3.286 \(\int \frac{(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{B \tan (c+d x)}{d} \]

[Out]

(B*Tan[c + d*x])/d

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Rubi [A]  time = 0.011593, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {21, 3767, 8} \[ \frac{B \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]

[Out]

(B*Tan[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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx &=B \int \sec ^2(c+d x) \, dx\\ &=-\frac{B \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{B \tan (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0043691, size = 11, normalized size = 1. \[ \frac{B \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]

[Out]

(B*Tan[c + d*x])/d

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Maple [A]  time = 0.056, size = 12, normalized size = 1.1 \begin{align*}{\frac{B\tan \left ( dx+c \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)^2/(a+b*cos(d*x+c)),x)

[Out]

B*tan(d*x+c)/d

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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)^2/(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31363, size = 45, normalized size = 4.09 \begin{align*} \frac{B \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \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)^2/(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

B*sin(d*x + c)/(d*cos(d*x + c))

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Sympy [A]  time = 8.56322, size = 32, normalized size = 2.91 \begin{align*} \begin{cases} \frac{B \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (B a + B b \cos{\left (c \right )}\right ) \sec ^{2}{\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)**2/(a+b*cos(d*x+c)),x)

[Out]

Piecewise((B*tan(c + d*x)/d, Ne(d, 0)), (x*(B*a + B*b*cos(c))*sec(c)**2/(a + b*cos(c)), True))

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Giac [A]  time = 1.53468, size = 15, normalized size = 1.36 \begin{align*} \frac{B \tan \left (d x + c\right )}{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)^2/(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

B*tan(d*x + c)/d