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3.242 \(\int \cos (a+b x) \sec ^2(c+b x) \, dx\)

Optimal. Leaf size=35 \[ \frac {\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac {\sin (a-c) \sec (b x+c)}{b} \]

[Out]

arctanh(sin(b*x+c))*cos(a-c)/b-sec(b*x+c)*sin(a-c)/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 = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4583, 2606, 8, 3770} \[ \frac {\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac {\sin (a-c) \sec (b x+c)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Sec[c + b*x]^2,x]

[Out]

(ArcTanh[Sin[c + b*x]]*Cos[a - c])/b - (Sec[c + b*x]*Sin[a - c])/b

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3770

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

Rule 4583

Int[Cos[v_]*Sec[w_]^(n_.), x_Symbol] :> -Dist[Sin[v - w], Int[Tan[w]*Sec[w]^(n - 1), x], x] + Dist[Cos[v - w],
 Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]

Rubi steps

\begin {align*} \int \cos (a+b x) \sec ^2(c+b x) \, dx &=\cos (a-c) \int \sec (c+b x) \, dx-\sin (a-c) \int \sec (c+b x) \tan (c+b x) \, dx\\ &=\frac {\tanh ^{-1}(\sin (c+b x)) \cos (a-c)}{b}-\frac {\sin (a-c) \operatorname {Subst}(\int 1 \, dx,x,\sec (c+b x))}{b}\\ &=\frac {\tanh ^{-1}(\sin (c+b x)) \cos (a-c)}{b}-\frac {\sec (c+b x) \sin (a-c)}{b}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 89, normalized size = 2.54 \[ -\frac {\sin (a-c) \sec (b x+c)}{b}-\frac {2 i \cos (a-c) \tan ^{-1}\left (\frac {(\sin (c)+i \cos (c)) \left (\sin (c) \cos \left (\frac {b x}{2}\right )+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \sin (c) \cos \left (\frac {b x}{2}\right )}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]*Sec[c + b*x]^2,x]

[Out]

((-2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[
(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Sec[c + b*x]*Sin[a - c])/b

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fricas [A]  time = 0.46, size = 69, normalized size = 1.97 \[ \frac {\cos \left (b x + c\right ) \cos \left (-a + c\right ) \log \left (\sin \left (b x + c\right ) + 1\right ) - \cos \left (b x + c\right ) \cos \left (-a + c\right ) \log \left (-\sin \left (b x + c\right ) + 1\right ) + 2 \, \sin \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos(b*x + c)*cos(-a + c)*log(sin(b*x + c) + 1) - cos(b*x + c)*cos(-a + c)*log(-sin(b*x + c) + 1) + 2*sin(
-a + c))/(b*cos(b*x + c))

