3.440 \(\int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=195 \[ \frac{a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac{a^4 (48 A+35 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(20 A+35 B+28 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac{(32 A+35 B+28 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+a^4 A x+\frac{a (5 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d} \]

[Out]

a^4*A*x + (a^4*(48*A + 35*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(40*A + 35*B + 28*C)*Tan[c + d*x])/(8*
d) + (a*(5*B + 4*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d
) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*S
ec[c + d*x])*Tan[c + d*x])/(24*d)

________________________________________________________________________________________

Rubi [A]  time = 0.304017, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4054, 3917, 3914, 3767, 8, 3770} \[ \frac{a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac{a^4 (48 A+35 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(20 A+35 B+28 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac{(32 A+35 B+28 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+a^4 A x+\frac{a (5 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

a^4*A*x + (a^4*(48*A + 35*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(40*A + 35*B + 28*C)*Tan[c + d*x])/(8*
d) + (a*(5*B + 4*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d
) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*S
ec[c + d*x])*Tan[c + d*x])/(24*d)

Rule 4054

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

Rule 3917

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> -Simp[(b*
d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1))/(f*m), x] + Dist[1/m, Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c*m
+ (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && Gt
Q[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 3914

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[a*c*x, x]
+ (Dist[b*d, Int[Csc[e + f*x]^2, x], x] + Dist[b*c + a*d, Int[Csc[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

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]

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 (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{\int (a+a \sec (c+d x))^4 (5 a A+a (5 B+4 C) \sec (c+d x)) \, dx}{5 a}\\ &=\frac{a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{\int (a+a \sec (c+d x))^3 \left (20 a^2 A+a^2 (20 A+35 B+28 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac{\int (a+a \sec (c+d x))^2 \left (60 a^3 A+5 a^3 (32 A+35 B+28 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac{a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac{(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac{\int (a+a \sec (c+d x)) \left (120 a^4 A+15 a^4 (40 A+35 B+28 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=a^4 A x+\frac{a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac{(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac{1}{8} \left (a^4 (40 A+35 B+28 C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{8} \left (a^4 (48 A+35 B+28 C)\right ) \int \sec (c+d x) \, dx\\ &=a^4 A x+\frac{a^4 (48 A+35 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac{(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}-\frac{\left (a^4 (40 A+35 B+28 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d}\\ &=a^4 A x+\frac{a^4 (48 A+35 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac{a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac{C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac{(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}\\ \end{align*}

