3.7.84 \(\int \frac {\sec ^3(c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^2} \, dx\) [684]

Optimal. Leaf size=271 \[ \frac {\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac {2 a \left (a^2 A b^2-2 A b^4+3 a^4 C-4 a^2 b^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^4 (a+b)^{3/2} d}-\frac {a \left (A b^2+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (2 A b^2+3 a^2 C-b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2+a^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.61, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4184, 4177, 4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\left (3 a^2 C+2 A b^2-b^2 C\right ) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left (a^2-b^2\right )}+\frac {\left (C \left (6 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (3 a^2 C+A b^2-2 b^2 C\right ) \tan (c+d x)}{b^3 d \left (a^2-b^2\right )}-\frac {2 a \left (3 a^4 C+a^2 A b^2-4 a^2 b^2 C-2 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*A*b^2 + (6*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^
2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(A*b^2 + 3*
a^2*C - 2*b^2*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(
2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^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 4167

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

Rule 4177

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

Rule 4184

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

Rubi steps

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

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Mathematica [A]
time = 4.05, size = 461, normalized size = 1.70 \begin {gather*} \frac {(b+a \cos (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {8 a \left (-2 A b^4+a^2 b^2 (A-4 C)+3 a^4 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))}{\left (a^2-b^2\right )^{3/2}}-2 \left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^2 C (b+a \cos (c+d x))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {8 a b C (b+a \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {b^2 C (b+a \cos (c+d x))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {8 a b C (b+a \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {4 a^2 b \left (A b^2+a^2 C\right ) \sin (c+d x)}{(-a+b) (a+b)}\right )}{2 b^4 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((8*a*(-2*A*b^4 + a^2*b^2*(A - 4*C) + 3*a^4*C)*ArcTanh[((-a + b)*
Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) - 2*(2*A*b^2 + (6*a^2 + b^2)*C)*(b
+ a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + (6*a^2 + b^2)*C)*(b + a*Cos[c + d*x]
)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*C*(b + a*Cos[c + d*x]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]
)^2 - (8*a*b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*C*(b + a*Co
s[c + d*x]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (8*a*b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c
+ d*x)/2] + Sin[(c + d*x)/2]) + (4*a^2*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((-a + b)*(a + b))))/(2*b^4*d*(A + 2*C
+ A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)

