3.9.97 \(\int (a+b \sec (c+d x))^3 (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)) \, dx\) [897]

Optimal. Leaf size=214 \[ a^4 (b B-a C) x+\frac {b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{6 d}+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]

[Out]

a^4*(B*b-C*a)*x+1/8*b*(32*B*a^3*b+16*B*a*b^3-24*C*a^4+8*C*a^2*b^2+3*C*b^4)*arctanh(sin(d*x+c))/d+1/6*b^2*(34*B
*a^2*b+4*B*b^3-15*C*a^3+12*C*a*b^2)*tan(d*x+c)/d+1/24*b^3*(32*B*a*b-6*C*a^2+9*C*b^2)*sec(d*x+c)*tan(d*x+c)/d+1
/12*b^2*(4*B*b+3*C*a)*(a+b*sec(d*x+c))^2*tan(d*x+c)/d+1/4*b^2*C*(a+b*sec(d*x+c))^3*tan(d*x+c)/d

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Rubi [A]
time = 0.35, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4126, 4003, 4141, 4133, 3855, 3852, 8} \begin {gather*} a^4 x (b B-a C)+\frac {b^3 \left (-6 a^2 C+32 a b B+9 b^2 C\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {b^2 \left (-15 a^3 C+34 a^2 b B+12 a b^2 C+4 b^3 B\right ) \tan (c+d x)}{6 d}+\frac {b \left (-24 a^4 C+32 a^3 b B+8 a^2 b^2 C+16 a b^3 B+3 b^4 C\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^2 (3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {b^2 C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*(b*B - a*C)*x + (b*(32*a^3*b*B + 16*a*b^3*B - 24*a^4*C + 8*a^2*b^2*C + 3*b^4*C)*ArcTanh[Sin[c + d*x]])/(8*
d) + (b^2*(34*a^2*b*B + 4*b^3*B - 15*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(6*d) + (b^3*(32*a*b*B - 6*a^2*C + 9*b^
2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (b^2*(4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b
^2*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 4003

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 - 2)*Simp[a^2
*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x],
x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 4126

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

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*} \int (a+b \sec (c+d x))^3 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx &=\frac {\int (a+b \sec (c+d x))^4 \left (b^2 (b B-a C)+b^3 C \sec (c+d x)\right ) \, dx}{b^2}\\ &=\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+b \sec (c+d x))^2 \left (4 a^2 b^2 (b B-a C)+b^3 \left (8 a b B-4 a^2 C+3 b^2 C\right ) \sec (c+d x)+b^4 (4 b B+3 a C) \sec ^2(c+d x)\right ) \, dx}{4 b^2}\\ &=\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+b \sec (c+d x)) \left (12 a^3 b^2 (b B-a C)+b^3 \left (36 a^2 b B+8 b^3 B-24 a^3 C+15 a b^2 C\right ) \sec (c+d x)+b^4 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx}{12 b^2}\\ &=\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int \left (24 a^4 b^2 (b B-a C)+3 b^3 \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \sec (c+d x)+4 b^4 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right ) \sec ^2(c+d x)\right ) \, dx}{24 b^2}\\ &=a^4 (b B-a C) x+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right )\right ) \int \sec (c+d x) \, dx\\ &=a^4 (b B-a C) x+\frac {b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=a^4 (b B-a C) x+\frac {b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^2 \left (34 a^2 b B+4 b^3 B-15 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{6 d}+\frac {b^3 \left (32 a b B-6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b^2 C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 1.34, size = 170, normalized size = 0.79 \begin {gather*} \frac {24 a^4 (b B-a C) d x+3 b \left (32 a^3 b B+16 a b^3 B-24 a^4 C+8 a^2 b^2 C+3 b^4 C\right ) \tanh ^{-1}(\sin (c+d x))+3 b^2 \left (8 \left (6 a^2 b B+b^3 B-2 a^3 C+3 a b^2 C\right )+b \left (16 a b B+8 a^2 C+3 b^2 C\right ) \sec (c+d x)+2 b^3 C \sec ^3(c+d x)\right ) \tan (c+d x)+8 b^4 (b B+3 a C) \tan ^3(c+d x)}{24 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a^4*(b*B - a*C)*d*x + 3*b*(32*a^3*b*B + 16*a*b^3*B - 24*a^4*C + 8*a^2*b^2*C + 3*b^4*C)*ArcTanh[Sin[c + d*x
]] + 3*b^2*(8*(6*a^2*b*B + b^3*B - 2*a^3*C + 3*a*b^2*C) + b*(16*a*b*B + 8*a^2*C + 3*b^2*C)*Sec[c + d*x] + 2*b^
3*C*Sec[c + d*x]^3)*Tan[c + d*x] + 8*b^4*(b*B + 3*a*C)*Tan[c + d*x]^3)/(24*d)

