3.7.55 \(\int \sec (c+d x) (a+b \sec (c+d x))^3 (A+C \sec ^2(c+d x)) \, dx\) [655]

Optimal. Leaf size=234 \[ \frac {a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {\left (3 a^4 C-4 b^4 (5 A+4 C)-4 a^2 b^2 (20 A+13 C)\right ) \tan (c+d x)}{30 b d}+\frac {a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{120 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d} \]

[Out]

1/8*a*(4*a^2*(2*A+C)+3*b^2*(4*A+3*C))*arctanh(sin(d*x+c))/d-1/30*(3*a^4*C-4*b^4*(5*A+4*C)-4*a^2*b^2*(20*A+13*C
))*tan(d*x+c)/b/d+1/120*a*(100*A*b^2-6*C*a^2+71*C*b^2)*sec(d*x+c)*tan(d*x+c)/d-1/60*(3*a^2*C-4*b^2*(5*A+4*C))*
(a+b*sec(d*x+c))^2*tan(d*x+c)/b/d-1/20*a*C*(a+b*sec(d*x+c))^3*tan(d*x+c)/b/d+1/5*C*(a+b*sec(d*x+c))^4*tan(d*x+
c)/b/d

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Rubi [A]
time = 0.34, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4168, 4087, 4082, 3872, 3855, 3852, 8} \begin {gather*} \frac {a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}+\frac {a \left (-6 a^2 C+100 A b^2+71 b^2 C\right ) \tan (c+d x) \sec (c+d x)}{120 d}-\frac {\left (3 a^4 C-4 a^2 b^2 (20 A+13 C)-4 b^4 (5 A+4 C)\right ) \tan (c+d x)}{30 b d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}-\frac {a C \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]

[Out]

(a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) - ((3*a^4*C - 4*b^4*(5*A + 4*C) - 4*a^2*
b^2*(20*A + 13*C))*Tan[c + d*x])/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(12
0*d) - ((3*a^2*C - 4*b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) - (a*C*(a + b*Sec[c + d*x]
)^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3872

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 4082

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
*Csc[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 4087

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

Rule 4168

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + 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 + 1)/(b*f*(m + 2))), x] + Dist[1/(b*(m + 2))
, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) - a*C*Csc[e + f*x], x], x], x] /; Fre
eQ[{a, b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 (b (5 A+4 C)-a C \sec (c+d x)) \, dx}{5 b}\\ &=-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (a b (20 A+13 C)-\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) \sec (c+d x)\right ) \, dx}{20 b}\\ &=-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (b \left (8 b^2 (5 A+4 C)+a^2 (60 A+33 C)\right )+a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \sec (c+d x)\right ) \, dx}{60 b}\\ &=\frac {a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{120 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\int \sec (c+d x) \left (15 a b \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right )-4 \left (3 a^4 C-4 b^4 (5 A+4 C)-4 a^2 b^2 (20 A+13 C)\right ) \sec (c+d x)\right ) \, dx}{120 b}\\ &=\frac {a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{120 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {1}{8} \left (a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right )\right ) \int \sec (c+d x) \, dx-\frac {\left (3 a^4 C-4 b^4 (5 A+4 C)-4 a^2 b^2 (20 A+13 C)\right ) \int \sec ^2(c+d x) \, dx}{30 b}\\ &=\frac {a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{120 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}+\frac {\left (3 a^4 C-4 b^4 (5 A+4 C)-4 a^2 b^2 (20 A+13 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{30 b d}\\ &=\frac {a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {\left (3 a^4 C-4 b^4 (5 A+4 C)-4 a^2 b^2 (20 A+13 C)\right ) \tan (c+d x)}{30 b d}+\frac {a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{120 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}-\frac {a C (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d}\\ \end {align*}

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Mathematica [A]
time = 1.89, size = 324, normalized size = 1.38 \begin {gather*} -\frac {\left (C+A \cos ^2(c+d x)\right ) \sec ^5(c+d x) \left (120 a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right ) \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 (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-2 \left (540 a^2 A b+200 A b^3+600 a^2 b C+256 b^3 C+45 a \left (12 A b^2+4 a^2 C+17 b^2 C\right ) \cos (c+d x)+48 b \left (15 a^2 (A+C)+b^2 (5 A+4 C)\right ) \cos (2 (c+d x))+180 a A b^2 \cos (3 (c+d x))+60 a^3 C \cos (3 (c+d x))+135 a b^2 C \cos (3 (c+d x))+180 a^2 A b \cos (4 (c+d x))+40 A b^3 \cos (4 (c+d x))+120 a^2 b C \cos (4 (c+d x))+32 b^3 C \cos (4 (c+d x))\right ) \sin (c+d x)\right )}{480 d (A+2 C+A \cos (2 (c+d x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]

