3.20 \(\int \frac {x}{a+b \sec (c+d x^2)} \, dx\)

Optimal. Leaf size=66 \[ \frac {x^2}{2 a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}} \]

[Out]

1/2*x^2/a-b*arctanh((a-b)^(1/2)*tan(1/2*d*x^2+1/2*c)/(a+b)^(1/2))/a/d/(a-b)^(1/2)/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4204, 3783, 2659, 208} \[ \frac {x^2}{2 a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sec[c + d*x^2]),x]

[Out]

x^2/(2*a) - (b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^2)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)

Rule 208

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

Rule 2659

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 3783

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Dist[1/a, Int[1/(1 + (a*Sin[c + d
*x])/b), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 4204

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 67, normalized size = 1.02 \[ \frac {\frac {2 b \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{d \sqrt {a^2-b^2}}+\frac {c}{d}+x^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sec[c + d*x^2]),x]

[Out]

(c/d + x^2 + (2*b*ArcTanh[((-a + b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d))/(2*a)

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 251, normalized size = 3.80 \[ \left [\frac {2 \, {\left (a^{2} - b^{2}\right )} d x^{2} + \sqrt {a^{2} - b^{2}} b \log \left (\frac {2 \, a b \cos \left (d x^{2} + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{2} + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x^{2} + c\right ) + a\right )} \sin \left (d x^{2} + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x^{2} + c\right )^{2} + 2 \, a b \cos \left (d x^{2} + c\right ) + b^{2}}\right )}{4 \, {\left (a^{3} - a b^{2}\right )} d}, \frac {{\left (a^{2} - b^{2}\right )} d x^{2} - \sqrt {-a^{2} + b^{2}} b \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x^{2} + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x^{2} + c\right )}\right )}{2 \, {\left (a^{3} - a b^{2}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(a^2 - b^2)*d*x^2 + sqrt(a^2 - b^2)*b*log((2*a*b*cos(d*x^2 + c) - (a^2 - 2*b^2)*cos(d*x^2 + c)^2 - 2*s
qrt(a^2 - b^2)*(b*cos(d*x^2 + c) + a)*sin(d*x^2 + c) + 2*a^2 - b^2)/(a^2*cos(d*x^2 + c)^2 + 2*a*b*cos(d*x^2 +
c) + b^2)))/((a^3 - a*b^2)*d), 1/2*((a^2 - b^2)*d*x^2 - sqrt(-a^2 + b^2)*b*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x
^2 + c) + a)/((a^2 - b^2)*sin(d*x^2 + c))))/((a^3 - a*b^2)*d)]

________________________________________________________________________________________

giac [B]  time = 0.50, size = 278, normalized size = 4.21 \[ \frac {{\left (\sqrt {-a^{2} + b^{2}} {\left (a - 2 \, b\right )} d {\left | -a + b \right |} - \sqrt {-a^{2} + b^{2}} {\left | a \right |} {\left | -a + b \right |} {\left | d \right |}\right )} {\left (\pi \left \lfloor \frac {d x^{2} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )}{\sqrt {-\frac {b d + \sqrt {b^{2} d^{2} + {\left (a d + b d\right )} {\left (a d - b d\right )}}}{a d - b d}}}\right )\right )}}{2 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} a^{2} d^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} d {\left | a \right |} {\left | d \right |}\right )}} + \frac {{\left (a d - 2 \, b d + {\left | a \right |} {\left | d \right |}\right )} {\left (\pi \left \lfloor \frac {d x^{2} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )}{\sqrt {-\frac {b d - \sqrt {b^{2} d^{2} + {\left (a d + b d\right )} {\left (a d - b d\right )}}}{a d - b d}}}\right )\right )}}{2 \, {\left (a^{2} d^{2} - b d {\left | a \right |} {\left | d \right |}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(-a^2 + b^2)*(a - 2*b)*d*abs(-a + b) - sqrt(-a^2 + b^2)*abs(a)*abs(-a + b)*abs(d))*(pi*floor(1/2*(d*x
^2 + c)/pi + 1/2) + arctan(tan(1/2*d*x^2 + 1/2*c)/sqrt(-(b*d + sqrt(b^2*d^2 + (a*d + b*d)*(a*d - b*d)))/(a*d -
b*d))))/((a^2 - 2*a*b + b^2)*a^2*d^2 + (a^2*b - 2*a*b^2 + b^3)*d*abs(a)*abs(d)) + 1/2*(a*d - 2*b*d + abs(a)*a
bs(d))*(pi*floor(1/2*(d*x^2 + c)/pi + 1/2) + arctan(tan(1/2*d*x^2 + 1/2*c)/sqrt(-(b*d - sqrt(b^2*d^2 + (a*d +
b*d)*(a*d - b*d)))/(a*d - b*d))))/(a^2*d^2 - b*d*abs(a)*abs(d))

