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

Optimal. Leaf size=106 \[ \frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 B+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d} \]

[Out]

(a*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(B + C)*Tan[c + d*x])/d + (a*(4*B + 3*C)*Sec[c + d*x]*Tan[c +
d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(B + C)*Tan[c + d*x]^3)/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.17563, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4072, 3997, 3787, 3768, 3770, 3767} \[ \frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 B+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(B + C)*Tan[c + d*x])/d + (a*(4*B + 3*C)*Sec[c + d*x]*Tan[c +
d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(B + C)*Tan[c + d*x]^3)/(3*d)

Rule 4072

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

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^3(c+d x) (a+a \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int \sec ^3(c+d x) (a (4 B+3 C)+4 a (B+C) \sec (c+d x)) \, dx\\ &=\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+(a (B+C)) \int \sec ^4(c+d x) \, dx+\frac{1}{4} (a (4 B+3 C)) \int \sec ^3(c+d x) \, dx\\ &=\frac{a (4 B+3 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} (a (4 B+3 C)) \int \sec (c+d x) \, dx-\frac{(a (B+C)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a (4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (4 B+3 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a (B+C) \tan ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [B]  time = 0.628324, size = 337, normalized size = 3.18 \[ -\frac{a \sec ^4(c+d x) \left (12 (4 B+3 C) \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+3 (4 B+3 C) \cos (4 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-24 B \sin (c+d x)-64 B \sin (2 (c+d x))-24 B \sin (3 (c+d x))-16 B \sin (4 (c+d x))+36 B \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-36 B \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-66 C \sin (c+d x)-64 C \sin (2 (c+d x))-18 C \sin (3 (c+d x))-16 C \sin (4 (c+d x))+27 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-27 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sec[c + d*x]^4*(36*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 27*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/
2]] + 12*(4*B + 3*C)*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(
c + d*x)/2]]) + 3*(4*B + 3*C)*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2
] + Sin[(c + d*x)/2]]) - 36*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 27*C*Log[Cos[(c + d*x)/2] + Sin[(c +
d*x)/2]] - 24*B*Sin[c + d*x] - 66*C*Sin[c + d*x] - 64*B*Sin[2*(c + d*x)] - 64*C*Sin[2*(c + d*x)] - 24*B*Sin[3*
(c + d*x)] - 18*C*Sin[3*(c + d*x)] - 16*B*Sin[4*(c + d*x)] - 16*C*Sin[4*(c + d*x)]))/(192*d)

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 171, normalized size = 1.6 \begin{align*}{\frac{Ba\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,aC\tan \left ( dx+c \right ) }{3\,d}}+{\frac{aC \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,Ba\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Ba\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{aC \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,aC\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(a+a*sec(d*x+c))*(B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

1/2/d*B*a*sec(d*x+c)*tan(d*x+c)+1/2/d*B*a*ln(sec(d*x+c)+tan(d*x+c))+2/3*a*C*tan(d*x+c)/d+1/3*a*C*sec(d*x+c)^2*
tan(d*x+c)/d+2/3/d*B*a*tan(d*x+c)+1/3/d*B*a*tan(d*x+c)*sec(d*x+c)^2+1/4*a*C*sec(d*x+c)^3*tan(d*x+c)/d+3/8*a*C*
sec(d*x+c)*tan(d*x+c)/d+3/8/d*a*C*ln(sec(d*x+c)+tan(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 0.944518, size = 220, normalized size = 2.08 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a - 3 \, C a{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B a{\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 )}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a + 16*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a - 3*C*a*(2*(3*sin(d*
x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x +
c) - 1)) - 12*B*a*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)))/d

________________________________________________________________________________________

Fricas [A]  time = 0.520872, size = 339, normalized size = 3.2 \begin{align*} \frac{3 \,{\left (4 \, B + 3 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (4 \, B + 3 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (B + C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (4 \, B + 3 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \,{\left (B + C\right )} a \cos \left (d x + c\right ) + 6 \, C a\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(3*(4*B + 3*C)*a*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(4*B + 3*C)*a*cos(d*x + c)^4*log(-sin(d*x + c)
+ 1) + 2*(16*(B + C)*a*cos(d*x + c)^3 + 3*(4*B + 3*C)*a*cos(d*x + c)^2 + 8*(B + C)*a*cos(d*x + c) + 6*C*a)*sin
(d*x + c))/(d*cos(d*x + c)^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int B \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(B*sec(c + d*x)**3, x) + Integral(B*sec(c + d*x)**4, x) + Integral(C*sec(c + d*x)**4, x) + Integral
(C*sec(c + d*x)**5, x))

________________________________________________________________________________________

Giac [A]  time = 1.14518, size = 254, normalized size = 2.4 \begin{align*} \frac{3 \,{\left (4 \, B a + 3 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (4 \, B a + 3 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (12 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 9 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 28 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 49 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 52 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 31 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 39 \, C a \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 )}^{4}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(4*B*a + 3*C*a)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(4*B*a + 3*C*a)*log(abs(tan(1/2*d*x + 1/2*c) -
1)) - 2*(12*B*a*tan(1/2*d*x + 1/2*c)^7 + 9*C*a*tan(1/2*d*x + 1/2*c)^7 - 28*B*a*tan(1/2*d*x + 1/2*c)^5 - 49*C*a
*tan(1/2*d*x + 1/2*c)^5 + 52*B*a*tan(1/2*d*x + 1/2*c)^3 + 31*C*a*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*tan(1/2*d*x +
1/2*c) - 39*C*a*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d