3.295 \(\int (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec (c+d x) \, dx\)

Optimal. Leaf size=27 \[ \frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c+d x)}{d} \]

[Out]

B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d

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Rubi [A]  time = 0.0521309, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3023, 2735, 3770} \[ \frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]

[Out]

B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{C \sin (c+d x)}{d}+\int (A+B \cos (c+d x)) \sec (c+d x) \, dx\\ &=B x+\frac{C \sin (c+d x)}{d}+A \int \sec (c+d x) \, dx\\ &=B x+\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \sin (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0198298, size = 38, normalized size = 1.41 \[ \frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c) \cos (d x)}{d}+\frac{C \cos (c) \sin (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]

[Out]

B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Cos[d*x]*Sin[c])/d + (C*Cos[c]*Sin[d*x])/d

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Maple [A]  time = 0.032, size = 41, normalized size = 1.5 \begin{align*} Bx+{\frac{A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Bc}{d}}+{\frac{C\sin \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x)

[Out]

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

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Maxima [A]  time = 0.973572, size = 49, normalized size = 1.81 \begin{align*} \frac{{\left (d x + c\right )} B + A \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + C \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="maxima")

[Out]

((d*x + c)*B + A*log(sec(d*x + c) + tan(d*x + c)) + C*sin(d*x + c))/d

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Fricas [A]  time = 1.89053, size = 120, normalized size = 4.44 \begin{align*} \frac{2 \, B d x + A \log \left (\sin \left (d x + c\right ) + 1\right ) - A \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B \cos{\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c),x)

[Out]

Integral((A + B*cos(c + d*x) + C*cos(c + d*x)**2)*sec(c + d*x), x)

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Giac [B]  time = 1.21668, size = 95, normalized size = 3.52 \begin{align*} \frac{{\left (d x + c\right )} B + 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 \, C \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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="giac")

[Out]

((d*x + c)*B + 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*C*tan(1/2*d*x +
1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1))/d