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Maple [A]
time = 0.12, size = 278, normalized size = 1.30 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(4*a*b^4*B*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+2*C*a^2*b^3*(1/2*sec(d*x+c)*tan(d*x+c
)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-b^5*B*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+C*b^5*(-(-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)))+6*a^2*b^3*B*tan(d*x+c)-2*a^3*b^2*C*tan(d*x+c)-3*b^4*C*a*(-2/3
-1/3*sec(d*x+c)^2)*tan(d*x+c)+4*a^3*b^2*B*ln(sec(d*x+c)+tan(d*x+c))-3*a^4*b*C*ln(sec(d*x+c)+tan(d*x+c))+a^4*b*
B*(d*x+c)-a^5*C*(d*x+c))

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Maxima [A]
time = 0.31, size = 320, normalized size = 1.50 \begin {gather*} -\frac {48 \, {\left (d x + c\right )} C a^{5} - 48 \, {\left (d x + c\right )} B a^{4} b - 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{4} - 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{5} + 3 \, C b^{5} {\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 )} + 24 \, C a^{2} b^{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 )} + 48 \, B a b^{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 )} + 144 \, C a^{4} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 192 \, B a^{3} b^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, C a^{3} b^{2} \tan \left (d x + c\right ) - 288 \, B a^{2} b^{3} \tan \left (d x + c\right )}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(48*(d*x + c)*C*a^5 - 48*(d*x + c)*B*a^4*b - 48*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a*b^4 - 16*(tan(d*x
+ c)^3 + 3*tan(d*x + c))*B*b^5 + 3*C*b^5*(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)) + 24*C*a^2*b^3*(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*b^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(
sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 144*C*a^4*b*log(sec(d*x + c) + tan(d*x + c)) - 192*B*a^3*b^2*log(
sec(d*x + c) + tan(d*x + c)) + 96*C*a^3*b^2*tan(d*x + c) - 288*B*a^2*b^3*tan(d*x + c))/d

