\(\int (a+b \sec (c+d x))^3 (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [786]

Optimal result

Integrand size = 32, antiderivative size = 180 \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{6 d}+\frac {b \left (20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]

[Out]

1/8*(8*B*a^3+12*B*a*b^2+12*C*a^2*b+3*C*b^3)*arctanh(sin(d*x+c))/d+1/6*(16*B*a^2*b+4*B*b^3+3*C*a^3+12*C*a*b^2)*
tan(d*x+c)/d+1/24*b*(20*B*a*b+6*C*a^2+9*C*b^2)*sec(d*x+c)*tan(d*x+c)/d+1/12*(4*B*b+3*C*a)*(a+b*sec(d*x+c))^2*t
an(d*x+c)/d+1/4*C*(a+b*sec(d*x+c))^3*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4141, 4133, 3855, 3852, 8} \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {b \left (6 a^2 C+20 a b B+9 b^2 C\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {\left (8 a^3 B+12 a^2 b C+12 a b^2 B+3 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (3 a^3 C+16 a^2 b B+12 a b^2 C+4 b^3 B\right ) \tan (c+d x)}{6 d}+\frac {(3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]

[In]

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

[Out]

((8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C
+ 12*a*b^2*C)*Tan[c + d*x])/(6*d) + (b*(20*a*b*B + 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4
*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*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 4133

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

Rule 4141

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/(m + 1), I
nt[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) +
a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left ((4 a B+3 b C) \sec (c+d x)+(4 b B+3 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (\left (12 a^2 B+8 b^2 B+15 a b C\right ) \sec (c+d x)+\left (20 a b B+6 a^2 C+9 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (3 \left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) \sec (c+d x)+4 \left (16 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (16 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = \frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{6 d}+\frac {b \left (20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.78 \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (24 \left (3 a^2 b B+b^3 B+a^3 C+3 a b^2 C\right )+9 b \left (4 a b B+4 a^2 C+b^2 C\right ) \sec (c+d x)+6 b^3 C \sec ^3(c+d x)+8 b^2 (b B+3 a C) \tan ^2(c+d x)\right )}{24 d} \]

[In]

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

[Out]

(3*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(3*a^2*b*B + b^3*B +
a^3*C + 3*a*b^2*C) + 9*b*(4*a*b*B + 4*a^2*C + b^2*C)*Sec[c + d*x] + 6*b^3*C*Sec[c + d*x]^3 + 8*b^2*(b*B + 3*a
*C)*Tan[c + d*x]^2))/(24*d)

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03

method	result	size
parts	\(-\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 B a \,b^{2}+3 a^{2} b C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 B \,a^{2} b +a^{3} C \right ) \tan \left (d x +c \right )}{d}+\frac {B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(185\)
derivativedivides	\(\frac {B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 B \tan \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 C a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(223\)
default	\(\frac {B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 B \tan \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 C a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(223\)
parallelrisch	\(\frac {-96 \left (B \,a^{3}+\frac {3}{2} B a \,b^{2}+\frac {3}{2} a^{2} b C +\frac {3}{8} C \,b^{3}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+96 \left (B \,a^{3}+\frac {3}{2} B a \,b^{2}+\frac {3}{2} a^{2} b C +\frac {3}{8} C \,b^{3}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (144 B \,a^{2} b +64 B \,b^{3}+48 a^{3} C +192 C a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (72 B \,a^{2} b +16 B \,b^{3}+24 a^{3} C +48 C a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+72 \left (\left (B a b +C \,a^{2}+\frac {1}{4} C \,b^{2}\right ) \sin \left (3 d x +3 c \right )+\sin \left (d x +c \right ) \left (B a b +C \,a^{2}+\frac {11}{12} C \,b^{2}\right )\right ) b}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\)	\(282\)
norman	\(\frac {-\frac {\left (24 B \,a^{2} b -12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C -12 a^{2} b C +24 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (24 B \,a^{2} b +12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C +12 a^{2} b C +24 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (216 B \,a^{2} b -36 B a \,b^{2}+40 B \,b^{3}+72 a^{3} C -36 a^{2} b C +120 C a \,b^{2}+9 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {\left (216 B \,a^{2} b +36 B a \,b^{2}+40 B \,b^{3}+72 a^{3} C +36 a^{2} b C +120 C a \,b^{2}-9 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {\left (8 B \,a^{3}+12 B a \,b^{2}+12 a^{2} b C +3 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (8 B \,a^{3}+12 B a \,b^{2}+12 a^{2} b C +3 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(357\)
risch	\(-\frac {i \left (-24 a^{3} C -48 C a \,b^{2}-16 B \,b^{3}-48 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-24 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+33 C \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-9 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-72 B \,a^{2} b -72 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-64 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-72 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-33 C \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+9 C \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-36 B a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-216 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+36 C \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-36 C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-192 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+36 C \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-72 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-36 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+36 B a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-216 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-144 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+36 B a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-36 B a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{3}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{2}}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b C}{2 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{3}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{2}}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b C}{2 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\)	\(570\)

[In]

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

[Out]

-(B*b^3+3*C*a*b^2)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(3*B*a*b^2+3*C*a^2*b)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2
*ln(sec(d*x+c)+tan(d*x+c)))+(3*B*a^2*b+C*a^3)/d*tan(d*x+c)+B*a^3/d*ln(sec(d*x+c)+tan(d*x+c))+C*b^3/d*(-(-1/4*s
ec(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)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (8 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 3 \, C b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 3 \, C b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C b^{3} + 8 \, {\left (3 \, C a^{3} + 9 \, B a^{2} b + 6 \, C a b^{2} + 2 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 9 \, {\left (4 \, C a^{2} b + 4 \, B a b^{2} + C b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]

