\(\int \sec (c+d x) (a+a \sec (c+d x))^3 (A+C \sec ^2(c+d x)) \, dx\) [103]

Optimal result

Integrand size = 31, antiderivative size = 157 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (20 A+13 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (20 A+13 C) \tan (c+d x)}{5 d}+\frac {3 a^3 (20 A+13 C) \sec (c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {a^3 (20 A+13 C) \tan ^3(c+d x)}{60 d} \]

[Out]

1/8*a^3*(20*A+13*C)*arctanh(sin(d*x+c))/d+1/5*a^3*(20*A+13*C)*tan(d*x+c)/d+3/40*a^3*(20*A+13*C)*sec(d*x+c)*tan
(d*x+c)/d-1/20*C*(a+a*sec(d*x+c))^3*tan(d*x+c)/d+1/5*C*(a+a*sec(d*x+c))^4*tan(d*x+c)/a/d+1/60*a^3*(20*A+13*C)*
tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4168, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (20 A+13 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (20 A+13 C) \tan ^3(c+d x)}{60 d}+\frac {a^3 (20 A+13 C) \tan (c+d x)}{5 d}+\frac {3 a^3 (20 A+13 C) \tan (c+d x) \sec (c+d x)}{40 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d}-\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d} \]

[In]

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

[Out]

(a^3*(20*A + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(20*A + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C
)*Sec[c + d*x]*Tan[c + d*x])/(40*d) - (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*a*d) + (a^3*(20*A + 13*C)*Tan[c + d*x]^3)/(60*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 3853

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 3855

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

Rule 3876

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rule 4086

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[(a*B*m + A*b*(m + 1))/(b
*(m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B
, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] &&  !LtQ[m, -2^(-1)]

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*} \text {integral}& = \frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^3 (a (5 A+4 C)-a C \sec (c+d x)) \, dx}{5 a} \\ & = -\frac {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 a d}+\frac {1}{20} (20 A+13 C) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx \\ & = -\frac {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 a d}+\frac {1}{20} (20 A+13 C) \int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx \\ & = -\frac {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 a d}+\frac {1}{20} \left (a^3 (20 A+13 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{20} \left (a^3 (20 A+13 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{20} \left (3 a^3 (20 A+13 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{20} \left (3 a^3 (20 A+13 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^3 (20 A+13 C) \text {arctanh}(\sin (c+d x))}{20 d}+\frac {3 a^3 (20 A+13 C) \sec (c+d x) \tan (c+d x)}{40 d}-\frac {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 a d}+\frac {1}{40} \left (3 a^3 (20 A+13 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (20 A+13 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{20 d}-\frac {\left (3 a^3 (20 A+13 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{20 d} \\ & = \frac {a^3 (20 A+13 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (20 A+13 C) \tan (c+d x)}{5 d}+\frac {3 a^3 (20 A+13 C) \sec (c+d x) \tan (c+d x)}{40 d}-\frac {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 a d}+\frac {a^3 (20 A+13 C) \tan ^3(c+d x)}{60 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.24 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {5 a^3 A \text {arctanh}(\sin (c+d x))}{2 d}+\frac {13 a^3 C \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 A \tan (c+d x)}{d}+\frac {4 a^3 C \tan (c+d x)}{d}+\frac {3 a^3 A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {13 a^3 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a^3 C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^3 A \tan ^3(c+d x)}{3 d}+\frac {5 a^3 C \tan ^3(c+d x)}{3 d}+\frac {a^3 C \tan ^5(c+d x)}{5 d} \]

[In]

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

[Out]

(5*a^3*A*ArcTanh[Sin[c + d*x]])/(2*d) + (13*a^3*C*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*A*Tan[c + d*x])/d + (4
*a^3*C*Tan[c + d*x])/d + (3*a^3*A*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (13*a^3*C*Sec[c + d*x]*Tan[c + d*x])/(8*d
) + (3*a^3*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*A*Tan[c + d*x]^3)/(3*d) + (5*a^3*C*Tan[c + d*x]^3)/(3*d
) + (a^3*C*Tan[c + d*x]^5)/(5*d)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.28

