3.185 \(\int (a+a \cos (x)) (A+B \sec (x)) \, dx\)

Optimal. Leaf size=18 \[ a x (A+B)+a A \sin (x)+a B \tanh ^{-1}(\sin (x)) \]

[Out]

a*(A + B)*x + a*B*ArcTanh[Sin[x]] + a*A*Sin[x]

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Rubi [A]  time = 0.0983354, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2828, 2968, 3023, 2735, 3770} \[ a x (A+B)+a A \sin (x)+a B \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[x])*(A + B*Sec[x]),x]

[Out]

a*(A + B)*x + a*B*ArcTanh[Sin[x]] + a*A*Sin[x]

Rule 2828

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> In
t[((a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^n, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Int
egerQ[n]

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

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

Mathematica [B]  time = 0.0152075, size = 51, normalized size = 2.83 \[ a A x+a A \sin (x)+a B x-a B \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+a B \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[x])*(A + B*Sec[x]),x]

[Out]

a*A*x + a*B*x - a*B*Log[Cos[x/2] - Sin[x/2]] + a*B*Log[Cos[x/2] + Sin[x/2]] + a*A*Sin[x]

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Maple [A]  time = 0.038, size = 24, normalized size = 1.3 \begin{align*} aA\sin \left ( x \right ) +Bax+aAx+Ba\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(x))*(A+B*sec(x)),x)

[Out]

a*A*sin(x)+B*a*x+a*A*x+B*a*ln(sec(x)+tan(x))

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Maxima [A]  time = 0.97508, size = 31, normalized size = 1.72 \begin{align*} A a x + B a x + B a \log \left (\sec \left (x\right ) + \tan \left (x\right )\right ) + A a \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(x))*(A+B*sec(x)),x, algorithm="maxima")

[Out]

A*a*x + B*a*x + B*a*log(sec(x) + tan(x)) + A*a*sin(x)

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Fricas [A]  time = 2.53537, size = 107, normalized size = 5.94 \begin{align*}{\left (A + B\right )} a x + \frac{1}{2} \, B a \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, B a \log \left (-\sin \left (x\right ) + 1\right ) + A a \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(x))*(A+B*sec(x)),x, algorithm="fricas")

[Out]

(A + B)*a*x + 1/2*B*a*log(sin(x) + 1) - 1/2*B*a*log(-sin(x) + 1) + A*a*sin(x)

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Sympy [A]  time = 3.57851, size = 27, normalized size = 1.5 \begin{align*} A a x + A a \sin{\left (x \right )} + B a x + B a \log{\left (\tan{\left (x \right )} + \sec{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(x))*(A+B*sec(x)),x)

[Out]

A*a*x + A*a*sin(x) + B*a*x + B*a*log(tan(x) + sec(x))

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Giac [B]  time = 1.17056, size = 69, normalized size = 3.83 \begin{align*} B a \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) - B a \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) +{\left (A a + B a\right )} x + \frac{2 \, A a \tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(x))*(A+B*sec(x)),x, algorithm="giac")

[Out]

B*a*log(abs(tan(1/2*x) + 1)) - B*a*log(abs(tan(1/2*x) - 1)) + (A*a + B*a)*x + 2*A*a*tan(1/2*x)/(tan(1/2*x)^2 +
1)