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giac [B]  time = 1.83, size = 1341, normalized size = 38.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)
 + 5*tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^3 - 5*tan(1/2*a)^2*tan(1/2*c) + 5*tan(1/
2*a)*tan(1/2*c)^2 - tan(1/2*c)^3 - tan(1/2*a)^2 + 5*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2 - tan(1/2*a) + tan(1/
2*c) + 1)*log(abs(-tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a)*tan(1/2*a) - tan(1/2*b*x
+ 1/2*a)*tan(1/2*c) + tan(1/2*a)*tan(1/2*c) - tan(1/2*b*x + 1/2*a) + tan(1/2*a) - tan(1/2*c) + 1))/(tan(1/2*a)
^3*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c) + tan(1/2*a)
^2*tan(1/2*c)^2 + tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^3 + tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 +
 tan(1/2*c)^3 + tan(1/2*a)^2 + tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 - tan(1/2*a) + tan(1/2*c) + 1) - (tan(1/2*
a)^3*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c) + 5*tan(1/
2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^3 + 5*tan(1/2*a)^2*tan(1/2*c) - 5*tan(1/2*a)*tan(1/
2*c)^2 + tan(1/2*c)^3 - tan(1/2*a)^2 + 5*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c) + 1)*l
og(abs(-tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c) - tan(1/2*b*x + 1/2*a)*tan(1/2*a) + tan(1/2*b*x + 1/2*a)*ta
n(1/2*c) - tan(1/2*a)*tan(1/2*c) - tan(1/2*b*x + 1/2*a) + tan(1/2*a) - tan(1/2*c) - 1))/(tan(1/2*a)^3*tan(1/2*
c)^3 + tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c) + tan(1/2*a)^2*tan(1/2*
c)^2 + tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^3 - tan(1/2*a)^2*tan(1/2*c) + tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*c)
^3 + tan(1/2*a)^2 + tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c) + 1) - 4*(2*tan(1/2*b*x + 1
/2*a)*tan(1/2*a)^4*tan(1/2*c)^2 - 4*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^3*tan(1/2*c)^3 + tan(1/2*a)^4*tan(1/2*c)^3
 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^4 - tan(1/2*a)^3*tan(1/2*c)^4 + 4*tan(1/2*b*x + 1/2*a)*tan(1
/2*a)^3*tan(1/2*c) - tan(1/2*a)^4*tan(1/2*c) - 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^2 + 6*tan(1/2*a)
^3*tan(1/2*c)^2 + 4*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^3 - 6*tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)*ta
n(1/2*c)^4 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2 - tan(1/2*a)^3 - 4*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)
 + 6*tan(1/2*a)^2*tan(1/2*c) + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c)^2 - 6*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*c)^3
+ tan(1/2*a) - tan(1/2*c))/((tan(1/2*b*x + 1/2*a)^2*tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*b*x + 1/2*a)^2*tan(1/2
*a)^2 + 4*tan(1/2*b*x + 1/2*a)^2*tan(1/2*a)*tan(1/2*c) - 4*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c) - tan(
1/2*b*x + 1/2*a)^2*tan(1/2*c)^2 + 4*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^2 +
 tan(1/2*b*x + 1/2*a)^2 - 4*tan(1/2*b*x + 1/2*a)*tan(1/2*a) + tan(1/2*a)^2 + 4*tan(1/2*b*x + 1/2*a)*tan(1/2*c)
 - 4*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 - 1)*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(1/
2*c) - tan(1/2*c)^2 + 1)))/b

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maple [B]  time = 2.25, size = 1049, normalized size = 29.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*sec(b*x+c)^2,x)

[Out]

2/b/(cos(a)*cos(c)*tan(1/2*b*x+1/2*a)^2+sin(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*sin(c)-
2*tan(1/2*b*x+1/2*a)*sin(a)*cos(c)-cos(a)*cos(c)-sin(a)*sin(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+cos(c)^2*
sin(a)^2+sin(a)^2*sin(c)^2)/(cos(a)*cos(c)+sin(a)*sin(c))*tan(1/2*b*x+1/2*a)*cos(a)^2*sin(c)^2-4/b/(cos(a)*cos
(c)*tan(1/2*b*x+1/2*a)^2+sin(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*sin(c)-2*tan(1/2*b*x+1
/2*a)*sin(a)*cos(c)-cos(a)*cos(c)-sin(a)*sin(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+cos(c)^2*sin(a)^2+sin(a)
^2*sin(c)^2)/(cos(a)*cos(c)+sin(a)*sin(c))*tan(1/2*b*x+1/2*a)*cos(a)*cos(c)*sin(a)*sin(c)+2/b/(cos(a)*cos(c)*t
an(1/2*b*x+1/2*a)^2+sin(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*sin(c)-2*tan(1/2*b*x+1/2*a)
*sin(a)*cos(c)-cos(a)*cos(c)-sin(a)*sin(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+cos(c)^2*sin(a)^2+sin(a)^2*si
n(c)^2)/(cos(a)*cos(c)+sin(a)*sin(c))*tan(1/2*b*x+1/2*a)*cos(c)^2*sin(a)^2-2/b/(cos(a)*cos(c)*tan(1/2*b*x+1/2*
a)^2+sin(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*sin(c)-2*tan(1/2*b*x+1/2*a)*sin(a)*cos(c)-
cos(a)*cos(c)-sin(a)*sin(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)*cos(a)*
sin(c)+2/b/(cos(a)*cos(c)*tan(1/2*b*x+1/2*a)^2+sin(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*
sin(c)-2*tan(1/2*b*x+1/2*a)*sin(a)*cos(c)-cos(a)*cos(c)-sin(a)*sin(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+co
s(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)*sin(a)*cos(c)-2/b/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+cos(c)^2*sin(a)^2+si
n(a)^2*sin(c)^2)/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2)*arctan(1/2*(
2*(cos(a)*cos(c)+sin(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*sin(c)-2*sin(a)*cos(c))/(-cos(a)^2*cos(c)^2-cos(a)
^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2))*cos(a)*cos(c)-2/b/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2
+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^
2)^(1/2)*arctan(1/2*(2*(cos(a)*cos(c)+sin(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*sin(c)-2*sin(a)*cos(c))/(-cos
(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2))*sin(a)*sin(c)

