3.65 \(\int \sec (a+b x) \sec (2 (a+b x)) \, dx\)

Optimal. Leaf size=35 \[ \frac {\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin (a+b x)\right )}{b}-\frac {\tanh ^{-1}(\sin (a+b x))}{b} \]

[Out]

-arctanh(sin(b*x+a))/b+arctanh(sin(b*x+a)*2^(1/2))*2^(1/2)/b

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Rubi [A]  time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4364, 1093, 207} \[ \frac {\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin (a+b x)\right )}{b}-\frac {\tanh ^{-1}(\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]*Sec[2*(a + b*x)],x]

[Out]

-(ArcTanh[Sin[a + b*x]]/b) + (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[a + b*x]])/b

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 4364

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist
[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d],
x] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (
EqQ[F, Cos] || EqQ[F, cos])

Rubi steps

\begin {align*} \int \sec (a+b x) \sec (2 (a+b x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-3 x^2+2 x^4} \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-2+2 x^2} \, dx,x,\sin (a+b x)\right )}{b}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\sin (a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\sin (a+b x))}{b}+\frac {\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin (a+b x)\right )}{b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 35, normalized size = 1.00 \[ \frac {\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin (a+b x)\right )}{b}-\frac {\tanh ^{-1}(\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]*Sec[2*(a + b*x)],x]

[Out]

-(ArcTanh[Sin[a + b*x]]/b) + (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[a + b*x]])/b

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fricas [B]  time = 0.98, size = 72, normalized size = 2.06 \[ \frac {\sqrt {2} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} - 2 \, \sqrt {2} \sin \left (b x + a\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} - 1}\right ) - \log \left (\sin \left (b x + a\right ) + 1\right ) + \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sec(2*b*x+2*a),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*log(-(2*cos(b*x + a)^2 - 2*sqrt(2)*sin(b*x + a) - 3)/(2*cos(b*x + a)^2 - 1)) - log(sin(b*x + a) +
1) + log(-sin(b*x + a) + 1))/b

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giac [B]  time = 1.19, size = 948, normalized size = 27.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sec(2*b*x+2*a),x, algorithm="giac")

[Out]

1/2*(sqrt(2)*log(abs(2*tan(1/2*b*x + 2*a)*tan(1/2*a)^6 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 - 2*tan(1/2*a)^6 -
30*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 12*tan(1/2*a)^5 - 40*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 30*tan(1/2*a)^4 +
30*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - 40*tan(1/2*a)^3 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a) - 30*tan(1/2*a)^2 - 2
*sqrt(2)*(tan(1/2*a)^6 + 3*tan(1/2*a)^4 + 3*tan(1/2*a)^2 + 1) - 2*tan(1/2*b*x + 2*a) + 12*tan(1/2*a) + 2)/abs(
2*tan(1/2*b*x + 2*a)*tan(1/2*a)^6 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 - 2*tan(1/2*a)^6 - 30*tan(1/2*b*x + 2*a
)*tan(1/2*a)^4 + 12*tan(1/2*a)^5 - 40*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 30*tan(1/2*a)^4 + 30*tan(1/2*b*x + 2*a
)*tan(1/2*a)^2 - 40*tan(1/2*a)^3 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a) - 30*tan(1/2*a)^2 + 2*sqrt(2)*(tan(1/2*a)^
6 + 3*tan(1/2*a)^4 + 3*tan(1/2*a)^2 + 1) - 2*tan(1/2*b*x + 2*a) + 12*tan(1/2*a) + 2)) + sqrt(2)*log(abs(2*tan(
1/2*b*x + 2*a)*tan(1/2*a)^6 - 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 + 2*tan(1/2*a)^6 - 30*tan(1/2*b*x + 2*a)*tan(
1/2*a)^4 + 12*tan(1/2*a)^5 + 40*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 - 30*tan(1/2*a)^4 + 30*tan(1/2*b*x + 2*a)*tan(
1/2*a)^2 - 40*tan(1/2*a)^3 - 12*tan(1/2*b*x + 2*a)*tan(1/2*a) + 30*tan(1/2*a)^2 - 2*sqrt(2)*(tan(1/2*a)^6 + 3*
tan(1/2*a)^4 + 3*tan(1/2*a)^2 + 1) - 2*tan(1/2*b*x + 2*a) + 12*tan(1/2*a) - 2)/abs(2*tan(1/2*b*x + 2*a)*tan(1/
2*a)^6 - 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 + 2*tan(1/2*a)^6 - 30*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 12*tan(1/2
*a)^5 + 40*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 - 30*tan(1/2*a)^4 + 30*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - 40*tan(1/2
*a)^3 - 12*tan(1/2*b*x + 2*a)*tan(1/2*a) + 30*tan(1/2*a)^2 + 2*sqrt(2)*(tan(1/2*a)^6 + 3*tan(1/2*a)^4 + 3*tan(
1/2*a)^2 + 1) - 2*tan(1/2*b*x + 2*a) + 12*tan(1/2*a) - 2)) - 2*log(abs(tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 3*tan
(1/2*b*x + 2*a)*tan(1/2*a)^2 - tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*tan(1/2*a)^2 - tan(1/2*b*x +
2*a) + 3*tan(1/2*a) - 1)) + 2*log(abs(tan(1/2*b*x + 2*a)*tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 + t
an(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*tan(1/2*a)^2 + tan(1/2*b*x + 2*a) - 3*tan(1/2*a) - 1)))/b