________________________________________________________________________________________

maple [A]  time = 0.54, size = 70, normalized size = 1.06 \[ -\frac {b \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d a \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\arctan \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{d a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/a*b/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d*x^2+1/2*c)/((a-b)*(a+b))^(1/2))+1/d/a*arctan(tan(1/2*d*x^
2+1/2*c))

________________________________________________________________________________________

maxima [B]  time = 63.90, size = 7945, normalized size = 120.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(-a^2 + b^2)*d*x^2 - b*arctan2(2*(4*(a^6 - a^4*b^2)*cos(d*x^2 + 2*c)^4*cos(c)*sin(c) - 4*(a^6 - a^4*b
^2)*cos(c)*sin(d*x^2 + 2*c)^4*sin(c) + 4*(3*(a^5*b - a^3*b^3)*cos(c)^2*sin(c) + (a^5*b - a^3*b^3)*sin(c)^3)*co
s(d*x^2 + 2*c)^3 - 4*((a^5*b - a^3*b^3)*cos(c)^3 + 3*(a^5*b - a^3*b^3)*cos(c)*sin(c)^2 + ((a^6 - a^4*b^2)*cos(
c)^2 - (a^6 - a^4*b^2)*sin(c)^2)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^3 - 4*((a^6 - 5*a^4*b^2 + 4*a^2*b^4)*cos(c
)^3*sin(c) + (a^6 - 5*a^4*b^2 + 4*a^2*b^4)*cos(c)*sin(c)^3)*cos(d*x^2 + 2*c)^2 + 4*((a^6 - 5*a^4*b^2 + 4*a^2*b
^4)*cos(c)^3*sin(c) + (a^6 - 5*a^4*b^2 + 4*a^2*b^4)*cos(c)*sin(c)^3 - 3*((a^5*b - a^3*b^3)*cos(c)^2*sin(c) - (
a^5*b - a^3*b^3)*sin(c)^3)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^2 - 4*((a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(c)^4*si
n(c) + 2*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(c)^2*sin(c)^3 + (a^5*b - 3*a^3*b^3 + 2*a*b^5)*sin(c)^5)*cos(d*x^2 +
2*c) + 4*((a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(c)^5 + 2*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(c)^3*sin(c)^2 + (a^5*b
- 3*a^3*b^3 + 2*a*b^5)*cos(c)*sin(c)^4 - ((a^6 - a^4*b^2)*cos(c)^2 - (a^6 - a^4*b^2)*sin(c)^2)*cos(d*x^2 + 2*
c)^3 - 3*((a^5*b - a^3*b^3)*cos(c)^3 - (a^5*b - a^3*b^3)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c)^2 + ((a^6 - 5*a^4*b