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Fricas [A]
time = 3.31, size = 268, normalized size = 1.25 \begin {gather*} -\frac {48 \, {\left (C a^{5} - B a^{4} b\right )} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (6 \, C b^{5} - 16 \, {\left (3 \, C a^{3} b^{2} - 9 \, B a^{2} b^{3} - 3 \, C a b^{4} - B b^{5}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, C a^{2} b^{3} + 16 \, B a b^{4} + 3 \, C b^{5}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(48*(C*a^5 - B*a^4*b)*d*x*cos(d*x + c)^4 + 3*(24*C*a^4*b - 32*B*a^3*b^2 - 8*C*a^2*b^3 - 16*B*a*b^4 - 3*C
*b^5)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(24*C*a^4*b - 32*B*a^3*b^2 - 8*C*a^2*b^3 - 16*B*a*b^4 - 3*C*b^5
)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*(6*C*b^5 - 16*(3*C*a^3*b^2 - 9*B*a^2*b^3 - 3*C*a*b^4 - B*b^5)*cos(
d*x + c)^3 + 3*(8*C*a^2*b^3 + 16*B*a*b^4 + 3*C*b^5)*cos(d*x + c)^2 + 8*(3*C*a*b^4 + B*b^5)*cos(d*x + c))*sin(d
*x + c))/(d*cos(d*x + c)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int C a^{5}\, dx - \int \left (- B a^{4} b\right )\, dx - \int \left (- B b^{5} \sec ^{4}{\left (c + d x \right )}\right )\, dx - \int \left (- C b^{5} \sec ^{5}{\left (c + d x \right )}\right )\, dx - \int \left (- 4 B a b^{4} \sec ^{3}{\left (c + d x \right )}\right )\, dx - \int \left (- 6 B a^{2} b^{3} \sec ^{2}{\left (c + d x \right )}\right )\, dx - \int \left (- 4 B a^{3} b^{2} \sec {\left (c + d x \right )}\right )\, dx - \int \left (- 3 C a b^{4} \sec ^{4}{\left (c + d x \right )}\right )\, dx - \int \left (- 2 C a^{2} b^{3} \sec ^{3}{\left (c + d x \right )}\right )\, dx - \int 2 C a^{3} b^{2} \sec ^{2}{\left (c + d x \right )}\, dx - \int 3 C a^{4} b \sec {\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(C*a**5, x) - Integral(-B*a**4*b, x) - Integral(-B*b**5*sec(c + d*x)**4, x) - Integral(-C*b**5*sec(c
+ d*x)**5, x) - Integral(-4*B*a*b**4*sec(c + d*x)**3, x) - Integral(-6*B*a**2*b**3*sec(c + d*x)**2, x) - Integ
ral(-4*B*a**3*b**2*sec(c + d*x), x) - Integral(-3*C*a*b**4*sec(c + d*x)**4, x) - Integral(-2*C*a**2*b**3*sec(c
+ d*x)**3, x) - Integral(2*C*a**3*b**2*sec(c + d*x)**2, x) - Integral(3*C*a**4*b*sec(c + d*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (205) = 410\).
time = 0.56, size = 658, normalized size = 3.07 \begin {gather*} -\frac {24 \, {\left (C a^{5} - B a^{4} b\right )} {\left (d x + c\right )} + 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 8 \, C a^{2} b^{3} - 16 \, B a b^{4} - 3 \, C b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (48 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 432 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 120 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 144 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 432 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, C b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 144 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, C a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C b^{5} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(24*(C*a^5 - B*a^4*b)*(d*x + c) + 3*(24*C*a^4*b - 32*B*a^3*b^2 - 8*C*a^2*b^3 - 16*B*a*b^4 - 3*C*b^5)*log
(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(24*C*a^4*b - 32*B*a^3*b^2 - 8*C*a^2*b^3 - 16*B*a*b^4 - 3*C*b^5)*log(abs(t
an(1/2*d*x + 1/2*c) - 1)) - 2*(48*C*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 144*B*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 24
*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 48*B*a*b^4*tan(1/2*d*x + 1/2*c)^7 - 72*C*a*b^4*tan(1/2*d*x + 1/2*c)^7 - 24
*B*b^5*tan(1/2*d*x + 1/2*c)^7 + 15*C*b^5*tan(1/2*d*x + 1/2*c)^7 - 144*C*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 + 432*B
*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 48*B*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 12
0*C*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 40*B*b^5*tan(1/2*d*x + 1/2*c)^5 + 9*C*b^5*tan(1/2*d*x + 1/2*c)^5 + 144*C*a^
3*b^2*tan(1/2*d*x + 1/2*c)^3 - 432*B*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 24*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 48
*B*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 120*C*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 40*B*b^5*tan(1/2*d*x + 1/2*c)^3 + 9*C*b
^5*tan(1/2*d*x + 1/2*c)^3 - 48*C*a^3*b^2*tan(1/2*d*x + 1/2*c) + 144*B*a^2*b^3*tan(1/2*d*x + 1/2*c) + 24*C*a^2*
b^3*tan(1/2*d*x + 1/2*c) + 48*B*a*b^4*tan(1/2*d*x + 1/2*c) + 72*C*a*b^4*tan(1/2*d*x + 1/2*c) + 24*B*b^5*tan(1/
2*d*x + 1/2*c) + 15*C*b^5*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d