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Maple [A]
time = 0.54, size = 322, normalized size = 1.19 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^3*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,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. 539 vs. \(2 (258) = 516\).
time = 25.80, size = 1135, normalized size = 4.19 \begin {gather*} \left [\frac {2 \, {\left ({\left (3 \, C a^{6} + {\left (A - 4 \, C\right )} a^{4} b^{2} - 2 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, C a^{5} b + {\left (A - 4 \, C\right )} a^{3} b^{3} - 2 \, A a b^{5}\right )} \cos \left (d x + c\right )^{2}\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 ) + {\left ({\left (6 \, C a^{7} + {\left (2 \, A - 11 \, C\right )} a^{5} b^{2} - 4 \, {\left (A - C\right )} a^{3} b^{4} + {\left (2 \, A + C\right )} a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (6 \, C a^{6} b + {\left (2 \, A - 11 \, C\right )} a^{4} b^{3} - 4 \, {\left (A - C\right )} a^{2} b^{5} + {\left (2 \, A + C\right )} b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (6 \, C a^{7} + {\left (2 \, A - 11 \, C\right )} a^{5} b^{2} - 4 \, {\left (A - C\right )} a^{3} b^{4} + {\left (2 \, A + C\right )} a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (6 \, C a^{6} b + {\left (2 \, A - 11 \, C\right )} a^{4} b^{3} - 4 \, {\left (A - C\right )} a^{2} b^{5} + {\left (2 \, A + C\right )} b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{4} b^{3} - 2 \, C a^{2} b^{5} + C b^{7} - 2 \, {\left (3 \, C a^{6} b + {\left (A - 5 \, C\right )} a^{4} b^{3} - {\left (A - 2 \, C\right )} a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (C a^{5} b^{2} - 2 \, C a^{3} b^{4} + C a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{2}\right )}}, -\frac {4 \, {\left ({\left (3 \, C a^{6} + {\left (A - 4 \, C\right )} a^{4} b^{2} - 2 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, C a^{5} b + {\left (A - 4 \, C\right )} a^{3} b^{3} - 2 \, A a b^{5}\right )} \cos \left (d x + c\right )^{2}\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 ({\left (6 \, C a^{7} + {\left (2 \, A - 11 \, C\right )} a^{5} b^{2} - 4 \, {\left (A - C\right )} a^{3} b^{4} + {\left (2 \, A + C\right )} a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (6 \, C a^{6} b + {\left (2 \, A - 11 \, C\right )} a^{4} b^{3} - 4 \, {\left (A - C\right )} a^{2} b^{5} + {\left (2 \, A + C\right )} b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (6 \, C a^{7} + {\left (2 \, A - 11 \, C\right )} a^{5} b^{2} - 4 \, {\left (A - C\right )} a^{3} b^{4} + {\left (2 \, A + C\right )} a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (6 \, C a^{6} b + {\left (2 \, A - 11 \, C\right )} a^{4} b^{3} - 4 \, {\left (A - C\right )} a^{2} b^{5} + {\left (2 \, A + C\right )} b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (C a^{4} b^{3} - 2 \, C a^{2} b^{5} + C b^{7} - 2 \, {\left (3 \, C a^{6} b + {\left (A - 5 \, C\right )} a^{4} b^{3} - {\left (A - 2 \, C\right )} a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (C a^{5} b^{2} - 2 \, C a^{3} b^{4} + C a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.50, size = 358, normalized size = 1.32 \begin {gather*} -\frac {\frac {4 \, {\left (3 \, C a^{5} + A a^{3} b^{2} - 4 \, C a^{3} b^{2} - 2 \, A a b^{4}\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^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {4 \, {\left (C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\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 )}} - \frac {{\left (6 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} + \frac {{\left (6 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {2 \, {\left (4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b \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 )}^{2} b^{3}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*(3*C*a^5 + A*a^3*b^2 - 4*C*a^3*b^2 - 2*A*a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + ar
ctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^2*b^4 - b^6)*sqrt(-a^2 + b^2))
- 4*(C*a^4*tan(1/2*d*x + 1/2*c) + A*a^2*b^2*tan(1/2*d*x + 1/2*c))/((a^2*b^3 - b^5)*(a*tan(1/2*d*x + 1/2*c)^2 -
b*tan(1/2*d*x + 1/2*c)^2 - a - b)) - (6*C*a^2 + 2*A*b^2 + C*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 + (6*
C*a^2 + 2*A*b^2 + C*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^4 - 2*(4*C*a*tan(1/2*d*x + 1/2*c)^3 + C*b*tan(1/
2*d*x + 1/2*c)^3 - 4*C*a*tan(1/2*d*x + 1/2*c) + C*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*b^3)
)/d

