3.48 \(\int \cos (c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=32 \[ a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0473172, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {3996, 3770} \[ a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3996

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.)
+ (A_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x
])^(n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B},
x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]

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

Mathematica [A]  time = 0.0266833, size = 46, normalized size = 1.44 \[ \frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.064, size = 56, normalized size = 1.8 \begin{align*} aAx+Bax+{\frac{Aa\sin \left ( dx+c \right ) }{d}}+{\frac{Aac}{d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Bac}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02222, size = 78, normalized size = 2.44 \begin{align*} \frac{2 \,{\left (d x + c\right )} A a + 2 \,{\left (d x + c\right )} B a + B a{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.48656, size = 139, normalized size = 4.34 \begin{align*} \frac{2 \,{\left (A + B\right )} a d x + B a \log \left (\sin \left (d x + c\right ) + 1\right ) - B a \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A a \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.2413, size = 107, normalized size = 3.34 \begin{align*} \frac{B a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - B a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) +{\left (A a + B a\right )}{\left (d x + c\right )} + \frac{2 \, A a \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(cos(d*x+c)*(a+a*sec(d*x+c))*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

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