Mathematica [B]  time = 5.09068, size = 538, normalized size = 2.76 \[ \frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (\sec (c) (-3120 A \sin (2 c+d x)+480 A \sin (c+2 d x)+480 A \sin (3 c+2 d x)+3280 A \sin (2 c+3 d x)-720 A \sin (4 c+3 d x)+240 A \sin (3 c+4 d x)+240 A \sin (5 c+4 d x)+800 A \sin (4 c+5 d x)+600 A d x \cos (2 c+d x)+300 A d x \cos (2 c+3 d x)+300 A d x \cos (4 c+3 d x)+60 A d x \cos (4 c+5 d x)+60 A d x \cos (6 c+5 d x)+4880 A \sin (d x)+600 A d x \cos (d x)-2880 B \sin (2 c+d x)+930 B \sin (c+2 d x)+930 B \sin (3 c+2 d x)+3520 B \sin (2 c+3 d x)-480 B \sin (4 c+3 d x)+405 B \sin (3 c+4 d x)+405 B \sin (5 c+4 d x)+800 B \sin (4 c+5 d x)+5120 B \sin (d x)-1920 C \sin (2 c+d x)+1320 C \sin (c+2 d x)+1320 C \sin (3 c+2 d x)+3200 C \sin (2 c+3 d x)-120 C \sin (4 c+3 d x)+420 C \sin (3 c+4 d x)+420 C \sin (5 c+4 d x)+664 C \sin (4 c+5 d x)+4720 C \sin (d x))-240 (48 A+35 B+28 C) \cos ^5(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{15360 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^4*(1 + Cos[c + d*x])^4*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^5*(-240*(48*
A + 35*B + 28*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*
x)/2]]) + Sec[c]*(600*A*d*x*Cos[d*x] + 600*A*d*x*Cos[2*c + d*x] + 300*A*d*x*Cos[2*c + 3*d*x] + 300*A*d*x*Cos[4
*c + 3*d*x] + 60*A*d*x*Cos[4*c + 5*d*x] + 60*A*d*x*Cos[6*c + 5*d*x] + 4880*A*Sin[d*x] + 5120*B*Sin[d*x] + 4720
*C*Sin[d*x] - 3120*A*Sin[2*c + d*x] - 2880*B*Sin[2*c + d*x] - 1920*C*Sin[2*c + d*x] + 480*A*Sin[c + 2*d*x] + 9
30*B*Sin[c + 2*d*x] + 1320*C*Sin[c + 2*d*x] + 480*A*Sin[3*c + 2*d*x] + 930*B*Sin[3*c + 2*d*x] + 1320*C*Sin[3*c
+ 2*d*x] + 3280*A*Sin[2*c + 3*d*x] + 3520*B*Sin[2*c + 3*d*x] + 3200*C*Sin[2*c + 3*d*x] - 720*A*Sin[4*c + 3*d*
x] - 480*B*Sin[4*c + 3*d*x] - 120*C*Sin[4*c + 3*d*x] + 240*A*Sin[3*c + 4*d*x] + 405*B*Sin[3*c + 4*d*x] + 420*C
*Sin[3*c + 4*d*x] + 240*A*Sin[5*c + 4*d*x] + 405*B*Sin[5*c + 4*d*x] + 420*C*Sin[5*c + 4*d*x] + 800*A*Sin[4*c +
5*d*x] + 800*B*Sin[4*c + 5*d*x] + 664*C*Sin[4*c + 5*d*x])))/(15360*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c
+ d*x)]))

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Maple [A]  time = 0.074, size = 331, normalized size = 1.7 \begin{align*}{a}^{4}Ax+{\frac{A{a}^{4}c}{d}}+{\frac{35\,B{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{83\,{a}^{4}C\tan \left ( dx+c \right ) }{15\,d}}+6\,{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{20\,B{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{7\,{a}^{4}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{7\,{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{20\,A{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{27\,B{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{34\,{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+2\,{\frac{A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{4\,B{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{B{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{{a}^{4}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

a^4*A*x+1/d*A*a^4*c+35/8/d*B*a^4*ln(sec(d*x+c)+tan(d*x+c))+83/15/d*a^4*C*tan(d*x+c)+6/d*A*a^4*ln(sec(d*x+c)+ta
n(d*x+c))+20/3/d*B*a^4*tan(d*x+c)+7/2/d*a^4*C*sec(d*x+c)*tan(d*x+c)+7/2/d*a^4*C*ln(sec(d*x+c)+tan(d*x+c))+20/3
/d*A*a^4*tan(d*x+c)+27/8/d*B*a^4*sec(d*x+c)*tan(d*x+c)+34/15/d*a^4*C*tan(d*x+c)*sec(d*x+c)^2+2/d*A*a^4*sec(d*x
+c)*tan(d*x+c)+4/3/d*B*a^4*tan(d*x+c)*sec(d*x+c)^2+1/d*a^4*C*tan(d*x+c)*sec(d*x+c)^3+1/3/d*A*a^4*tan(d*x+c)*se
c(d*x+c)^2+1/4/d*B*a^4*tan(d*x+c)*sec(d*x+c)^3+1/5/d*a^4*C*tan(d*x+c)*sec(d*x+c)^4