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maxima [B]  time = 0.52, size = 391, normalized size = 11.17 \[ -\frac {2 \, {\left (\sin \left (b x + 2 \, a\right ) - \sin \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) + 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \relax (a) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right ) \sin \relax (a) + {\left (\cos \relax (a)^{2} + \sin \relax (a)^{2}\right )} \cos \left (-a + c\right )\right )} \log \left (\frac {\cos \left (b x + 2 \, c\right )^{2} + \cos \relax (c)^{2} - 2 \, \cos \relax (c) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} + 2 \, \cos \left (b x + 2 \, c\right ) \sin \relax (c) + \sin \relax (c)^{2}}{\cos \left (b x + 2 \, c\right )^{2} + \cos \relax (c)^{2} + 2 \, \cos \relax (c) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} - 2 \, \cos \left (b x + 2 \, c\right ) \sin \relax (c) + \sin \relax (c)^{2}}\right ) - 2 \, {\left (\cos \left (b x + 2 \, a\right ) - \cos \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) + 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) - 2 \, \cos \relax (a) \sin \left (b x + 2 \, c\right ) - 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + 2 \, \cos \left (b x + 2 \, c\right ) \sin \relax (a)}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \relax (a) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \relax (a) + {\left (\cos \relax (a)^{2} + \sin \relax (a)^{2}\right )} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(sin(b*x + 2*a) - sin(b*x + 2*c))*cos(2*b*x + a + 2*c) + (cos(2*b*x + a + 2*c)^2*cos(-a + c) + 2*cos(2
*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(2*b*x + a + 2*c)^2 + 2*cos(-a + c)*sin(2*b*x + a + 2*c)*s
in(a) + (cos(a)^2 + sin(a)^2)*cos(-a + c))*log((cos(b*x + 2*c)^2 + cos(c)^2 - 2*cos(c)*sin(b*x + 2*c) + sin(b*
x + 2*c)^2 + 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)/(cos(b*x + 2*c)^2 + cos(c)^2 + 2*cos(c)*sin(b*x + 2*c) + sin(
b*x + 2*c)^2 - 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)) - 2*(cos(b*x + 2*a) - cos(b*x + 2*c))*sin(2*b*x + a + 2*c)
 + 2*cos(a)*sin(b*x + 2*a) - 2*cos(a)*sin(b*x + 2*c) - 2*cos(b*x + 2*a)*sin(a) + 2*cos(b*x + 2*c)*sin(a))/(b*c
os(2*b*x + a + 2*c)^2 + 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b*x + a + 2*c)^2 + 2*b*sin(2*b*x + a + 2*c)*
sin(a) + (cos(a)^2 + sin(a)^2)*b)

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mupad [B]  time = 6.52, size = 246, normalized size = 7.03 \[ \frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)/cos(c + b*x)^2,x)

[Out]

(log(- exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i) + 1) - (exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) + 1)*1i)
/(exp(a*2i)*exp(-c*2i))^(1/2))*(exp(a*2i - c*2i) + 1))/(2*b*exp(a*2i - c*2i)^(1/2)) - (log((exp(a*2i)*exp(-c*2
i)*(exp(a*2i)*exp(-c*2i) + 1)*1i)/(exp(a*2i)*exp(-c*2i))^(1/2) - exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i) +
 1))*(exp(a*2i - c*2i) + 1))/(2*b*exp(a*2i - c*2i)^(1/2)) + (exp(a*1i + b*x*1i)*(exp(a*2i - c*2i) - 1)*1i)/(b*
(exp(a*2i - c*2i) + exp(a*2i + b*x*2i)))