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maple [A]  time = 0.00, size = 48, normalized size = 1.37 \[ \frac {\ln \left (\sin \left (b x +a \right )-1\right )}{2 b}+\frac {\arctanh \left (\sin \left (b x +a \right ) \sqrt {2}\right ) \sqrt {2}}{b}-\frac {\ln \left (1+\sin \left (b x +a \right )\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)*sec(2*b*x+2*a),x)

[Out]

1/2/b*ln(sin(b*x+a)-1)+arctanh(sin(b*x+a)*2^(1/2))*2^(1/2)/b-1/2/b*ln(1+sin(b*x+a))

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maxima [B]  time = 117.00, size = 6257, normalized size = 178.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sec(2*b*x+2*a),x, algorithm="maxima")

[Out]

-1/8*(2*(sqrt(2)*cos(3*a)*cos(3/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(a)*cos(1/4*arctan2(sin(4*a), cos(
4*a))) + sqrt(2)*sin(3*a)*sin(3/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(a)*sin(1/4*arctan2(sin(4*a), cos(
4*a))))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(b*x) + sqrt(2)*cos(b*x)*sin(1/4*arctan2(sin(4
*a), cos(4*a))) + cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(2*b*x) + cos(2*b*x)*sin(1/2*arctan2(sin(4*a), cos(4
*a))), sqrt(2)*cos(b*x)*cos(1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(b*x)*sin(1/4*arctan2(sin(4*a), cos(
4*a))) + cos(2*b*x)*cos(1/2*arctan2(sin(4*a), cos(4*a))) - sin(2*b*x)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + 1
) - 2*(sqrt(2)*cos(3*a)*cos(3/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(a)*cos(1/4*arctan2(sin(4*a), cos(4*
a))) + sqrt(2)*sin(3*a)*sin(3/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(a)*sin(1/4*arctan2(sin(4*a), cos(4*
a))))*arctan2(-sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(b*x) - sqrt(2)*cos(b*x)*sin(1/4*arctan2(sin(4*
a), cos(4*a))) + cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(2*b*x) + cos(2*b*x)*sin(1/2*arctan2(sin(4*a), cos(4*
a))), -sqrt(2)*cos(b*x)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(b*x)*sin(1/4*arctan2(sin(4*a), cos(
4*a))) + cos(2*b*x)*cos(1/2*arctan2(sin(4*a), cos(4*a))) - sin(2*b*x)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + 1
) - 2*((sqrt(2)*cos(3*a)*cos(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(3*a)*sin(1/2*arctan2(sin(4*a), cos
(4*a))) + sqrt(2)*cos(a))*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) + (sqrt(2)*cos(1/2*arctan2(sin(4*a), c
os(4*a)))*sin(3*a) - sqrt(2)*cos(3*a)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(a))*sin(1/2*pi + 1/4*
arctan2(sin(4*a), cos(4*a))))*arctan2(-2*((sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/2*pi + 1/4*arcta
n2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(
4*a))))*cos(b*x) + cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - (sqrt(
2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*cos(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*pi + 1
/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*sin(b*x) - cos(1/2*pi + 1/4*arctan2(sin(
4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))/(2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*ar
ctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + 2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a
), cos(4*a)))^2)*sin(b*x)^2 + cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*pi + 1/4*ar
ctan2(sin(4*a), cos(4*a)))*cos(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))
*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(
sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(
b*x) + 2*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + cos(1/4*arctan2(
sin(4*a), cos(4*a)))^2 + sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*arctan2(sin(4*a)
, cos(4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*
a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4
*a), cos(4*a))) + sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x)
+ 2*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))) + sin(1/4*arctan2(sin(4
*a), cos(4*a)))^2), (2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(b
*x)^2 + 2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 - cos(1
/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(s
in(4*a), cos(4*a))) + sqrt(2)*sin(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b
*x) + cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 - sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 - 2*(sqrt(2)*co
s(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), co
s(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)/(2*(cos(1/2*
arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + 2*(cos(1/2*arctan2(sin(4
*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 + cos(1/2*pi + 1/4*arctan2(sin(4*a), co
s(4*a)))^2 + 2*(sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*cos(1/2*arctan2(sin(4*a), cos(4*a))) + s
qrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*pi + 1/4*ar
ctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*arctan2(sin(4*a), cos(4*a)))
*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + 2*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arct
an2(sin(4*a), cos(4*a))) + cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a
)))^2 + 2*(sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2
)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/4*arctan2
(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(
1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x) + 2*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(s