^2 + 4*a^2*b^4)*cos(c)^4 - (a^6 - 5*a^4*b^2 + 4*a^2*b^4)*sin(c)^4)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c) + (a^5*c
os(c)*sin(d*x^2 + 2*c)^5 - a^5*cos(d*x^2 + 2*c)^5*sin(c) - 4*a^4*b*cos(d*x^2 + 2*c)^4*cos(c)*sin(c) - (a^5*cos
(d*x^2 + 2*c)*sin(c) - 4*a^4*b*cos(c)*sin(c))*sin(d*x^2 + 2*c)^4 + 2*(3*(a^5 - 2*a^3*b^2)*cos(c)^2*sin(c) + (a
^5 - 2*a^3*b^2)*sin(c)^3)*cos(d*x^2 + 2*c)^3 + 2*(a^5*cos(d*x^2 + 2*c)^2*cos(c) - (a^5 - 2*a^3*b^2)*cos(c)^3 -
3*(a^5 - 2*a^3*b^2)*cos(c)*sin(c)^2 + 2*(a^4*b*cos(c)^2 - a^4*b*sin(c)^2)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^
3 + 4*((3*a^4*b - 4*a^2*b^3)*cos(c)^3*sin(c) + (3*a^4*b - 4*a^2*b^3)*cos(c)*sin(c)^3)*cos(d*x^2 + 2*c)^2 - 2*(
a^5*cos(d*x^2 + 2*c)^3*sin(c) + 2*(3*a^4*b - 4*a^2*b^3)*cos(c)^3*sin(c) + 2*(3*a^4*b - 4*a^2*b^3)*cos(c)*sin(c
)^3 + 3*((a^5 - 2*a^3*b^2)*cos(c)^2*sin(c) - (a^5 - 2*a^3*b^2)*sin(c)^3)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^2
- ((a^5 - 8*a^3*b^2 + 8*a*b^4)*cos(c)^4*sin(c) + 2*(a^5 - 8*a^3*b^2 + 8*a*b^4)*cos(c)^2*sin(c)^3 + (a^5 - 8*a^
3*b^2 + 8*a*b^4)*sin(c)^5)*cos(d*x^2 + 2*c) + (a^5*cos(d*x^2 + 2*c)^4*cos(c) + (a^5 - 8*a^3*b^2 + 8*a*b^4)*cos
(c)^5 + 2*(a^5 - 8*a^3*b^2 + 8*a*b^4)*cos(c)^3*sin(c)^2 + (a^5 - 8*a^3*b^2 + 8*a*b^4)*cos(c)*sin(c)^4 + 4*(a^4
*b*cos(c)^2 - a^4*b*sin(c)^2)*cos(d*x^2 + 2*c)^3 - 6*((a^5 - 2*a^3*b^2)*cos(c)^3 - (a^5 - 2*a^3*b^2)*cos(c)*si
n(c)^2)*cos(d*x^2 + 2*c)^2 - 4*((3*a^4*b - 4*a^2*b^3)*cos(c)^4 - (3*a^4*b - 4*a^2*b^3)*sin(c)^4)*cos(d*x^2 + 2
*c))*sin(d*x^2 + 2*c))*sqrt(-a^2 + b^2))/(a^6*cos(d*x^2 + 2*c)^6 + 6*a^5*b*cos(d*x^2 + 2*c)^5*cos(c) + a^6*sin
(d*x^2 + 2*c)^6 + 6*a^5*b*sin(d*x^2 + 2*c)^5*sin(c) - (a^6 - 18*a^4*b^2 + 48*a^2*b^4 - 32*b^6)*cos(c)^6 - 3*(a
^6 - 18*a^4*b^2 + 48*a^2*b^4 - 32*b^6)*cos(c)^4*sin(c)^2 - 3*(a^6 - 18*a^4*b^2 + 48*a^2*b^4 - 32*b^6)*cos(c)^2
*sin(c)^4 - (a^6 - 18*a^4*b^2 + 48*a^2*b^4 - 32*b^6)*sin(c)^6 - 3*(5*(a^6 - 2*a^4*b^2)*cos(c)^2 + (a^6 - 2*a^4
*b^2)*sin(c)^2)*cos(d*x^2 + 2*c)^4 + 3*(a^6*cos(d*x^2 + 2*c)^2 + 2*a^5*b*cos(d*x^2 + 2*c)*cos(c) - (a^6 - 2*a^
