3.24 \(\int \frac{3+3 \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\)

Optimal. Leaf size=21 \[ \frac{6 \sin (c+d x) \sqrt{\sec (c+d x)}}{d} \]

[Out]

(6*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d

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Rubi [A]  time = 0.02323, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {4043} \[ \frac{6 \sin (c+d x) \sqrt{\sec (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d

Rule 4043

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]

Rubi steps

\begin{align*} \int \frac{3+3 \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx &=\frac{6 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.132904, size = 21, normalized size = 1. \[ \frac{6 \sin (c+d x) \sqrt{\sec (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d

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Maple [B]  time = 0.684, size = 41, normalized size = 2. \begin{align*} 12\,{\frac{\sin \left ( 1/2\,dx+c/2 \right ) \cos \left ( 1/2\,dx+c/2 \right ) }{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12*sin(1/2*d*x+1/2*c)*cos(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.480884, size = 53, normalized size = 2.52 \begin{align*} \frac{6 \, \sin \left (d x + c\right )}{d \sqrt{\cos \left (d x + c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*sin(d*x + c)/(d*sqrt(cos(d*x + c)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(Integral(1/sqrt(sec(c + d*x)), x) + Integral(sec(c + d*x)**(3/2), x))

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Giac [B]  time = 1.39341, size = 63, normalized size = 3. \begin{align*} -\frac{12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1} d \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*tan(1/2*d*x + 1/2*c)/(sqrt(-tan(1/2*d*x + 1/2*c)^4 + 1)*d*sgn(tan(1/2*d*x + 1/2*c)^2 - 1))