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Maxima [B]  time = 0.966995, size = 651, normalized size = 3.34 \begin{align*} \frac{80 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 240 \,{\left (d x + c\right )} A a^{4} + 320 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 16 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{4} + 480 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 15 \, B a^{4}{\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 )} - 60 \, C a^{4}{\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 )} - 240 \, A a^{4}{\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 )} - 360 \, B a^{4}{\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 )} - 240 \, C a^{4}{\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 )} + 960 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1440 \, A a^{4} \tan \left (d x + c\right ) + 960 \, B a^{4} \tan \left (d x + c\right ) + 240 \, C a^{4} \tan \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/240*(80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 240*(d*x + c)*A*a^4 + 320*(tan(d*x + c)^3 + 3*tan(d*x + c)
)*B*a^4 + 16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 480*(tan(d*x + c)^3 + 3*tan(d*x
+ c))*C*a^4 - 15*B*a^4*(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)) - 60*C*a^4*(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)) - 240*A*a^4*(2*sin(d*x + c)/(sin
(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 360*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2
- 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 240*C*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(s
in(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 960*A*a^4*log(sec(d*x + c) + tan(d*x + c)) + 240*B*a^4*log(sec(d*x
+ c) + tan(d*x + c)) + 1440*A*a^4*tan(d*x + c) + 960*B*a^4*tan(d*x + c) + 240*C*a^4*tan(d*x + c))/d

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Fricas [A]  time = 0.561003, size = 521, normalized size = 2.67 \begin{align*} \frac{240 \, A a^{4} d x \cos \left (d x + c\right )^{5} + 15 \,{\left (48 \, A + 35 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (48 \, A + 35 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (100 \, A + 100 \, B + 83 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \,{\left (16 \, A + 27 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, A + 20 \, B + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 30 \,{\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, C a^{4}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(240*A*a^4*d*x*cos(d*x + c)^5 + 15*(48*A + 35*B + 28*C)*a^4*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(4
8*A + 35*B + 28*C)*a^4*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(8*(100*A + 100*B + 83*C)*a^4*cos(d*x + c)^4
+ 15*(16*A + 27*B + 28*C)*a^4*cos(d*x + c)^3 + 8*(5*A + 20*B + 34*C)*a^4*cos(d*x + c)^2 + 30*(B + 4*C)*a^4*cos
(d*x + c) + 24*C*a^4)*sin(d*x + c))/(d*cos(d*x + c)^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{4} \left (\int A\, dx + \int 4 A \sec{\left (c + d x \right )}\, dx + \int 6 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec{\left (c + d x \right )}\, dx + \int 4 B \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{4}{\left (c + d x \right )}\, dx + \int B \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{6}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

a**4*(Integral(A, x) + Integral(4*A*sec(c + d*x), x) + Integral(6*A*sec(c + d*x)**2, x) + Integral(4*A*sec(c +
d*x)**3, x) + Integral(A*sec(c + d*x)**4, x) + Integral(B*sec(c + d*x), x) + Integral(4*B*sec(c + d*x)**2, x)
+ Integral(6*B*sec(c + d*x)**3, x) + Integral(4*B*sec(c + d*x)**4, x) + Integral(B*sec(c + d*x)**5, x) + Inte
gral(C*sec(c + d*x)**2, x) + Integral(4*C*sec(c + d*x)**3, x) + Integral(6*C*sec(c + d*x)**4, x) + Integral(4*
C*sec(c + d*x)**5, x) + Integral(C*sec(c + d*x)**6, x))

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Giac [A]  time = 1.29353, size = 475, normalized size = 2.44 \begin{align*} \frac{120 \,{\left (d x + c\right )} A a^{4} + 15 \,{\left (48 \, A a^{4} + 35 \, B a^{4} + 28 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (48 \, A a^{4} + 35 \, B a^{4} + 28 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (600 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 420 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 2720 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 2450 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1960 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 4720 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3584 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3680 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3950 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3160 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1080 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1500 \, C a^{4} \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 )}^{5}}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/120*(120*(d*x + c)*A*a^4 + 15*(48*A*a^4 + 35*B*a^4 + 28*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(48*A
*a^4 + 35*B*a^4 + 28*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(600*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 525*B*a
^4*tan(1/2*d*x + 1/2*c)^9 + 420*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 2720*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 2450*B*a^4*
tan(1/2*d*x + 1/2*c)^7 - 1960*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 4720*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 4480*B*a^4*ta
n(1/2*d*x + 1/2*c)^5 + 3584*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 3680*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 3950*B*a^4*tan(
1/2*d*x + 1/2*c)^3 - 3160*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 1080*A*a^4*tan(1/2*d*x + 1/2*c) + 1395*B*a^4*tan(1/2*
d*x + 1/2*c) + 1500*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d