4*b^2)*cos(c)^2 - 5*(a^6 - 2*a^4*b^2)*sin(c)^2)*sin(d*x^2 + 2*c)^4 - 4*(5*(3*a^5*b - 4*a^3*b^3)*cos(c)^3 + 3*(
3*a^5*b - 4*a^3*b^3)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c)^3 + 4*(3*a^5*b*cos(d*x^2 + 2*c)^2*sin(c) - 6*(a^6 - 2*a
^4*b^2)*cos(d*x^2 + 2*c)*cos(c)*sin(c) - 3*(3*a^5*b - 4*a^3*b^3)*cos(c)^2*sin(c) - 5*(3*a^5*b - 4*a^3*b^3)*sin
(c)^3)*sin(d*x^2 + 2*c)^3 + 3*(5*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^4 + 6*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(
c)^2*sin(c)^2 + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*sin(c)^4)*cos(d*x^2 + 2*c)^2 + 3*(a^6*cos(d*x^2 + 2*c)^4 + 4*a^5
*b*cos(d*x^2 + 2*c)^3*cos(c) + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^4 + 6*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)
^2*sin(c)^2 + 5*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*sin(c)^4 - 6*((a^6 - 2*a^4*b^2)*cos(c)^2 + (a^6 - 2*a^4*b^2)*sin
(c)^2)*cos(d*x^2 + 2*c)^2 - 4*((3*a^5*b - 4*a^3*b^3)*cos(c)^3 + 3*(3*a^5*b - 4*a^3*b^3)*cos(c)*sin(c)^2)*cos(d
*x^2 + 2*c))*sin(d*x^2 + 2*c)^2 + 6*((5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)^5 + 2*(5*a^5*b - 20*a^3*b^3 + 16
*a*b^5)*cos(c)^3*sin(c)^2 + (5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)*sin(c)^4)*cos(d*x^2 + 2*c) + 6*(a^5*b*cos
(d*x^2 + 2*c)^4*sin(c) - 4*(a^6 - 2*a^4*b^2)*cos(d*x^2 + 2*c)^3*cos(c)*sin(c) + (5*a^5*b - 20*a^3*b^3 + 16*a*b
^5)*cos(c)^4*sin(c) + 2*(5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)^2*sin(c)^3 + (5*a^5*b - 20*a^3*b^3 + 16*a*b^5
)*sin(c)^5 - 2*(3*(3*a^5*b - 4*a^3*b^3)*cos(c)^2*sin(c) + (3*a^5*b - 4*a^3*b^3)*sin(c)^3)*cos(d*x^2 + 2*c)^2 +
4*((a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^3*sin(c) + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)*sin(c)^3)*cos(d*x^2 +
2*c))*sin(d*x^2 + 2*c) + 2*(3*a^5*cos(d*x^2 + 2*c)^5*cos(c) + 3*a^5*sin(d*x^2 + 2*c)^5*sin(c) + (3*a^4*b - 16
*a^2*b^3 + 16*b^5)*cos(c)^6 + 3*(3*a^4*b - 16*a^2*b^3 + 16*b^5)*cos(c)^4*sin(c)^2 + 3*(3*a^4*b - 16*a^2*b^3 +
16*b^5)*cos(c)^2*sin(c)^4 + (3*a^4*b - 16*a^2*b^3 + 16*b^5)*sin(c)^6 + 3*(5*a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*c
