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

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.17, size = 21, normalized size = 1.00 \[ \frac {6 \sin (c+d x) \sqrt {\sec (c+d x)}}{d} \]

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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fricas [A]  time = 0.41, size = 19, normalized size = 0.90 \[ \frac {6 \, \sin \left (d x + c\right )}{d \sqrt {\cos \left (d x + c\right )}} \]

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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giac [B]  time = 0.36, size = 47, normalized size = 2.24 \[ -\frac {12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1} d \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} \]

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))

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maple [B]  time = 2.58, size = 41, normalized size = 1.95 \[ \frac {12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ 3 \, \int \frac {\sec \left (d x + c\right )^{2} + 1}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]

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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mupad [B]  time = 0.21, size = 21, normalized size = 1.00 \[ \frac {6\,\sin \left (c+d\,x\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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