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

Optimal. Leaf size=69 \[ \frac{(2 a B+2 A b+b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a A x+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0706864, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {4048, 3770, 3767, 8} \[ \frac{(2 a B+2 A b+b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a A x+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4048

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

Mathematica [A]  time = 0.021202, size = 92, normalized size = 1.33 \[ a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d}+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tan (c+d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 117, normalized size = 1.7 \begin{align*} aAx+{\frac{Aac}{d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{aC\tan \left ( dx+c \right ) }{d}}+{\frac{Ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Bb\tan \left ( dx+c \right ) }{d}}+{\frac{Cb\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Cb\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((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.15323, size = 157, normalized size = 2.28 \begin{align*} \frac{4 \,{\left (d x + c\right )} A a - C 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 \, B a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, A b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, C a \tan \left (d x + c\right ) + 4 \, B b \tan \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(d*x + c)*A*a - C*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*B*a*log(sec(d*x + c) + tan(d*x + c)) + 4*A*b*log(sec(d*x + c) + tan(d*x + c)) + 4*C*a*tan(d*x + c) + 4*
B*b*tan(d*x + c))/d

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Fricas [A]  time = 0.592188, size = 305, normalized size = 4.42 \begin{align*} \frac{4 \, A a d x \cos \left (d x + c\right )^{2} +{\left (2 \, B a +{\left (2 \, A + C\right )} b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, B a +{\left (2 \, A + C\right )} b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (C b + 2 \,{\left (C a + B 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((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/4*(4*A*a*d*x*cos(d*x + c)^2 + (2*B*a + (2*A + C)*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*B*a + (2*A + C
)*b)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(C*b + 2*(C*a + B*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 )} + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23104, size = 230, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (d x + c\right )} A a +{\left (2 \, B a + 2 \, A b + C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, B a + 2 \, A b + C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C 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((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

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