os(d*x^2 + 2*c)^4 + 3*(a^5*cos(d*x^2 + 2*c)*cos(c) + a^4*b*cos(c)^2 + 5*a^4*b*sin(c)^2)*sin(d*x^2 + 2*c)^4 - 2
*(5*(a^5 - 4*a^3*b^2)*cos(c)^3 + 3*(a^5 - 4*a^3*b^2)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c)^3 + 2*(3*a^5*cos(d*x^2
+ 2*c)^2*sin(c) + 12*a^4*b*cos(d*x^2 + 2*c)*cos(c)*sin(c) - 3*(a^5 - 4*a^3*b^2)*cos(c)^2*sin(c) - 5*(a^5 - 4*a
^3*b^2)*sin(c)^3)*sin(d*x^2 + 2*c)^3 - 6*(5*(a^4*b - 2*a^2*b^3)*cos(c)^4 + 6*(a^4*b - 2*a^2*b^3)*cos(c)^2*sin(
c)^2 + (a^4*b - 2*a^2*b^3)*sin(c)^4)*cos(d*x^2 + 2*c)^2 + 6*(a^5*cos(d*x^2 + 2*c)^3*cos(c) - (a^4*b - 2*a^2*b^
3)*cos(c)^4 - 6*(a^4*b - 2*a^2*b^3)*cos(c)^2*sin(c)^2 - 5*(a^4*b - 2*a^2*b^3)*sin(c)^4 + 3*(a^4*b*cos(c)^2 + a
^4*b*sin(c)^2)*cos(d*x^2 + 2*c)^2 - ((a^5 - 4*a^3*b^2)*cos(c)^3 + 3*(a^5 - 4*a^3*b^2)*cos(c)*sin(c)^2)*cos(d*x
^2 + 2*c))*sin(d*x^2 + 2*c)^2 + 3*((a^5 - 12*a^3*b^2 + 16*a*b^4)*cos(c)^5 + 2*(a^5 - 12*a^3*b^2 + 16*a*b^4)*co
s(c)^3*sin(c)^2 + (a^5 - 12*a^3*b^2 + 16*a*b^4)*cos(c)*sin(c)^4)*cos(d*x^2 + 2*c) + 3*(a^5*cos(d*x^2 + 2*c)^4*
sin(c) + 8*a^4*b*cos(d*x^2 + 2*c)^3*cos(c)*sin(c) + (a^5 - 12*a^3*b^2 + 16*a*b^4)*cos(c)^4*sin(c) + 2*(a^5 - 1
2*a^3*b^2 + 16*a*b^4)*cos(c)^2*sin(c)^3 + (a^5 - 12*a^3*b^2 + 16*a*b^4)*sin(c)^5 - 2*(3*(a^5 - 4*a^3*b^2)*cos(
c)^2*sin(c) + (a^5 - 4*a^3*b^2)*sin(c)^3)*cos(d*x^2 + 2*c)^2 - 16*((a^4*b - 2*a^2*b^3)*cos(c)^3*sin(c) + (a^4*
b - 2*a^2*b^3)*cos(c)*sin(c)^3)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c))*sqrt(-a^2 + b^2)), (a^6*cos(d*x^2 + 2*c)^6
+ 6*a^5*b*cos(d*x^2 + 2*c)^5*cos(c) + a^6*sin(d*x^2 + 2*c)^6 + 6*a^5*b*sin(d*x^2 + 2*c)^5*sin(c) + (a^6 - 8*a
^4*b^2 + 8*a^2*b^4)*cos(c)^6 + 3*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^4*sin(c)^2 + 3*(a^6 - 8*a^4*b^2 + 8*a^2*
b^4)*cos(c)^2*sin(c)^4 + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*sin(c)^6 - (5*(a^6 - 4*a^4*b^2)*cos(c)^2 + (a^6 - 4*a^4
*b^2)*sin(c)^2)*cos(d*x^2 + 2*c)^4 + (3*a^6*cos(d*x^2 + 2*c)^2 + 6*a^5*b*cos(d*x^2 + 2*c)*cos(c) - (a^6 - 4*a^
4*b^2)*cos(c)^2 - 5*(a^6 - 4*a^4*b^2)*sin(c)^2)*sin(d*x^2 + 2*c)^4 - 4*(5*(a^5*b - 2*a^3*b^3)*cos(c)^3 + 3*(a^
