3.9.9 \(\int \frac {\sec ^2(c+d x) (B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [809]

Optimal. Leaf size=220 \[ \frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-6 a b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^3 (a+b)^{5/2} d}-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.55, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4157, 4113, 4165, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {a \left (-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {\left (-2 a^5 C+5 a^3 b^2 C+a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(C*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 6*a*b^4*C)*ArcTanh[(Sqrt[a
- b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(b*B - a*C)*Tan[c + d*x])/(2*
b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Tan[c + d*x])/(2*b^2*
(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3855

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

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4083

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]

Rule 4113

Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_
)), x_Symbol] :> Simp[(-a^2)*(A*b - a*B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2)
)), x] + Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[a*b*(A*b - a*B)*
(m + 1) - (A*b - a*B)*(a^2 + b^2*(m + 1))*Csc[e + f*x] + b*B*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 4157

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

Rule 4165

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

Rubi steps

\begin {align*} \int \frac {\sec ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\int \frac {\sec ^3(c+d x) (B+C \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\\ &=-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\sec (c+d x) \left (-2 a b (b B-a C)-\left (a^2-2 b^2\right ) (b B-a C) \sec (c+d x)-2 b \left (a^2-b^2\right ) C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (b^2 \left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right )+2 b \left (a^2-b^2\right )^2 C \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {C \int \sec (c+d x) \, dx}{b^3}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-6 a b^4 C\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-6 a b^4 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-6 a b^4 C\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 \left (a^2-b^2\right )^2 d}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac {\left (a^2 b^3 B+2 b^5 B-2 a^5 C+5 a^3 b^2 C-6 a b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^3 (a+b)^{5/2} d}-\frac {a^2 (b B-a C) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 1.82, size = 270, normalized size = 1.23 \begin {gather*} \frac {\cos (c+d x) (B+C \sec (c+d x)) \left (\frac {2 \left (-a^2 b^3 B-2 b^5 B+2 a^5 C-5 a^3 b^2 C+6 a b^4 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-2 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a b^2 (-b B+a C) \sin (c+d x)}{(-a+b) (a+b) (b+a \cos (c+d x))^2}+\frac {a b \left (-3 b^3 B-2 a^3 C+5 a b^2 C\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}\right )}{2 b^3 d (C+B \cos (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x]*(B + C*Sec[c + d*x])*((2*(-(a^2*b^3*B) - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 6*a*b^4*C)*ArcTanh[((
-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] +
2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^2*(-(b*B) + a*C)*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*C
os[c + d*x])^2) + (a*b*(-3*b^3*B - 2*a^3*C + 5*a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x]
))))/(2*b^3*d*(C + B*Cos[c + d*x]))