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Mupad [B]
time = 6.70, size = 2500, normalized size = 11.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*(2*B*b^5 + (5*C*b^5)/4 + 12*B*a^2*b^3 + 2*C*a^2*b^3 - 4*C*a^3*b^2 + 4*B*a*b^4 + 6*C*a*b^4)
+ tan(c/2 + (d*x)/2)^7*((5*C*b^5)/4 - 2*B*b^5 - 12*B*a^2*b^3 + 2*C*a^2*b^3 + 4*C*a^3*b^2 + 4*B*a*b^4 - 6*C*a*
b^4) - tan(c/2 + (d*x)/2)^3*((10*B*b^5)/3 - (3*C*b^5)/4 + 36*B*a^2*b^3 + 2*C*a^2*b^3 - 12*C*a^3*b^2 + 4*B*a*b^
4 + 10*C*a*b^4) + tan(c/2 + (d*x)/2)^5*((10*B*b^5)/3 + (3*C*b^5)/4 + 36*B*a^2*b^3 - 2*C*a^2*b^3 - 12*C*a^3*b^2
- 4*B*a*b^4 + 10*C*a*b^4))/(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)) - (atan(((tan(c/2 + (d*x)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2*a^2*b^8 + 512*B^2
*a^4*b^6 + 512*B^2*a^6*b^4 + 32*B^2*a^8*b^2 + 24*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*b^4 + 288*C^2*a^8*
b^2 + 48*B*C*a*b^9 - 64*B*C*a^9*b + 224*B*C*a^3*b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) + ((3*C*b^5)/8 + 4*B*
a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b)*(12*C*b^5 - 32*C*a^5 + 128*B*a^3*b^2 + 32*C*a^2*b^3 + 64*B*a*b^4
+ 32*B*a^4*b - 96*C*a^4*b))*((3*C*b^5)/8 + 4*B*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b)*1i + (tan(c/2 + (d
*x)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2*a^2*b^8 + 512*B^2*a^4*b^6 + 512*B^2*a^6*b^4 + 32*B^2*a^8*b^2 +
24*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*b^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b^9 - 64*B*C*a^9*b + 224*B*C*a^
3*b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) - ((3*C*b^5)/8 + 4*B*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b)*(
12*C*b^5 - 32*C*a^5 + 128*B*a^3*b^2 + 32*C*a^2*b^3 + 64*B*a*b^4 + 32*B*a^4*b - 96*C*a^4*b))*((3*C*b^5)/8 + 4*B
*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b)*1i)/((tan(c/2 + (d*x)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2
*a^2*b^8 + 512*B^2*a^4*b^6 + 512*B^2*a^6*b^4 + 32*B^2*a^8*b^2 + 24*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*
b^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b^9 - 64*B*C*a^9*b + 224*B*C*a^3*b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) + (
(3*C*b^5)/8 + 4*B*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b)*(12*C*b^5 - 32*C*a^5 + 128*B*a^3*b^2 + 32*C*a^2
*b^3 + 64*B*a*b^4 + 32*B*a^4*b - 96*C*a^4*b))*((3*C*b^5)/8 + 4*B*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b)
- (tan(c/2 + (d*x)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2*a^2*b^8 + 512*B^2*a^4*b^6 + 512*B^2*a^6*b^4 + 32
*B^2*a^8*b^2 + 24*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*b^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b^9 - 64*B*C*a^9
*b + 224*B*C*a^3*b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) - ((3*C*b^5)/8 + 4*B*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4