5*b - 2*a^3*b^3)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c)^3 + 4*(3*a^5*b*cos(d*x^2 + 2*c)^2*sin(c) - 2*(a^6 - 4*a^4*b
^2)*cos(d*x^2 + 2*c)*cos(c)*sin(c) - 3*(a^5*b - 2*a^3*b^3)*cos(c)^2*sin(c) - 5*(a^5*b - 2*a^3*b^3)*sin(c)^3)*s
in(d*x^2 + 2*c)^3 - (5*(a^6 + 4*a^4*b^2 - 8*a^2*b^4)*cos(c)^4 + 6*(a^6 + 4*a^4*b^2 - 8*a^2*b^4)*cos(c)^2*sin(c
)^2 + (a^6 + 4*a^4*b^2 - 8*a^2*b^4)*sin(c)^4)*cos(d*x^2 + 2*c)^2 + (3*a^6*cos(d*x^2 + 2*c)^4 + 12*a^5*b*cos(d*
x^2 + 2*c)^3*cos(c) - (a^6 + 4*a^4*b^2 - 8*a^2*b^4)*cos(c)^4 - 6*(a^6 + 4*a^4*b^2 - 8*a^2*b^4)*cos(c)^2*sin(c)
^2 - 5*(a^6 + 4*a^4*b^2 - 8*a^2*b^4)*sin(c)^4 - 6*((a^6 - 4*a^4*b^2)*cos(c)^2 + (a^6 - 4*a^4*b^2)*sin(c)^2)*co
s(d*x^2 + 2*c)^2 - 12*((a^5*b - 2*a^3*b^3)*cos(c)^3 + 3*(a^5*b - 2*a^3*b^3)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c))
*sin(d*x^2 + 2*c)^2 - 2*((5*a^5*b - 8*a*b^5)*cos(c)^5 + 2*(5*a^5*b - 8*a*b^5)*cos(c)^3*sin(c)^2 + (5*a^5*b - 8
*a*b^5)*cos(c)*sin(c)^4)*cos(d*x^2 + 2*c) + 2*(3*a^5*b*cos(d*x^2 + 2*c)^4*sin(c) - 4*(a^6 - 4*a^4*b^2)*cos(d*x
^2 + 2*c)^3*cos(c)*sin(c) - (5*a^5*b - 8*a*b^5)*cos(c)^4*sin(c) - 2*(5*a^5*b - 8*a*b^5)*cos(c)^2*sin(c)^3 - (5
*a^5*b - 8*a*b^5)*sin(c)^5 - 6*(3*(a^5*b - 2*a^3*b^3)*cos(c)^2*sin(c) + (a^5*b - 2*a^3*b^3)*sin(c)^3)*cos(d*x^
2 + 2*c)^2 - 4*((a^6 + 4*a^4*b^2 - 8*a^2*b^4)*cos(c)^3*sin(c) + (a^6 + 4*a^4*b^2 - 8*a^2*b^4)*cos(c)*sin(c)^3)
*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c) + 4*(a^5*cos(d*x^2 + 2*c)^5*cos(c) + a^5*sin(d*x^2 + 2*c)^5*sin(c) - (a^4*
b - 2*a^2*b^3)*cos(c)^6 - 3*(a^4*b - 2*a^2*b^3)*cos(c)^4*sin(c)^2 - 3*(a^4*b - 2*a^2*b^3)*cos(c)^2*sin(c)^4 -
(a^4*b - 2*a^2*b^3)*sin(c)^6 + (5*a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*cos(d*x^2 + 2*c)^4 + (a^5*cos(d*x^2 + 2*c)*
cos(c) + a^4*b*cos(c)^2 + 5*a^4*b*sin(c)^2)*sin(d*x^2 + 2*c)^4 + 2*(5*a^3*b^2*cos(c)^3 + 3*a^3*b^2*cos(c)*sin(
c)^2)*cos(d*x^2 + 2*c)^3 + 2*(a^5*cos(d*x^2 + 2*c)^2*sin(c) + 4*a^4*b*cos(d*x^2 + 2*c)*cos(c)*sin(c) + 3*a^3*b
^2*cos(c)^2*sin(c) + 5*a^3*b^2*sin(c)^3)*sin(d*x^2 + 2*c)^3 + 2*(5*a^2*b^3*cos(c)^4 + 6*a^2*b^3*cos(c)^2*sin(c