[Out]

-1/480*((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(120*a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*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]]) - 2*(540*a^2*A*b + 200*A*b^3
+ 600*a^2*b*C + 256*b^3*C + 45*a*(12*A*b^2 + 4*a^2*C + 17*b^2*C)*Cos[c + d*x] + 48*b*(15*a^2*(A + C) + b^2*(5
*A + 4*C))*Cos[2*(c + d*x)] + 180*a*A*b^2*Cos[3*(c + d*x)] + 60*a^3*C*Cos[3*(c + d*x)] + 135*a*b^2*C*Cos[3*(c
+ d*x)] + 180*a^2*A*b*Cos[4*(c + d*x)] + 40*A*b^3*Cos[4*(c + d*x)] + 120*a^2*b*C*Cos[4*(c + d*x)] + 32*b^3*C*C
os[4*(c + d*x)])*Sin[c + d*x]))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))

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Maple [A]
time = 0.12, size = 246, normalized size = 1.05 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)*(a+b*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(-A*b^3*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)-C*b^3*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+3*a
*A*b^2*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+3*C*b^2*a*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c)
)*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))+3*A*a^2*b*tan(d*x+c)-3*a^2*b*C*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+
A*a^3*ln(sec(d*x+c)+tan(d*x+c))+a^3*C*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c))))

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Maxima [A]
time = 0.29, size = 289, normalized size = 1.24 \begin {gather*} \frac {240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{3} + 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 b^{3} - 45 \, C a b^{2} {\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^{3} {\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 )} - 180 \, A a b^{2} {\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 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 720 \, A a^{2} b \tan \left (d x + c\right )}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/240*(240*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^2*b + 80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*b^3 + 16*(3*tan(
d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*b^3 - 45*C*a*b^2*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(s
in(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^3*(2*sin(d
*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 180*A*a*b^2*(2*sin(d*x + c)/(s
in(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 240*A*a^3*log(sec(d*x + c) + tan(d*x + c
)) + 720*A*a^2*b*tan(d*x + c))/d

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Fricas [A]
time = 4.50, size = 225, normalized size = 0.96 \begin {gather*} \frac {15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (90 \, C a b^{2} \cos \left (d x + c\right ) + 8 \, {\left (15 \, {\left (3 \, A + 2 \, C\right )} a^{2} b + 2 \, {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 24 \, C b^{3} + 15 \, {\left (4 \, C a^{3} + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (15 \, C a^{2} b + {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(15*(4*(2*A + C)*a^3 + 3*(4*A + 3*C)*a*b^2)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(4*(2*A + C)*a^3 +
3*(4*A + 3*C)*a*b^2)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(90*C*a*b^2*cos(d*x + c) + 8*(15*(3*A + 2*C)*a
^2*b + 2*(5*A + 4*C)*b^3)*cos(d*x + c)^4 + 24*C*b^3 + 15*(4*C*a^3 + 3*(4*A + 3*C)*a*b^2)*cos(d*x + c)^3 + 8*(1
5*C*a^2*b + (5*A + 4*C)*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sec(d*x+c))**3*(A+C*sec(d*x+c)**2),x)

[Out]

