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

Optimal. Leaf size=12 \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

(B*ArcTanh[Sin[c + d*x]])/d

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Rubi [A]  time = 0.0068244, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {21, 3770} \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*ArcTanh[Sin[c + d*x]])/d

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0030115, size = 12, normalized size = 1. \[ \frac{B \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*ArcTanh[Sin[c + d*x]])/d

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Maple [A]  time = 0.056, size = 20, normalized size = 1.7 \begin{align*}{\frac{B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*B*ln(sec(d*x+c)+tan(d*x+c))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.41919, size = 81, normalized size = 6.75 \begin{align*} \frac{B \log \left (\sin \left (d x + c\right ) + 1\right ) - B \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*log(sin(d*x + c) + 1) - B*log(-sin(d*x + c) + 1))/d

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Sympy [A]  time = 4.50306, size = 39, normalized size = 3.25 \begin{align*} \begin{cases} \frac{B \log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (B a + B b \cos{\left (c \right )}\right ) \sec{\left (c \right )}}{a + b \cos{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((B*log(tan(c + d*x) + sec(c + d*x))/d, Ne(d, 0)), (x*(B*a + B*b*cos(c))*sec(c)/(a + b*cos(c)), True)
)

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Giac [B]  time = 1.47797, size = 63, normalized size = 5.25 \begin{align*} \frac{B \log \left ({\left | \frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - B \log \left ({\left | \frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(B*log(abs(1/sin(d*x + c) + sin(d*x + c) + 2)) - B*log(abs(1/sin(d*x + c) + sin(d*x + c) - 2)))/d