3.396 \(\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{4}}(c+d x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.118211, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4222, 2762, 8} \[ \frac{4 a^2 \sin (c+d x) \sqrt [4]{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/4),x]

[Out]

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

Rule 4222

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 8

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

Rubi steps

\begin{align*} \int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{4}}(c+d x) \, dx &=\left (\sqrt [4]{\cos (c+d x)} \sqrt [4]{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{4}}(c+d x)} \, dx\\ &=\frac{4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)}}-\left (4 a \sqrt [4]{\cos (c+d x)} \sqrt [4]{\sec (c+d x)}\right ) \int 0 \, dx\\ &=\frac{4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.114656, size = 51, normalized size = 1.34 \[ \frac{2 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt [4]{\sec (c+d x)} (a (\cos (c+d x)+1))^{3/2}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/4),x]

[Out]

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

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Maple [F]  time = 0.362, size = 0, normalized size = 0. \begin{align*} \int \left ( a+\cos \left ( dx+c \right ) a \right ) ^{{\frac{3}{2}}} \left ( \sec \left ( dx+c \right ) \right ) ^{{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+cos(d*x+c)*a)^(3/2)*sec(d*x+c)^(5/4),x)

[Out]

int((a+cos(d*x+c)*a)^(3/2)*sec(d*x+c)^(5/4),x)

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Maxima [B]  time = 1.49958, 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*cos(d*x+c))^(3/2)*sec(d*x+c)^(5/4),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 = 2.09714, size = 115, normalized size = 3.03 \begin{align*} \frac{4 \, \sqrt{a \cos \left (d x + c\right ) + a} a \sin \left (d x + c\right )}{{\left (d \cos \left (d x + c\right ) + d\right )} \cos \left (d x + c\right )^{\frac{1}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(3/2)*sec(d*x+c)^(5/4),x, algorithm="fricas")

[Out]

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

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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((a+a*cos(d*x+c))**(3/2)*sec(d*x+c)**(5/4),x)

[Out]

Timed out

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Giac [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*cos(d*x+c))^(3/2)*sec(d*x+c)^(5/4),x, algorithm="giac")

[Out]

Timed out