)^2 + a^2*b^3*sin(c)^4)*cos(d*x^2 + 2*c)^2 + 2*(a^5*cos(d*x^2 + 2*c)^3*cos(c) + a^2*b^3*cos(c)^4 + 6*a^2*b^3*c
os(c)^2*sin(c)^2 + 5*a^2*b^3*sin(c)^4 + 3*(a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*cos(d*x^2 + 2*c)^2 + 3*(a^3*b^2*co
s(c)^3 + 3*a^3*b^2*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^2 - ((a^5 - 2*a^3*b^2 - 4*a*b^4)*cos(c)
^5 + 2*(a^5 - 2*a^3*b^2 - 4*a*b^4)*cos(c)^3*sin(c)^2 + (a^5 - 2*a^3*b^2 - 4*a*b^4)*cos(c)*sin(c)^4)*cos(d*x^2
+ 2*c) + (a^5*cos(d*x^2 + 2*c)^4*sin(c) + 8*a^4*b*cos(d*x^2 + 2*c)^3*cos(c)*sin(c) - (a^5 - 2*a^3*b^2 - 4*a*b^
4)*cos(c)^4*sin(c) - 2*(a^5 - 2*a^3*b^2 - 4*a*b^4)*cos(c)^2*sin(c)^3 - (a^5 - 2*a^3*b^2 - 4*a*b^4)*sin(c)^5 +
6*(3*a^3*b^2*cos(c)^2*sin(c) + a^3*b^2*sin(c)^3)*cos(d*x^2 + 2*c)^2 + 16*(a^2*b^3*cos(c)^3*sin(c) + a^2*b^3*co
s(c)*sin(c)^3)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c))*sqrt(-a^2 + b^2))/(a^6*cos(d*x^2 + 2*c)^6 + 6*a^5*b*cos(d*x
^2 + 2*c)^5*cos(c) + a^6*sin(d*x^2 + 2*c)^6 + 6*a^5*b*sin(d*x^2 + 2*c)^5*sin(c) - (a^6 - 18*a^4*b^2 + 48*a^2*b
^4 - 32*b^6)*cos(c)^6 - 3*(a^6 - 18*a^4*b^2 + 48*a^2*b^4 - 32*b^6)*cos(c)^4*sin(c)^2 - 3*(a^6 - 18*a^4*b^2 + 4
8*a^2*b^4 - 32*b^6)*cos(c)^2*sin(c)^4 - (a^6 - 18*a^4*b^2 + 48*a^2*b^4 - 32*b^6)*sin(c)^6 - 3*(5*(a^6 - 2*a^4*
b^2)*cos(c)^2 + (a^6 - 2*a^4*b^2)*sin(c)^2)*cos(d*x^2 + 2*c)^4 + 3*(a^6*cos(d*x^2 + 2*c)^2 + 2*a^5*b*cos(d*x^2
+ 2*c)*cos(c) - (a^6 - 2*a^4*b^2)*cos(c)^2 - 5*(a^6 - 2*a^4*b^2)*sin(c)^2)*sin(d*x^2 + 2*c)^4 - 4*(5*(3*a^5*b
- 4*a^3*b^3)*cos(c)^3 + 3*(3*a^5*b - 4*a^3*b^3)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c)^3 + 4*(3*a^5*b*cos(d*x^2 +
2*c)^2*sin(c) - 6*(a^6 - 2*a^4*b^2)*cos(d*x^2 + 2*c)*cos(c)*sin(c) - 3*(3*a^5*b - 4*a^3*b^3)*cos(c)^2*sin(c) -
5*(3*a^5*b - 4*a^3*b^3)*sin(c)^3)*sin(d*x^2 + 2*c)^3 + 3*(5*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^4 + 6*(a^6 -
8*a^4*b^2 + 8*a^2*b^4)*cos(c)^2*sin(c)^2 + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*sin(c)^4)*cos(d*x^2 + 2*c)^2 + 3*(a^
6*cos(d*x^2 + 2*c)^4 + 4*a^5*b*cos(d*x^2 + 2*c)^3*cos(c) + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^4 + 6*(a^6 - 8