method	result	size
norman	\(\frac {-\frac {32 a^{3} \left (20 A +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {7 a^{3} \left (20 A +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}-\frac {a^{3} \left (20 A +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {a^{3} \left (44 A +51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (212 A +133 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {a^{3} \left (20 A +13 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{3} \left (20 A +13 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(201\)
parts	\(-\frac {\left (a^{3} A +3 a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a^{3} A +a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{3}}{d}-\frac {a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {3 a^{3} A \tan \left (d x +c \right )}{d}+\frac {3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(211\)
parallelrisch	\(\frac {26 a^{3} \left (-\frac {75 \left (A +\frac {13 C}{20}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{26}+\frac {75 \left (A +\frac {13 C}{20}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{26}+\left (\frac {9 A}{13}+\frac {75 C}{52}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {37 A}{26}+\frac {19 C}{13}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {9 A}{26}+\frac {3 C}{8}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {11 A}{26}+\frac {19 C}{65}\right ) \sin \left (5 d x +5 c \right )+\sin \left (d x +c \right ) \left (A +\frac {20 C}{13}\right )\right )}{3 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\)	\(217\)
derivativedivides	\(\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(242\)
default	\(\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(242\)
risch	\(-\frac {i a^{3} \left (180 A \,{\mathrm e}^{9 i \left (d x +c \right )}+195 C \,{\mathrm e}^{9 i \left (d x +c \right )}-360 A \,{\mathrm e}^{8 i \left (d x +c \right )}+360 A \,{\mathrm e}^{7 i \left (d x +c \right )}+750 C \,{\mathrm e}^{7 i \left (d x +c \right )}-1680 A \,{\mathrm e}^{6 i \left (d x +c \right )}-720 C \,{\mathrm e}^{6 i \left (d x +c \right )}-2720 A \,{\mathrm e}^{4 i \left (d x +c \right )}-2320 C \,{\mathrm e}^{4 i \left (d x +c \right )}-360 A \,{\mathrm e}^{3 i \left (d x +c \right )}-750 C \,{\mathrm e}^{3 i \left (d x +c \right )}-1840 A \,{\mathrm e}^{2 i \left (d x +c \right )}-1520 C \,{\mathrm e}^{2 i \left (d x +c \right )}-180 A \,{\mathrm e}^{i \left (d x +c \right )}-195 C \,{\mathrm e}^{i \left (d x +c \right )}-440 A -304 C \right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\)	\(299\)

[In]

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

[Out]

(-32/15*a^3*(20*A+13*C)/d*tan(1/2*d*x+1/2*c)^5+7/6*a^3*(20*A+13*C)/d*tan(1/2*d*x+1/2*c)^7-1/4*a^3*(20*A+13*C)/
d*tan(1/2*d*x+1/2*c)^9-1/4*a^3*(44*A+51*C)/d*tan(1/2*d*x+1/2*c)+1/6*a^3*(212*A+133*C)/d*tan(1/2*d*x+1/2*c)^3)/
(tan(1/2*d*x+1/2*c)^2-1)^5-1/8*a^3*(20*A+13*C)/d*ln(tan(1/2*d*x+1/2*c)-1)+1/8*a^3*(20*A+13*C)/d*ln(tan(1/2*d*x
+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.03 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (20 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (20 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (55 \, A + 38 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (12 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 90 \, C a^{3} \cos \left (d x + c\right ) + 24 \, C a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]

[In]

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

[Out]

1/240*(15*(20*A + 13*C)*a^3*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(20*A + 13*C)*a^3*cos(d*x + c)^5*log(-si
n(d*x + c) + 1) + 2*(8*(55*A + 38*C)*a^3*cos(d*x + c)^4 + 15*(12*A + 13*C)*a^3*cos(d*x + c)^3 + 8*(5*A + 19*C)
*a^3*cos(d*x + c)^2 + 90*C*a^3*cos(d*x + c) + 24*C*a^3)*sin(d*x + c))/(d*cos(d*x + c)^5)

Sympy [F]

\[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.82 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 45 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 720 \, A a^{3} \tan \left (d x + c\right )}{240 \, d} \]

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))^3*(A+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^3 + 16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c)
)*C*a^3 + 240*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 - 45*C*a^3*(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)) - 180*A*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)) - 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)) + 240*A*a^3*log(sec(d*x + c) + tan(d*x + c)) + 7
20*A*a^3*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.57 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (20 \, A a^{3} + 13 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (20 \, A a^{3} + 13 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (300 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 195 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1400 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 910 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2560 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1664 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2120 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1330 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 660 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 765 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]

[In]

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

[Out]

1/120*(15*(20*A*a^3 + 13*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(20*A*a^3 + 13*C*a^3)*log(abs(tan(1/2*
d*x + 1/2*c) - 1)) - 2*(300*A*a^3*tan(1/2*d*x + 1/2*c)^9 + 195*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 1400*A*a^3*tan(1
/2*d*x + 1/2*c)^7 - 910*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 2560*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 1664*C*a^3*tan(1/2*
d*x + 1/2*c)^5 - 2120*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 1330*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 660*A*a^3*tan(1/2*d*x
+ 1/2*c) + 765*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d

Mupad [B] (verification not implemented)

Time = 18.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.43 \[ \int \sec (c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (20\,A+13\,C\right )}{4\,d}-\frac {\left (5\,A\,a^3+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {70\,A\,a^3}{3}-\frac {91\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {128\,A\,a^3}{3}+\frac {416\,C\,a^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {106\,A\,a^3}{3}-\frac {133\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A\,a^3+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

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

[Out]

(a^3*atanh(tan(c/2 + (d*x)/2))*(20*A + 13*C))/(4*d) - (tan(c/2 + (d*x)/2)*(11*A*a^3 + (51*C*a^3)/4) + tan(c/2
+ (d*x)/2)^9*(5*A*a^3 + (13*C*a^3)/4) - tan(c/2 + (d*x)/2)^7*((70*A*a^3)/3 + (91*C*a^3)/6) - tan(c/2 + (d*x)/2
)^3*((106*A*a^3)/3 + (133*C*a^3)/6) + tan(c/2 + (d*x)/2)^5*((128*A*a^3)/3 + (416*C*a^3)/15))/(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))