in(4*a), cos(4*a))) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)) - 2*((sqrt(2)*cos(3*a)*cos(1/2*arctan2(sin(4*a)
, cos(4*a))) + sqrt(2)*sin(3*a)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(a))*cos(1/2*pi + 1/4*arctan
2(sin(4*a), cos(4*a))) + (sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(3*a) - sqrt(2)*cos(3*a)*sin(1/2*arc
tan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(a))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))))*arctan2(-2*((sqrt(2
)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*pi + 1/
4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*cos(b*x) - cos(1/4*arctan2(sin(4*a), cos(
4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - (sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*
cos(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(s
in(4*a), cos(4*a))))*sin(b*x) + cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*
a))))/(2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + 2*(cos
(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 + cos(1/2*pi + 1/4*ar
ctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*cos(1/2*arctan2(sin(4*
a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*s
in(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(1/2*arctan2(si
n(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b*x) - 2*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4
*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*pi + 1/4*arctan2
(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), c
os(4*a))) - sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(
2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a
), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x) - 2*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))
*sin(1/4*arctan2(sin(4*a), cos(4*a))) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2), (2*(cos(1/2*arctan2(sin(4*a),
cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + 2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2
+ sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 - cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 - 2*(s
qrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*arctan2(sin
(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 - si
n(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan
2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*si
n(b*x) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)/(2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(
sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + 2*(cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos
(4*a)))^2)*sin(b*x)^2 + cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*pi + 1/4*arctan2(
sin(4*a), cos(4*a)))*cos(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1
/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*
a), cos(4*a))) - sqrt(2)*sin(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b*x) -
2*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + cos(1/4*arctan2(sin(4*
a), cos(4*a)))^2 + sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))^2 + 2*(sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(
4*a)))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*s
in(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(sin(4*a), c
os(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x) - 2*si
n(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))) + sin(1/4*arctan2(sin(4*a), c
os(4*a)))^2)) + (sqrt(2)*cos(3/4*arctan2(sin(4*a), cos(4*a)))*sin(3*a) - sqrt(2)*cos(1/4*arctan2(sin(4*a), cos
(4*a)))*sin(a) - sqrt(2)*cos(3*a)*sin(3/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(a)*sin(1/4*arctan2(sin(4*
a), cos(4*a))))*log((cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(2*b*
x)^2 + 2*(cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + (cos(1
/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(2*b*x)^2 + 2*(cos(1/4*arctan2(
sin(4*a), cos(4*a)))^2 + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 + 2*sqrt(2)*cos(b*x)*cos(1/4*arcta
n2(sin(4*a), cos(4*a))) - 2*sqrt(2)*sin(b*x)*sin(1/4*arctan2(sin(4*a), cos(4*a))) + 2*((sqrt(2)*cos(1/2*arctan
2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*arctan2(sin(4*a), cos(4*a)))*sin
(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1/2*arctan2(si
n(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*
x) + cos(1/2*arctan2(sin(4*a), cos(4*a))))*cos(2*b*x) - 2*((sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*sin(1
/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4
*a))))*cos(b*x) - (sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)
*sin(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x) + sin(1/2*arctan2(sin(4*a
), cos(4*a))))*sin(2*b*x) + 1) - (sqrt(2)*cos(3/4*arctan2(sin(4*a), cos(4*a)))*sin(3*a) - sqrt(2)*cos(1/4*arct
an2(sin(4*a), cos(4*a)))*sin(a) - sqrt(2)*cos(3*a)*sin(3/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(a)*sin(1
/4*arctan2(sin(4*a), cos(4*a))))*log((cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4
*a)))^2)*cos(2*b*x)^2 + 2*(cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)*co
s(b*x)^2 + (cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(2*b*x)^2 + 2*
(cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 - 2*sqrt(2)*cos(b
*x)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + 2*sqrt(2)*sin(b*x)*sin(1/4*arctan2(sin(4*a), cos(4*a))) - 2*((sqrt(
