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

Optimal. Leaf size=6 \[ \frac{B x}{b} \]

[Out]

(B*x)/b

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Rubi [A]  time = 0.0013912, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {21, 8} \[ \frac{B x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x)/b

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\frac{a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{B \int 1 \, dx}{b}\\ &=\frac{B x}{b}\\ \end{align*}

Mathematica [A]  time = 0.0005538, size = 6, normalized size = 1. \[ \frac{B x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x)/b

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Maple [A]  time = 0.007, size = 7, normalized size = 1.2 \begin{align*}{\frac{Bx}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.429939, size = 9, normalized size = 1.5 \begin{align*} \frac{B x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Sympy [A]  time = 6.06857, size = 3, normalized size = 0.5 \begin{align*} \frac{B x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b

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Giac [B]  time = 1.38539, size = 18, normalized size = 3. \begin{align*} \frac{{\left (d x + c\right )} B}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)*B/(b*d)