- 3*C*a^4*b)*(12*C*b^5 - 32*C*a^5 + 128*B*a^3*b^2 + 32*C*a^2*b^3 + 64*B*a*b^4 + 32*B*a^4*b - 96*C*a^4*b))*((3
*C*b^5)/8 + 4*B*a^3*b^2 + C*a^2*b^3 + 2*B*a*b^4 - 3*C*a^4*b) - 192*C^3*a^14*b - 256*B^3*a^6*b^9 - 1024*B^3*a^8
*b^7 + 128*B^3*a^9*b^6 - 1024*B^3*a^10*b^5 + 256*B^3*a^11*b^4 + 9*C^3*a^5*b^10 + 48*C^3*a^7*b^8 - 80*C^3*a^9*b
^6 + 24*C^3*a^10*b^5 - 384*C^3*a^11*b^4 + 64*C^3*a^12*b^3 + 576*C^3*a^13*b^2 - 9*B*C^2*a^4*b^11 + 48*B*C^2*a^6
*b^9 + 528*B*C^2*a^8*b^7 - 48*B*C^2*a^9*b^6 + 128*B*C^2*a^10*b^5 - 2112*B*C^2*a^12*b^3 + 640*B*C^2*a^13*b^2 -
96*B^2*C*a^5*b^10 - 192*B^2*C*a^7*b^8 + 24*B^2*C*a^8*b^7 + 1280*B^2*C*a^9*b^6 - 192*B^2*C*a^10*b^5 + 2560*B^2*
C*a^11*b^4 - 704*B^2*C*a^12*b^3))*((C*b^5*3i)/4 + B*a^3*b^2*8i + C*a^2*b^3*2i + B*a*b^4*4i - C*a^4*b*6i))/d -
(2*a^4*atan((a^4*(tan(c/2 + (d*x)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2*a^2*b^8 + 512*B^2*a^4*b^6 + 512*B
^2*a^6*b^4 + 32*B^2*a^8*b^2 + 24*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*b^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b
^9 - 64*B*C*a^9*b + 224*B*C*a^3*b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) - a^4*(B*b - C*a)*(12*C*b^5 - 32*C*a^
5 + 128*B*a^3*b^2 + 32*C*a^2*b^3 + 64*B*a*b^4 + 32*B*a^4*b - 96*C*a^4*b)*1i)*(B*b - C*a) + a^4*(tan(c/2 + (d*x
)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2*a^2*b^8 + 512*B^2*a^4*b^6 + 512*B^2*a^6*b^4 + 32*B^2*a^8*b^2 + 24
*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*b^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b^9 - 64*B*C*a^9*b + 224*B*C*a^3*
b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) + a^4*(B*b - C*a)*(12*C*b^5 - 32*C*a^5 + 128*B*a^3*b^2 + 32*C*a^2*b^3
+ 64*B*a*b^4 + 32*B*a^4*b - 96*C*a^4*b)*1i)*(B*b - C*a))/(128*B^3*a^9*b^6 - 256*B^3*a^6*b^9 - 1024*B^3*a^8*b^
7 - 192*C^3*a^14*b - 1024*B^3*a^10*b^5 + 256*B^3*a^11*b^4 + 9*C^3*a^5*b^10 + 48*C^3*a^7*b^8 - 80*C^3*a^9*b^6 +
24*C^3*a^10*b^5 - 384*C^3*a^11*b^4 + 64*C^3*a^12*b^3 + 576*C^3*a^13*b^2 - a^4*(tan(c/2 + (d*x)/2)*(32*C^2*a^1
0 + (9*C^2*b^10)/2 + 128*B^2*a^2*b^8 + 512*B^2*a^4*b^6 + 512*B^2*a^6*b^4 + 32*B^2*a^8*b^2 + 24*C^2*a^2*b^8 - 4
0*C^2*a^4*b^6 - 192*C^2*a^6*b^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b^9 - 64*B*C*a^9*b + 224*B*C*a^3*b^7 - 128*B*C*a^
5*b^5 - 768*B*C*a^7*b^3) - a^4*(B*b - C*a)*(12*C*b^5 - 32*C*a^5 + 128*B*a^3*b^2 + 32*C*a^2*b^3 + 64*B*a*b^4 +
32*B*a^4*b - 96*C*a^4*b)*1i)*(B*b - C*a)*1i + a^4*(tan(c/2 + (d*x)/2)*(32*C^2*a^10 + (9*C^2*b^10)/2 + 128*B^2*
a^2*b^8 + 512*B^2*a^4*b^6 + 512*B^2*a^6*b^4 + 32*B^2*a^8*b^2 + 24*C^2*a^2*b^8 - 40*C^2*a^4*b^6 - 192*C^2*a^6*b
^4 + 288*C^2*a^8*b^2 + 48*B*C*a*b^9 - 64*B*C*a^9*b + 224*B*C*a^3*b^7 - 128*B*C*a^5*b^5 - 768*B*C*a^7*b^3) + a^
4*(B*b - C*a)*(12*C*b^5 - 32*C*a^5 + 128*B*a^3*...

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