2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/2*arctan2(sin(4*a
), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a)))*s
in(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), c
os(4*a))))*sin(b*x) - cos(1/2*arctan2(sin(4*a), cos(4*a))))*cos(2*b*x) + 2*((sqrt(2)*cos(1/4*arctan2(sin(4*a),
cos(4*a)))*sin(1/2*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan
2(sin(4*a), cos(4*a))))*cos(b*x) - (sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*cos(1/4*arctan2(sin(4*a), cos
(4*a))) + sqrt(2)*sin(1/2*arctan2(sin(4*a), cos(4*a)))*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(b*x) - sin(1/
2*arctan2(sin(4*a), cos(4*a))))*sin(2*b*x) + 1) - ((sqrt(2)*cos(1/2*arctan2(sin(4*a), cos(4*a)))*sin(3*a) - sq
rt(2)*cos(3*a)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(a))*cos(1/2*pi + 1/4*arctan2(sin(4*a), cos(4
*a))) - (sqrt(2)*cos(3*a)*cos(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(3*a)*sin(1/2*arctan2(sin(4*a), co
s(4*a))) + sqrt(2)*cos(a))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))))*log(((cos(1/2*arctan2(sin(4*a), cos(
4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + (cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(
1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 + ((sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(
1/4*arctan2(sin(4*a), cos(4*a))))*cos(1/2*arctan2(sin(4*a), cos(4*a))) - (sqrt(2)*cos(1/4*arctan2(sin(4*a), co
s(4*a))) - sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + cos(
1/4*arctan2(sin(4*a), cos(4*a)))^2 - ((sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(1/4*arctan2(
sin(4*a), cos(4*a))))*cos(1/2*arctan2(sin(4*a), cos(4*a))) + (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + s
qrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*sin(b*x) + sin(1/4*arctan2(
sin(4*a), cos(4*a)))^2)/((cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos
(b*x)^2 + (cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 + ((sqr
t(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(1/2*arctan2(sin(
4*a), cos(4*a))) + (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))
))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 + ((sqrt(2)*cos(1/4
*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(1/2*arctan2(sin(4*a), cos(4*
a))) - (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(1/2*a
rctan2(sin(4*a), cos(4*a))))*sin(b*x) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)) - ((sqrt(2)*cos(1/2*arctan2(s
in(4*a), cos(4*a)))*sin(3*a) - sqrt(2)*cos(3*a)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(a))*cos(1/2
*pi + 1/4*arctan2(sin(4*a), cos(4*a))) - (sqrt(2)*cos(3*a)*cos(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(
3*a)*sin(1/2*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*cos(a))*sin(1/2*pi + 1/4*arctan2(sin(4*a), cos(4*a))))*log
(((cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + (cos(1/2*arct
an2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a), cos(4*a)))^2)*sin(b*x)^2 - ((sqrt(2)*cos(1/4*arctan2(si
n(4*a), cos(4*a))) - sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(1/2*arctan2(sin(4*a), cos(4*a))) + (sqr
t(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(1/2*arctan2(sin(
4*a), cos(4*a))))*cos(b*x) + cos(1/4*arctan2(sin(4*a), cos(4*a)))^2 - ((sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(
4*a))) + sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*cos(1/2*arctan2(sin(4*a), cos(4*a))) - (sqrt(2)*cos(1/4
*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))*sin(1/2*arctan2(sin(4*a), cos(4*
a))))*sin(b*x) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^2)/((cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*ar
ctan2(sin(4*a), cos(4*a)))^2)*cos(b*x)^2 + (cos(1/2*arctan2(sin(4*a), cos(4*a)))^2 + sin(1/2*arctan2(sin(4*a),
cos(4*a)))^2)*sin(b*x)^2 - ((sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/4*arctan2(sin(4*a),
cos(4*a))))*cos(1/2*arctan2(sin(4*a), cos(4*a))) - (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*si
n(1/4*arctan2(sin(4*a), cos(4*a))))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*cos(b*x) + cos(1/4*arctan2(sin(4*a),
cos(4*a)))^2 + ((sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) - sqrt(2)*sin(1/4*arctan2(sin(4*a), cos(4*a))))
*cos(1/2*arctan2(sin(4*a), cos(4*a))) + (sqrt(2)*cos(1/4*arctan2(sin(4*a), cos(4*a))) + sqrt(2)*sin(1/4*arctan
2(sin(4*a), cos(4*a))))*sin(1/2*arctan2(sin(4*a), cos(4*a))))*sin(b*x) + sin(1/4*arctan2(sin(4*a), cos(4*a)))^
2)) - 4*log((cos(b*x + 2*a)^2 + cos(a)^2 - 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 + 2*cos(b*x + 2*a)*sin(a
) + sin(a)^2)/(cos(b*x + 2*a)^2 + cos(a)^2 + 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*sin
(a) + sin(a)^2)))/b

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mupad [B]  time = 0.00, size = 29, normalized size = 0.83 \[ -\frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )-\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (a+b\,x\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(a + b*x)*cos(2*a + 2*b*x)),x)

[Out]

-(atanh(sin(a + b*x)) - 2^(1/2)*atanh(2^(1/2)*sin(a + b*x)))/b

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec {\left (a + b x \right )} \sec {\left (2 a + 2 b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sec(2*b*x+2*a),x)

[Out]

Integral(sec(a + b*x)*sec(2*a + 2*b*x), x)

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