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Mupad [B]
time = 13.26, size = 2500, normalized size = 9.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((tan(c/2 + (d*x)/2)^5*(6*C*a^4 + C*b^4 + 2*A*a^2*b^2 - 5*C*a^2*b^2 + 3*C*a*b^3 - 3*C*a^3*b))/((a*b^3 - b^4)
*(a + b)) + (tan(c/2 + (d*x)/2)*(6*C*a^4 + C*b^4 + 2*A*a^2*b^2 - 5*C*a^2*b^2 - 3*C*a*b^3 + 3*C*a^3*b))/(b^3*(a
+ b)*(a - b)) - (2*tan(c/2 + (d*x)/2)^3*(6*C*a^4 - C*b^4 + 2*A*a^2*b^2 - 3*C*a^2*b^2))/(b*(a*b^2 - b^3)*(a +
b)))/(d*(a + b - tan(c/2 + (d*x)/2)^2*(3*a + b) - tan(c/2 + (d*x)/2)^6*(a - b) + tan(c/2 + (d*x)/2)^4*(3*a - b
))) - (atan(-(((((3*C*a^2 + b^2*(A + C/2))*((8*(4*A*b^15 + 2*C*b^15 - 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b
^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))
/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*tan(c/2 + (d*x)/2)*(3*C*a^2 + b^2*(A + C/2))*(8*a*b^13 - 8*a^2*b^12
- 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6))))/b^4 - (8*tan(
c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^2*a^2*b^
8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2
*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*C*a*b^9 + 20
*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 + 48*A*C*a^
8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6))*(3*C*a^2 + b^2*(A + C/2))*1i)/b^4 - ((((3*C*a^2 + b^2*(A + C/2))*((
8*(4*A*b^15 + 2*C*b^15 - 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4
*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (8*ta
n(c/2 + (d*x)/2)*(3*C*a^2 + b^2*(A + C/2))*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*
a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6))))/b^4 + (8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^
2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a
^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 +
120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^
6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 + 48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6))*(3
*C*a^2 + b^2*(A + C/2))*1i)/b^4)/((16*(108*C^3*a^11 + 8*A^3*a*b^10 - 54*C^3*a^10*b + 8*A^3*a^2*b^9 - 12*A^3*a^
3*b^8 - 4*A^3*a^4*b^7 + 4*A^3*a^5*b^6 + 4*C^3*a^3*b^8 - 4*C^3*a^4*b^7 + 41*C^3*a^5*b^6 - 9*C^3*a^6*b^5 + 63*C^
3*a^7*b^4 + 81*C^3*a^8*b^3 - 216*C^3*a^9*b^2 + 2*A*C^2*a*b^10 + 8*A^2*C*a*b^10 - 2*A*C^2*a^2*b^9 + 37*A*C^2*a^
3*b^8 - 5*A*C^2*a^4*b^7 + 105*A*C^2*a^5*b^6 + 111*A*C^2*a^6*b^5 - 252*A*C^2*a^7*b^4 - 72*A*C^2*a^8*b^3 + 108*A
*C^2*a^9*b^2 + 52*A^2*C*a^3*b^8 + 52*A^2*C*a^4*b^7 - 96*A^2*C*a^5*b^6 - 30*A^2*C*a^6*b^5 + 36*A^2*C*a^7*b^4))/
(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + ((((3*C*a^2 + b^2*(A + C/2))*((8*(4*A*b^15 + 2*C*b^15 - 4*A*a^2*b^13 +
12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12
*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*tan(c/2 + (d*x)/2)*(3*C*a^2 + b^2*(A + C/2
))*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 -
a^3*b^6))))/b^4 - (8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*
C^2*a^9*b + 12*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8
- 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*
C*b^10 - 8*A*C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 4
8*A*C*a^7*b^3 + 48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6))*(3*C*a^2 + b^2*(A + C/2)))/b^4 + ((((3*C*a
^2 + b^2*(A + C/2))*((8*(4*A*b^15 + 2*C*b^15 - 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16
*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b
^10 - a^3*b^9) + (8*tan(c/2 + (d*x)/2)*(3*C*a^2 + b^2*(A + C/2))*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4
*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6))))/b^4 + (8*tan(c/2 + (d*x)/2)*(4*A^2*b
^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20
*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b
^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a
^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 + 48*A*C*a^8*b^2))/(a*b^8 + b^9 -
a^2*b^7 - a^3*b^6))*(3*C*a^2 + b^2*(A + C/2)))/b^4))*(3*C*a^2 + b^2*(A + C/2))*2i)/(b^4*d) - (a*atan(((a*((8*t
an(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^2*a^2
*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*...

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