3.279 \(\int \sec (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=61 \[ \frac{(a B+A b) \tan (c+d x)}{d}+\frac{(2 a A+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B \tan (c+d x) \sec (c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0775628, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3997, 3787, 3770, 3767, 8} \[ \frac{(a B+A b) \tan (c+d x)}{d}+\frac{(2 a A+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B \tan (c+d x) \sec (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3997

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*C
sc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*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 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)) (A+B \sec (c+d x)) \, dx &=\frac{b B \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int \sec (c+d x) (2 a A+b B+2 (A b+a B) \sec (c+d x)) \, dx\\ &=\frac{b B \sec (c+d x) \tan (c+d x)}{2 d}+(A b+a B) \int \sec ^2(c+d x) \, dx+\frac{1}{2} (2 a A+b B) \int \sec (c+d x) \, dx\\ &=\frac{(2 a A+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B \sec (c+d x) \tan (c+d x)}{2 d}-\frac{(A b+a B) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{(2 a A+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(A b+a B) \tan (c+d x)}{d}+\frac{b B \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0220697, size = 75, normalized size = 1.23 \[ \frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+\frac{A b \tan (c+d x)}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B \tan (c+d x) \sec (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 86, normalized size = 1.4 \begin{align*}{\frac{Aa\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Ba\tan \left ( dx+c \right ) }{d}}+{\frac{Ab\tan \left ( dx+c \right ) }{d}}+{\frac{Bb\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Bb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,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))*(A+B*sec(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.962849, size = 119, normalized size = 1.95 \begin{align*} -\frac{B b{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 4 \, B a \tan \left (d x + c\right ) - 4 \, A b \tan \left (d x + c\right )}{4 \, 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))*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.80849, size = 247, normalized size = 4.05 \begin{align*} \frac{{\left (2 \, A a + B b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, A a + B b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (B b + 2 \,{\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 ) \sec{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.24083, size = 207, normalized size = 3.39 \begin{align*} \frac{{\left (2 \, A a + B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, A a + B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, 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))*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

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