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Maple [A]
time = 0.58, size = 305, normalized size = 1.39 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (210) = 420\).
time = 29.10, size = 1419, normalized size = 6.45 \begin {gather*} \left [-\frac {{\left (2 \, C a^{5} b^{2} - 5 \, C a^{3} b^{4} - B a^{2} b^{5} + 6 \, C a b^{6} - 2 \, B b^{7} + {\left (2 \, C a^{7} - 5 \, C a^{5} b^{2} - B a^{4} b^{3} + 6 \, C a^{3} b^{4} - 2 \, B a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{6} b - 5 \, C a^{4} b^{3} - B a^{3} b^{4} + 6 \, C a^{2} b^{5} - 2 \, B a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} - 9 \, C a^{4} b^{4} + 5 \, B a^{3} b^{5} + 6 \, C a^{2} b^{6} - 4 \, B a b^{7} + {\left (2 \, C a^{7} b - 7 \, C a^{5} b^{3} + 3 \, B a^{4} b^{4} + 5 \, C a^{3} b^{5} - 3 \, B a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b^{3} - 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} - a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} d\right )}}, -\frac {{\left (2 \, C a^{5} b^{2} - 5 \, C a^{3} b^{4} - B a^{2} b^{5} + 6 \, C a b^{6} - 2 \, B b^{7} + {\left (2 \, C a^{7} - 5 \, C a^{5} b^{2} - B a^{4} b^{3} + 6 \, C a^{3} b^{4} - 2 \, B a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{6} b - 5 \, C a^{4} b^{3} - B a^{3} b^{4} + 6 \, C a^{2} b^{5} - 2 \, B a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{6} b^{2} - 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} - C b^{8} + {\left (C a^{8} - 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} - C a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b - 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} - C a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} - 9 \, C a^{4} b^{4} + 5 \, B a^{3} b^{5} + 6 \, C a^{2} b^{6} - 4 \, B a b^{7} + {\left (2 \, C a^{7} b - 7 \, C a^{5} b^{3} + 3 \, B a^{4} b^{4} + 5 \, C a^{3} b^{5} - 3 \, B a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} b^{3} - 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} - a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((2*C*a^5*b^2 - 5*C*a^3*b^4 - B*a^2*b^5 + 6*C*a*b^6 - 2*B*b^7 + (2*C*a^7 - 5*C*a^5*b^2 - B*a^4*b^3 + 6*C
*a^3*b^4 - 2*B*a^2*b^5)*cos(d*x + c)^2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 - B*a^3*b^4 + 6*C*a^2*b^5 - 2*B*a*b^6)*cos
(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d
*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*(C*a^6*b^2 - 3*C
*a^4*b^4 + 3*C*a^2*b^6 - C*b^8 + (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*cos(d*x + c)^2 + 2*(C*a^7*b -
3*C*a^5*b^3 + 3*C*a^3*b^5 - C*a*b^7)*cos(d*x + c))*log(sin(d*x + c) + 1) + 2*(C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a
^2*b^6 - C*b^8 + (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*cos(d*x + c)^2 + 2*(C*a^7*b - 3*C*a^5*b^3 + 3
*C*a^3*b^5 - C*a*b^7)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(3*C*a^6*b^2 - B*a^5*b^3 - 9*C*a^4*b^4 + 5*B*a^
3*b^5 + 6*C*a^2*b^6 - 4*B*a*b^7 + (2*C*a^7*b - 7*C*a^5*b^3 + 3*B*a^4*b^4 + 5*C*a^3*b^5 - 3*B*a^2*b^6)*cos(d*x
+ c))*sin(d*x + c))/((a^8*b^3 - 3*a^6*b^5 + 3*a^4*b^7 - a^2*b^9)*d*cos(d*x + c)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3
*a^3*b^8 - a*b^10)*d*cos(d*x + c) + (a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*d), -1/2*((2*C*a^5*b^2 - 5*C*a^3*
b^4 - B*a^2*b^5 + 6*C*a*b^6 - 2*B*b^7 + (2*C*a^7 - 5*C*a^5*b^2 - B*a^4*b^3 + 6*C*a^3*b^4 - 2*B*a^2*b^5)*cos(d*
x + c)^2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 - B*a^3*b^4 + 6*C*a^2*b^5 - 2*B*a*b^6)*cos(d*x + c))*sqrt(-a^2 + b^2)*ar
ctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b
^6 - C*b^8 + (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*cos(d*x + c)^2 + 2*(C*a^7*b - 3*C*a^5*b^3 + 3*C*a
^3*b^5 - C*a*b^7)*cos(d*x + c))*log(sin(d*x + c) + 1) + (C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8 + (C*a^
8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*cos(d*x + c)^2 + 2*(C*a^7*b - 3*C*a^5*b^3 + 3*C*a^3*b^5 - C*a*b^7)*
cos(d*x + c))*log(-sin(d*x + c) + 1) + (3*C*a^6*b^2 - B*a^5*b^3 - 9*C*a^4*b^4 + 5*B*a^3*b^5 + 6*C*a^2*b^6 - 4*
B*a*b^7 + (2*C*a^7*b - 7*C*a^5*b^3 + 3*B*a^4*b^4 + 5*C*a^3*b^5 - 3*B*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^
8*b^3 - 3*a^6*b^5 + 3*a^4*b^7 - a^2*b^9)*d*cos(d*x + c)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*d*cos
(d*x + c) + (a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (210) = 420\).
time = 0.56, size = 486, normalized size = 2.21 \begin {gather*} -\frac {\frac {{\left (2 \, C a^{5} - 5 \, C a^{3} b^{2} - B a^{2} b^{3} + 6 \, C a b^{4} - 2 \, B b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{3}} + \frac {C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{3}} - \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((2*C*a^5 - 5*C*a^3*b^2 - B*a^2*b^3 + 6*C*a*b^4 - 2*B*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b)
+ arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^4*b^3 - 2*a^2*b^5 + b^7)*sq
rt(-a^2 + b^2)) - C*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^3 + C*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^3 - (2*C*a
^5*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^4*b*tan(1/2*d*x + 1/2*c)^3 + B*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*C*a^3*b^2*
tan(1/2*d*x + 1/2*c)^3 + 3*B*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 4*B*a*b^4*t
an(1/2*d*x + 1/2*c)^3 - 2*C*a^5*tan(1/2*d*x + 1/2*c) - 3*C*a^4*b*tan(1/2*d*x + 1/2*c) + B*a^3*b^2*tan(1/2*d*x
+ 1/2*c) + 5*C*a^3*b^2*tan(1/2*d*x + 1/2*c) - 3*B*a^2*b^3*tan(1/2*d*x + 1/2*c) + 6*C*a^2*b^3*tan(1/2*d*x + 1/2
*c) - 4*B*a*b^4*tan(1/2*d*x + 1/2*c))/((a^4*b^2 - 2*a^2*b^4 + b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x +
1/2*c)^2 - a - b)^2))/d

