3.86 \(\int \frac{A+B \sec (c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=35 \[ \frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]

[Out]

(A*x)/a - ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))

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Rubi [A]  time = 0.0591123, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3919, 3794} \[ \frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]),x]

[Out]

(A*x)/a - ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{A+B \sec (c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{A x}{a}-(A-B) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}

Mathematica [B]  time = 0.13754, size = 72, normalized size = 2.06 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (2 (B-A) \sin \left (\frac{d x}{2}\right )+A d x \cos \left (c+\frac{d x}{2}\right )+A d x \cos \left (\frac{d x}{2}\right )\right )}{a d (\cos (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]),x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(A*d*x*Cos[(d*x)/2] + A*d*x*Cos[c + (d*x)/2] + 2*(-A + B)*Sin[(d*x)/2]))/(a*d*(1 +
Cos[c + d*x]))

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Maple [A]  time = 0.05, size = 56, normalized size = 1.6 \begin{align*} 2\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{ad}}-{\frac{A}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sec(d*x+c))/(a+a*sec(d*x+c)),x)

[Out]

2/a/d*A*arctan(tan(1/2*d*x+1/2*c))-1/a/d*A*tan(1/2*d*x+1/2*c)+1/a/d*B*tan(1/2*d*x+1/2*c)

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Maxima [B]  time = 1.46919, size = 99, normalized size = 2.83 \begin{align*} \frac{A{\left (\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + \frac{B \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

(A*(2*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - sin(d*x + c)/(a*(cos(d*x + c) + 1))) + B*sin(d*x + c)/(a*(co
s(d*x + c) + 1)))/d

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Fricas [A]  time = 0.450558, size = 105, normalized size = 3. \begin{align*} \frac{A d x \cos \left (d x + c\right ) + A d x -{\left (A - B\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

(A*d*x*cos(d*x + c) + A*d*x - (A - B)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+a*sec(d*x+c)),x)

[Out]

(Integral(A/(sec(c + d*x) + 1), x) + Integral(B*sec(c + d*x)/(sec(c + d*x) + 1), x))/a

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Giac [A]  time = 1.18523, size = 59, normalized size = 1.69 \begin{align*} \frac{\frac{{\left (d x + c\right )} A}{a} - \frac{A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

((d*x + c)*A/a - (A*tan(1/2*d*x + 1/2*c) - B*tan(1/2*d*x + 1/2*c))/a)/d