3.289 \(\int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=60 \[ \frac{a^2 A \sin (c+d x)}{d}+\frac{b (2 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a B+2 A b)+\frac{b^2 B \tan (c+d x)}{d} \]

[Out]

a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*A*Sin[c + d*x])/d + (b^2*B*Tan[c + d*x])/
d

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Rubi [A]  time = 0.102278, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {4024, 3770, 3767, 8} \[ \frac{a^2 A \sin (c+d x)}{d}+\frac{b (2 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a B+2 A b)+\frac{b^2 B \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*A*Sin[c + d*x])/d + (b^2*B*Tan[c + d*x])/
d

Rule 4024

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2*(csc[(e_.) + (f_.)*(x_)]*(B_
.) + (A_)), x_Symbol] :> Simp[(a^2*A*Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1))/(d*f*n), x] + Dist[1/(d*n), Int[(d
*Csc[e + f*x])^(n + 1)*(a*(2*A*b + a*B)*n + (2*a*b*B*n + A*(b^2*n + a^2*(n + 1)))*Csc[e + f*x] + b^2*B*n*Csc[e
+ f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 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]

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

Mathematica [A]  time = 0.478934, size = 109, normalized size = 1.82 \[ \frac{a^2 A \sin (c+d x)+a (c+d x) (a B+2 A b)-b (2 a B+A b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+b (2 a B+A b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+b^2 B \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*A*b + a*B)*(c + d*x) - b*(A*b + 2*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(A*b + 2*a*B)*Log[Co
s[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2*A*Sin[c + d*x] + b^2*B*Tan[c + d*x])/d

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Maple [A]  time = 0.045, size = 104, normalized size = 1.7 \begin{align*} 2\,Aabx+B{a}^{2}x+{\frac{A{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}A\sin \left ( dx+c \right ) }{d}}+2\,{\frac{Aabc}{d}}+2\,{\frac{Bab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{B{b}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{B{a}^{2}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*b*x+B*a^2*x+1/d*A*b^2*ln(sec(d*x+c)+tan(d*x+c))+a^2*A*sin(d*x+c)/d+2/d*A*a*b*c+2/d*B*a*b*ln(sec(d*x+c)+t
an(d*x+c))+b^2*B*tan(d*x+c)/d+1/d*B*a^2*c

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Maxima [A]  time = 0.989084, size = 139, normalized size = 2.32 \begin{align*} \frac{2 \,{\left (d x + c\right )} B a^{2} + 4 \,{\left (d x + c\right )} A a b + 2 \, B a b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + A b^{2}{\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^{2} \sin \left (d x + c\right ) + 2 \, B b^{2} \tan \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+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/2*(2*(d*x + c)*B*a^2 + 4*(d*x + c)*A*a*b + 2*B*a*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + A*b^2*(
log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 2*A*a^2*sin(d*x + c) + 2*B*b^2*tan(d*x + c))/d

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Fricas [A]  time = 0.503146, size = 294, normalized size = 4.9 \begin{align*} \frac{2 \,{\left (B a^{2} + 2 \, A a b\right )} d x \cos \left (d x + c\right ) +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (A a^{2} \cos \left (d x + c\right ) + B b^{2}\right )} \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(cos(d*x+c)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23978, size = 208, normalized size = 3.47 \begin{align*} \frac{{\left (B a^{2} + 2 \, A a b\right )}{\left (d x + c\right )} +{\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

((B*a^2 + 2*A*a*b)*(d*x + c) + (2*B*a*b + A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (2*B*a*b + A*b^2)*log(ab
s(tan(1/2*d*x + 1/2*c) - 1)) + 2*(A*a^2*tan(1/2*d*x + 1/2*c)^3 - B*b^2*tan(1/2*d*x + 1/2*c)^3 - A*a^2*tan(1/2*
d*x + 1/2*c) - B*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^4 - 1))/d