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Mupad [B]
time = 14.00, size = 2500, normalized size = 11.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^3*(2*C*a^4 + B*a^2*b^2 - 6*C*a^2*b^2 + 4*B*a*b^3 - C*a^3*b))/((a*b^2 - b^3)*(a + b)^2) -
(tan(c/2 + (d*x)/2)*(2*C*a^4 - B*a^2*b^2 - 6*C*a^2*b^2 + 4*B*a*b^3 + C*a^3*b))/((a + b)*(b^4 - 2*a*b^3 + a^2*b
^2)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b + b^2) + a^2 + b^2)
) - (C*atan(-((C*((C*((8*(4*B*b^15 + 4*C*b^15 - 6*B*a^2*b^13 + 6*B*a^3*b^12 + 2*B*a^6*b^9 - 2*B*a^7*b^8 - 8*C*
a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b
^6 - 4*B*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a
^7*b^6) - (8*C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^
10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/(b^3*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^
7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))))/b^3 - (8*tan(c/2 + (d*x)/2)*(4*B^2*b^10 + 8*C^2*a^10 + 4*C^2*b^10 - 8*C^
2*a*b^9 - 8*C^2*a^9*b + 4*B^2*a^2*b^8 + B^2*a^4*b^6 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^
2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 - 24*B*C*a*b^9 + 8*B*C*a^3*b^7 + 2*B*C*a^5*b^5 -
4*B*C*a^7*b^3))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))*1i)/b^3 -
(C*((C*((8*(4*B*b^15 + 4*C*b^15 - 6*B*a^2*b^13 + 6*B*a^3*b^12 + 2*B*a^6*b^9 - 2*B*a^7*b^8 - 8*C*a^2*b^13 + 34
*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 4*B*a*b^
14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (8*
C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b
^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/(b^3*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6
- a^6*b^5 - a^7*b^4))))/b^3 + (8*tan(c/2 + (d*x)/2)*(4*B^2*b^10 + 8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C
^2*a^9*b + 4*B^2*a^2*b^8 + B^2*a^4*b^6 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 5
7*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 - 24*B*C*a*b^9 + 8*B*C*a^3*b^7 + 2*B*C*a^5*b^5 - 4*B*C*a^7*b^3
))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))*1i)/b^3)/((16*(4*C^3*a
^9 - 4*B*C^2*b^9 + 4*B^2*C*b^9 + 12*C^3*a*b^8 - 2*C^3*a^8*b + 24*C^3*a^2*b^7 - 34*C^3*a^3*b^6 - 26*C^3*a^4*b^5
+ 36*C^3*a^5*b^4 + 13*C^3*a^6*b^3 - 18*C^3*a^7*b^2 - 20*B*C^2*a*b^8 + 6*B*C^2*a^2*b^7 + 2*B*C^2*a^3*b^6 + 2*B
*C^2*a^5*b^4 - 2*B*C^2*a^6*b^3 - 2*B*C^2*a^7*b^2 + 4*B^2*C*a^2*b^7 + B^2*C*a^4*b^5))/(a*b^12 + b^13 - 3*a^2*b^
11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (C*((C*((8*(4*B*b^15 + 4*C*b^15 - 6*B*a^2*b^13
+ 6*B*a^3*b^12 + 2*B*a^6*b^9 - 2*B*a^7*b^8 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C
*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 4*B*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 -
3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (8*C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32
*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/(b^3*
(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))))/b^3 - (8*tan(c/2 + (d*x
)/2)*(4*B^2*b^10 + 8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 4*B^2*a^2*b^8 + B^2*a^4*b^6 + 24*C^2*
a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2
- 24*B*C*a*b^9 + 8*B*C*a^3*b^7 + 2*B*C*a^5*b^5 - 4*B*C*a^7*b^3))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^
4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))/b^3 + (C*((C*((8*(4*B*b^15 + 4*C*b^15 - 6*B*a^2*b^13 + 6*B*a^3*b^12 +
2*B*a^6*b^9 - 2*B*a^7*b^8 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*
a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 4*B*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*
a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (8*C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a
^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/(b^3*(a*b^10 + b^11 -
3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))))/b^3 + (8*tan(c/2 + (d*x)/2)*(4*B^2*b^10
+ 8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 4*B^2*a^2*b^8 + B^2*a^4*b^6 + 24*C^2*a^2*b^8 + 32*C^2
*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 - 24*B*C*a*b^9 +
8*B*C*a^3*b^7 + 2*B*C*a^5*b^5 - 4*B*C*a^7*b^3))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^
6 - a^6*b^5 - a^7*b^4)))/b^3))*2i)/(b^3*d) - (atan(((((8*tan(c/2 + (d*x)/2)*(4*B^2*b^10 + 8*C^2*a^10 + 4*C^2*b
^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 4*B^2*a^2*b^8 + B^2*a^4*b^6 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b
^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 - 24*B*C*a*b^9 + 8*B*C*a^3*b^7 + 2*B*C*
a^5*b^5 - 4*B*C*a^7*b^3))/(a*b^10 + b^11 - 3*a^...

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