3.242 \(\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt [4]{\sec (c+d x)}} \, dx\)

Optimal. Leaf size=38 \[ \frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0564871, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {3814, 8} \[ \frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/4),x]

[Out]

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

Rule 3814

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(b^2*
Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*(m + n - 1)), x] + Dist[b/(m + n - 1), Int[(a
+ b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /; Fr
eeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt [4]{\sec (c+d x)}} \, dx &=\frac{4 a^2 \sec ^{\frac{3}{4}}(c+d x) \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+(4 a) \int 0 \, dx\\ &=\frac{4 a^2 \sec ^{\frac{3}{4}}(c+d x) \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0806643, size = 45, normalized size = 1.18 \[ \frac{4 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}{d (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/4),x]

[Out]

(4*Sec[c + d*x]^(3/4)*(a*(1 + Sec[c + d*x]))^(3/2)*Sin[c + d*x])/(d*(1 + Sec[c + d*x])^2)

________________________________________________________________________________________

Maple [F]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sec \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{\sec \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^(3/2)/sec(d*x+c)^(1/4),x)

[Out]

int((a+a*sec(d*x+c))^(3/2)/sec(d*x+c)^(1/4),x)

________________________________________________________________________________________

Maxima [B]  time = 1.91563, size = 163, normalized size = 4.29 \begin{align*} \frac{4 \,{\left (\frac{\sqrt{2} a^{\frac{3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sqrt{2} a^{\frac{3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{4}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{4}}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac{1}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(3/2)/sec(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

4*(sqrt(2)*a^(3/2)*sin(d*x + c)/(cos(d*x + c) + 1) - sqrt(2)*a^(3/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(d*(
sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/4)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/4)*(sin(d*x + c)^2/(cos(d
*x + c) + 1)^2 + 1)^(1/4))

________________________________________________________________________________________

Fricas [A]  time = 2.08256, size = 132, normalized size = 3.47 \begin{align*} \frac{4 \, a \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{1}{4}} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(3/2)/sec(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

4*a*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)^(1/4)*sin(d*x + c)/(d*cos(d*x + c) + d)

________________________________________________________________________________________

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((a+a*sec(d*x+c))**(3/2)/sec(d*x+c)**(1/4),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\sec \left (d x + c\right )^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^(3/2)/sec(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(1/4), x)