Integral((A + C*sec(c + d*x)**2)*(a + b*sec(c + d*x))**3*sec(c + d*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (222) = 444\).
time = 0.55, size = 656, normalized size = 2.80 \begin {gather*} \frac {15 \, {\left (8 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 9 \, C a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (8 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 9 \, C a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (60 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 225 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 120 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 120 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1440 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 960 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 360 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 90 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 320 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 160 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2160 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1200 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 400 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 464 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 120 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1440 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 960 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 90 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 360 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 360 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 180 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 225 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, C b^{3} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/120*(15*(8*A*a^3 + 4*C*a^3 + 12*A*a*b^2 + 9*C*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(8*A*a^3 + 4*C*
a^3 + 12*A*a*b^2 + 9*C*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(60*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 360*A*
a^2*b*tan(1/2*d*x + 1/2*c)^9 - 360*C*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 180*A*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 225*C
*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 120*A*b^3*tan(1/2*d*x + 1/2*c)^9 - 120*C*b^3*tan(1/2*d*x + 1/2*c)^9 - 120*C*a^
3*tan(1/2*d*x + 1/2*c)^7 + 1440*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 960*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 360*A*a*
b^2*tan(1/2*d*x + 1/2*c)^7 - 90*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 320*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 160*C*b^3*
tan(1/2*d*x + 1/2*c)^7 - 2160*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 1200*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 400*A*b^3
*tan(1/2*d*x + 1/2*c)^5 - 464*C*b^3*tan(1/2*d*x + 1/2*c)^5 + 120*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 1440*A*a^2*b*t
an(1/2*d*x + 1/2*c)^3 + 960*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 360*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 90*C*a*b^2*t
an(1/2*d*x + 1/2*c)^3 + 320*A*b^3*tan(1/2*d*x + 1/2*c)^3 + 160*C*b^3*tan(1/2*d*x + 1/2*c)^3 - 60*C*a^3*tan(1/2
*d*x + 1/2*c) - 360*A*a^2*b*tan(1/2*d*x + 1/2*c) - 360*C*a^2*b*tan(1/2*d*x + 1/2*c) - 180*A*a*b^2*tan(1/2*d*x
+ 1/2*c) - 225*C*a*b^2*tan(1/2*d*x + 1/2*c) - 120*A*b^3*tan(1/2*d*x + 1/2*c) - 120*C*b^3*tan(1/2*d*x + 1/2*c))
/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d

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Mupad [B]
time = 7.52, size = 445, normalized size = 1.90 \begin {gather*} \frac {a\,\mathrm {atanh}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+12\,A\,b^2+4\,C\,a^2+9\,C\,b^2\right )}{2\,\left (4\,A\,a^3+2\,C\,a^3+6\,A\,a\,b^2+\frac {9\,C\,a\,b^2}{2}\right )}\right )\,\left (8\,A\,a^2+12\,A\,b^2+4\,C\,a^2+9\,C\,b^2\right )}{4\,d}-\frac {\left (2\,A\,b^3-C\,a^3+2\,C\,b^3-3\,A\,a\,b^2+6\,A\,a^2\,b-\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (2\,C\,a^3-\frac {16\,A\,b^3}{3}-\frac {8\,C\,b^3}{3}+6\,A\,a\,b^2-24\,A\,a^2\,b+\frac {3\,C\,a\,b^2}{2}-16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,b^3}{3}+\frac {116\,C\,b^3}{15}+36\,A\,a^2\,b+20\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {16\,A\,b^3}{3}-2\,C\,a^3-\frac {8\,C\,b^3}{3}-6\,A\,a\,b^2-24\,A\,a^2\,b-\frac {3\,C\,a\,b^2}{2}-16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,b^3+C\,a^3+2\,C\,b^3+3\,A\,a\,b^2+6\,A\,a^2\,b+\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^3)/cos(c + d*x),x)

[Out]

(a*atanh((a*tan(c/2 + (d*x)/2)*(8*A*a^2 + 12*A*b^2 + 4*C*a^2 + 9*C*b^2))/(2*(4*A*a^3 + 2*C*a^3 + 6*A*a*b^2 + (
9*C*a*b^2)/2)))*(8*A*a^2 + 12*A*b^2 + 4*C*a^2 + 9*C*b^2))/(4*d) - (tan(c/2 + (d*x)/2)^9*(2*A*b^3 - C*a^3 + 2*C
*b^3 - 3*A*a*b^2 + 6*A*a^2*b - (15*C*a*b^2)/4 + 6*C*a^2*b) - tan(c/2 + (d*x)/2)^3*((16*A*b^3)/3 + 2*C*a^3 + (8
*C*b^3)/3 + 6*A*a*b^2 + 24*A*a^2*b + (3*C*a*b^2)/2 + 16*C*a^2*b) - tan(c/2 + (d*x)/2)^7*((16*A*b^3)/3 - 2*C*a^
3 + (8*C*b^3)/3 - 6*A*a*b^2 + 24*A*a^2*b - (3*C*a*b^2)/2 + 16*C*a^2*b) + tan(c/2 + (d*x)/2)^5*((20*A*b^3)/3 +
(116*C*b^3)/15 + 36*A*a^2*b + 20*C*a^2*b) + tan(c/2 + (d*x)/2)*(2*A*b^3 + C*a^3 + 2*C*b^3 + 3*A*a*b^2 + 6*A*a^
2*b + (15*C*a*b^2)/4 + 6*C*a^2*b))/(d*(5*tan(c/2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2
)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 - 1))

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