3.3.42 \(\int \frac {(a+a \sec (c+d x))^{3/2}}{\sqrt [4]{\sec (c+d x)}} \, dx\) [242]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3899, 8} \begin {gather*} \frac {4 a^2 \sin (c+d x) \sec ^{\frac {3}{4}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}} \end {gather*}

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 8

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

Rule 3899

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] /;
FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]

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*}

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Mathematica [A]
time = 0.09, size = 45, normalized size = 1.18 \begin {gather*} \frac {4 \sec ^{\frac {3}{4}}(c+d x) (a (1+\sec (c+d x)))^{3/2} \sin (c+d x)}{d (1+\sec (c+d x))^2} \end {gather*}

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)

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{\sec \left (d x +c \right )^{\frac {1}{4}}}\, dx\]

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)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (34) = 68\).
time = 0.51, size = 121, normalized size = 3.18 \begin {gather*} \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 {gather*}

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))

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Fricas [A]
time = 2.62, size = 50, normalized size = 1.32 \begin {gather*} \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 {gather*}

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.75, size = 55, normalized size = 1.45 \begin {gather*} \frac {2\,a\,\sin \left (2\,c+2\,d\,x\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/4}\,\sqrt {\frac {a\,\left (\cos \left (c+d\,x\right )+1\right )}{\cos \left (c+d\,x\right )}}}{d\,\left (\cos \left (c+d\,x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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