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(d*x + c))/sec(d*x + c)^(7/2), x)