________________________________________________________________________________________

sympy [B]  time = 160.75, size = 5552, normalized size = 158.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/cos(c)**2, Eq(b, 0)), (-1/(b*sin(b*x)), Eq(c, -pi/2) | Eq(c, pi/2)), (-log(tan(b*x/2) - tan(c/2)/
(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**6*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*
b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c
/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/
2) - 1))*tan(c/2)**6/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4
*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan
(b*x/2)**2 - b) + 4*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**5*tan(b*x/2)/(b*tan
(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)
**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 3*log(tan
(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**4*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 -
 b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(
b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 3*log(tan(b*x/2) - tan(c/2)/(tan(
c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**4/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b
*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/
2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 8*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)
**3*tan(b*x/2)/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b
*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2
)**2 - b) - 3*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**2*tan(b*x/2)**2/(b*tan(c/
2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4
 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 3*log(tan(b*
x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 -
4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan
(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 4*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(ta
n(c/2) - 1))*tan(c/2)*tan(b*x/2)/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b
*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*
x/2) + b*tan(b*x/2)**2 - b) + log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(b*x/2)**2/(b*ta
n(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2
)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - log(tan(
b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/
2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 +
 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))
*tan(c/2)**6*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c
/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) +
 b*tan(b*x/2)**2 - b) - log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**6/(b*tan(c/2)**
6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b
*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 4*log(tan(b*x/2)
 + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**5*tan(b*x/2)/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2
)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2
+ b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 3*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1)
- 1/(tan(c/2) + 1))*tan(c/2)**4*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*t
an(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*ta
n(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 3*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(
c/2)**4/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**
2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 -
b) + 8*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**3*tan(b*x/2)/(b*tan(c/2)**6*tan(
b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c
/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 3*log(tan(b*x/2) + tan
(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**2*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**
6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b
*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 3*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1
/(tan(c/2) + 1))*tan(c/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan
(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2)
 + b*tan(b*x/2)**2 - b) - 4*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)*tan(b*x/2)/(
b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan
(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - log(
tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(b*x/2)**2/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/
2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2
 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) -
 1/(tan(c/2) + 1))/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*t
an(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b
*x/2)**2 - b) + 4*tan(c/2)**5/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*ta
n(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2
) + b*tan(b*x/2)**2 - b) + 8*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan(c/2)**6 - 4*b*tan(c/2
)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)**2 + b*tan(c/2)**2 +
4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) + 8*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**6*tan(b*x/2)**2 - b*tan
(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*tan(b*x/2)
**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b) - 4*tan(c/2)/(b*tan(c/2)**6*tan(b*x/2)**2
 - b*tan(c/2)**6 - 4*b*tan(c/2)**5*tan(b*x/2) - b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 - b*tan(c/2)**2*ta
n(b*x/2)**2 + b*tan(c/2)**2 + 4*b*tan(c/2)*tan(b*x/2) + b*tan(b*x/2)**2 - b), True))*cos(a) - Piecewise((log(t
an(b*x/2))/b, Eq(c, pi/2)), (0, Eq(b, 0)), (log(tan(b*x/2))/b, Eq(c, -pi/2)), (-2*log(tan(b*x/2) - tan(c/2)/(t
an(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*
tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2
) - 1) - 1/(tan(c/2) - 1))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/
2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 8*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2)
 - 1))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*
tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*ta
n(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*
tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) - 1) - 1/(tan(c/2) - 1))*tan(c/2)/(b
*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/
2)**2 + b) + 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c
/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2
+ b) - 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2)**2
 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 8*log(tan(b*x
/2) + tan(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(
c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + tan
(c/2)/(tan(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 -
 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + tan(c/2)/(ta
n(c/2) + 1) - 1/(tan(c/2) + 1))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*
x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*tan(c/2)**4/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2
)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 4*tan(c/2)**3*tan(b*x/2)/
(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*
x/2)**2 + b) - 4*tan(c/2)*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c/2)**3*tan(b*x/2)
 - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 - 4*b*tan(c
/2)**3*tan(b*x/2) - 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b), True))*sin(a)

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