[In]

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

[Out]

1/48*(3*(8*B*a^3 + 12*C*a^2*b + 12*B*a*b^2 + 3*C*b^3)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(8*B*a^3 + 12*C
*a^2*b + 12*B*a*b^2 + 3*C*b^3)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(6*C*b^3 + 8*(3*C*a^3 + 9*B*a^2*b + 6
*C*a*b^2 + 2*B*b^3)*cos(d*x + c)^3 + 9*(4*C*a^2*b + 4*B*a*b^2 + C*b^3)*cos(d*x + c)^2 + 8*(3*C*a*b^2 + B*b^3)*
cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4)

Sympy [F]

\[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.48 \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{2} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{3} - 3 \, C b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, C a^{3} \tan \left (d x + c\right ) + 144 \, B a^{2} b \tan \left (d x + c\right )}{48 \, d} \]

[In]

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

[Out]

1/48*(48*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a*b^2 + 16*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*b^3 - 3*C*b^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(s
in(d*x + c) - 1)) - 36*C*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c)
- 1)) - 36*B*a*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 48
*B*a^3*log(sec(d*x + c) + tan(d*x + c)) + 48*C*a^3*tan(d*x + c) + 144*B*a^2*b*tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (170) = 340\).

Time = 0.35 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.26 \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (8 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 3 \, C b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 3 \, C b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 216 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, 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 )}^{4}}}{24 \, d} \]

[In]

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

[Out]

1/24*(3*(8*B*a^3 + 12*C*a^2*b + 12*B*a*b^2 + 3*C*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(8*B*a^3 + 12*C*a
^2*b + 12*B*a*b^2 + 3*C*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(24*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 72*B*a^
2*b*tan(1/2*d*x + 1/2*c)^7 - 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 72*C*a*b^
2*tan(1/2*d*x + 1/2*c)^7 + 24*B*b^3*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^3*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^3*tan(1/
2*d*x + 1/2*c)^5 - 216*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*B*a*b^2*tan(1/2
*d*x + 1/2*c)^5 - 120*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 40*B*b^3*tan(1/2*d*x + 1/2*c)^5 - 9*C*b^3*tan(1/2*d*x +
1/2*c)^5 + 72*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 216*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^2*b*tan(1/2*d*x + 1/
2*c)^3 + 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 40*B*b^3*tan(1/2*d*x + 1/2*c
)^3 - 9*C*b^3*tan(1/2*d*x + 1/2*c)^3 - 24*C*a^3*tan(1/2*d*x + 1/2*c) - 72*B*a^2*b*tan(1/2*d*x + 1/2*c) - 36*C*
a^2*b*tan(1/2*d*x + 1/2*c) - 36*B*a*b^2*tan(1/2*d*x + 1/2*c) - 72*C*a*b^2*tan(1/2*d*x + 1/2*c) - 24*B*b^3*tan(
1/2*d*x + 1/2*c) - 15*C*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d

Mupad [B] (verification not implemented)

Time = 20.99 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.19 \[ \int (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B\,a^3+\frac {3\,C\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}+\frac {3\,C\,b^3}{8}\right )}{4\,B\,a^3+6\,C\,a^2\,b+6\,B\,a\,b^2+\frac {3\,C\,b^3}{2}}\right )\,\left (2\,B\,a^3+3\,C\,a^2\,b+3\,B\,a\,b^2+\frac {3\,C\,b^3}{4}\right )}{d}-\frac {\left (2\,B\,b^3+2\,C\,a^3-\frac {5\,C\,b^3}{4}-3\,B\,a\,b^2+6\,B\,a^2\,b+6\,C\,a\,b^2-3\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (3\,B\,a\,b^2-6\,C\,a^3-\frac {3\,C\,b^3}{4}-\frac {10\,B\,b^3}{3}-18\,B\,a^2\,b-10\,C\,a\,b^2+3\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {10\,B\,b^3}{3}+6\,C\,a^3-\frac {3\,C\,b^3}{4}+3\,B\,a\,b^2+18\,B\,a^2\,b+10\,C\,a\,b^2+3\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-2\,B\,b^3-2\,C\,a^3-\frac {5\,C\,b^3}{4}-3\,B\,a\,b^2-6\,B\,a^2\,b-6\,C\,a\,b^2-3\,C\,a^2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(atanh((4*tan(c/2 + (d*x)/2)*(B*a^3 + (3*C*b^3)/8 + (3*B*a*b^2)/2 + (3*C*a^2*b)/2))/(4*B*a^3 + (3*C*b^3)/2 + 6
*B*a*b^2 + 6*C*a^2*b))*(2*B*a^3 + (3*C*b^3)/4 + 3*B*a*b^2 + 3*C*a^2*b))/d - (tan(c/2 + (d*x)/2)^7*(2*B*b^3 + 2
*C*a^3 - (5*C*b^3)/4 - 3*B*a*b^2 + 6*B*a^2*b + 6*C*a*b^2 - 3*C*a^2*b) + tan(c/2 + (d*x)/2)^3*((10*B*b^3)/3 + 6
*C*a^3 - (3*C*b^3)/4 + 3*B*a*b^2 + 18*B*a^2*b + 10*C*a*b^2 + 3*C*a^2*b) - tan(c/2 + (d*x)/2)^5*((10*B*b^3)/3 +
6*C*a^3 + (3*C*b^3)/4 - 3*B*a*b^2 + 18*B*a^2*b + 10*C*a*b^2 - 3*C*a^2*b) - tan(c/2 + (d*x)/2)*(2*B*b^3 + 2*C*
a^3 + (5*C*b^3)/4 + 3*B*a*b^2 + 6*B*a^2*b + 6*C*a*b^2 + 3*C*a^2*b))/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (
d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1))