3.449 \(\int \sec (c+d x) (a+b \sec (c+d x)) \, dx\)

Optimal. Leaf size=24 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d} \]

[Out]

(a*ArcTanh[Sin[c + d*x]])/d + (b*Tan[c + d*x])/d

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Rubi [A]  time = 0.0260089, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3787, 3770, 3767, 8} \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcTanh[Sin[c + d*x]])/d + (b*Tan[c + d*x])/d

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, 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 \sec (c+d x) (a+b \sec (c+d x)) \, dx &=a \int \sec (c+d x) \, dx+b \int \sec ^2(c+d x) \, dx\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0122902, size = 24, normalized size = 1. \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcTanh[Sin[c + d*x]])/d + (b*Tan[c + d*x])/d

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Maple [A]  time = 0.018, size = 32, normalized size = 1.3 \begin{align*}{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{b\tan \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06224, size = 39, normalized size = 1.62 \begin{align*} \frac{a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + b \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*log(sec(d*x + c) + tan(d*x + c)) + b*tan(d*x + c))/d

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Fricas [B]  time = 1.97673, size = 162, normalized size = 6.75 \begin{align*} \frac{a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*cos(d*x + c)*log(sin(d*x + c) + 1) - a*cos(d*x + c)*log(-sin(d*x + c) + 1) + 2*b*sin(d*x + c))/(d*cos(d
*x + c))

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Sympy [A]  time = 3.75222, size = 37, normalized size = 1.54 \begin{align*} \begin{cases} \frac{a \log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )} + b \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \sec{\left (c \right )}\right ) \sec{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.25528, size = 85, normalized size = 3.54 \begin{align*} \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - a*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*b*tan(1/2*d*x + 1/2*c)/(tan(1
/2*d*x + 1/2*c)^2 - 1))/d