3.149 \(\int \frac {(b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=65 \[ \frac {b x \sqrt {b \sec (c+d x)}}{2 \sqrt {\sec (c+d x)}}+\frac {b \sin (c+d x) \sqrt {b \sec (c+d x)}}{2 d \sec ^{\frac {3}{2}}(c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2635, 8} \[ \frac {b x \sqrt {b \sec (c+d x)}}{2 \sqrt {\sec (c+d x)}}+\frac {b \sin (c+d x) \sqrt {b \sec (c+d x)}}{2 d \sec ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2),x]

[Out]

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

Rule 8

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 45, normalized size = 0.69 \[ \frac {(2 (c+d x)+\sin (2 (c+d x))) (b \sec (c+d x))^{3/2}}{4 d \sec ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2),x]

[Out]

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

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fricas [A]  time = 0.85, size = 161, normalized size = 2.48 \[ \left [\frac {2 \, b \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + \sqrt {-b} b \log \left (-2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{4 \, d}, \frac {b \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{2 \, d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c))^(3/2)/sec(d*x + c)^(7/2), x)

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maple [A]  time = 1.19, size = 54, normalized size = 0.83 \[ \frac {\left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{2 d \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \cos \left (d x +c \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/2),x)

[Out]

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

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maxima [A]  time = 0.67, size = 28, normalized size = 0.43 \[ \frac {{\left (2 \, {\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {b}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.43, size = 42, normalized size = 0.65 \[ \frac {b\,\left (\sin \left (2\,c+2\,d\,x\right )+2\,d\,x\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{4\,d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((b*sec(d*x+c))**(3/2)/sec(d*x+c)**(7/2),x)

[Out]

Timed out

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