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

Optimal. Leaf size=73 \[ \frac{a^2 (B-C) \sin (c+d x)}{d}+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 B+C)+\frac{C \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{d} \]

[Out]

a^2*(2*B + C)*x + (a^2*(B + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(B - C)*Sin[c + d*x])/d + (C*(a^2 + a^2*Sec[c
+ d*x])*Sin[c + d*x])/d

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Rubi [A]  time = 0.206269, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4072, 4018, 3996, 3770} \[ \frac{a^2 (B-C) \sin (c+d x)}{d}+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 B+C)+\frac{C \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*(2*B + C)*x + (a^2*(B + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(B - C)*Sin[c + d*x])/d + (C*(a^2 + a^2*Sec[c
+ d*x])*Sin[c + d*x])/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 4018

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

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

Mathematica [A]  time = 0.317977, size = 143, normalized size = 1.96 \[ \frac{a^2 \left (B \sin (c+d x)-B \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+B \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 B c+2 B d x+C \tan (c+d x)-2 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+c C+C d x\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(2*B*c + c*C + 2*B*d*x + C*d*x - B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*C*Log[Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] + B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]
+ B*Sin[c + d*x] + C*Tan[c + d*x]))/d

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Maple [A]  time = 0.072, size = 107, normalized size = 1.5 \begin{align*} 2\,{a}^{2}Bx+{a}^{2}Cx+{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{B{a}^{2}c}{d}}+{\frac{{a}^{2}C\tan \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{C{a}^{2}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.94135, size = 142, normalized size = 1.95 \begin{align*} \frac{4 \,{\left (d x + c\right )} B a^{2} + 2 \,{\left (d x + c\right )} C a^{2} + B a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a^{2} \sin \left (d x + c\right ) + 2 \, C a^{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)^2*(a+a*sec(d*x+c))^2*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.516101, size = 278, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (2 \, B + C\right )} a^{2} d x \cos \left (d x + c\right ) +{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (B a^{2} \cos \left (d x + c\right ) + C a^{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)^2*(a+a*sec(d*x+c))^2*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.16266, size = 212, normalized size = 2.9 \begin{align*} \frac{{\left (2 \, B a^{2} + C a^{2}\right )}{\left (d x + c\right )} +{\left (B a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (B a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C a^{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)^2*(a+a*sec(d*x+c))^2*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

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