*a^4*b^2 + 8*a^2*b^4)*cos(c)^2*sin(c)^2 + 5*(a^6 - 8*a^4*b^2 + 8*a^2*b^4)*sin(c)^4 - 6*((a^6 - 2*a^4*b^2)*cos(
c)^2 + (a^6 - 2*a^4*b^2)*sin(c)^2)*cos(d*x^2 + 2*c)^2 - 4*((3*a^5*b - 4*a^3*b^3)*cos(c)^3 + 3*(3*a^5*b - 4*a^3
*b^3)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^2 + 6*((5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)^5 +
2*(5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)^3*sin(c)^2 + (5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)*sin(c)^4)*cos
(d*x^2 + 2*c) + 6*(a^5*b*cos(d*x^2 + 2*c)^4*sin(c) - 4*(a^6 - 2*a^4*b^2)*cos(d*x^2 + 2*c)^3*cos(c)*sin(c) + (5
*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)^4*sin(c) + 2*(5*a^5*b - 20*a^3*b^3 + 16*a*b^5)*cos(c)^2*sin(c)^3 + (5*a
^5*b - 20*a^3*b^3 + 16*a*b^5)*sin(c)^5 - 2*(3*(3*a^5*b - 4*a^3*b^3)*cos(c)^2*sin(c) + (3*a^5*b - 4*a^3*b^3)*si
n(c)^3)*cos(d*x^2 + 2*c)^2 + 4*((a^6 - 8*a^4*b^2 + 8*a^2*b^4)*cos(c)^3*sin(c) + (a^6 - 8*a^4*b^2 + 8*a^2*b^4)*
cos(c)*sin(c)^3)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c) + 2*(3*a^5*cos(d*x^2 + 2*c)^5*cos(c) + 3*a^5*sin(d*x^2 + 2
*c)^5*sin(c) + (3*a^4*b - 16*a^2*b^3 + 16*b^5)*cos(c)^6 + 3*(3*a^4*b - 16*a^2*b^3 + 16*b^5)*cos(c)^4*sin(c)^2
+ 3*(3*a^4*b - 16*a^2*b^3 + 16*b^5)*cos(c)^2*sin(c)^4 + (3*a^4*b - 16*a^2*b^3 + 16*b^5)*sin(c)^6 + 3*(5*a^4*b*
cos(c)^2 + a^4*b*sin(c)^2)*cos(d*x^2 + 2*c)^4 + 3*(a^5*cos(d*x^2 + 2*c)*cos(c) + a^4*b*cos(c)^2 + 5*a^4*b*sin(
c)^2)*sin(d*x^2 + 2*c)^4 - 2*(5*(a^5 - 4*a^3*b^2)*cos(c)^3 + 3*(a^5 - 4*a^3*b^2)*cos(c)*sin(c)^2)*cos(d*x^2 +
2*c)^3 + 2*(3*a^5*cos(d*x^2 + 2*c)^2*sin(c) + 12*a^4*b*cos(d*x^2 + 2*c)*cos(c)*sin(c) - 3*(a^5 - 4*a^3*b^2)*co
s(c)^2*sin(c) - 5*(a^5 - 4*a^3*b^2)*sin(c)^3)*sin(d*x^2 + 2*c)^3 - 6*(5*(a^4*b - 2*a^2*b^3)*cos(c)^4 + 6*(a^4*
b - 2*a^2*b^3)*cos(c)^2*sin(c)^2 + (a^4*b - 2*a^2*b^3)*sin(c)^4)*cos(d*x^2 + 2*c)^2 + 6*(a^5*cos(d*x^2 + 2*c)^
3*cos(c) - (a^4*b - 2*a^2*b^3)*cos(c)^4 - 6*(a^4*b - 2*a^2*b^3)*cos(c)^2*sin(c)^2 - 5*(a^4*b - 2*a^2*b^3)*sin(
c)^4 + 3*(a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*cos(d*x^2 + 2*c)^2 - ((a^5 - 4*a^3*b^2)*cos(c)^3 + 3*(a^5 - 4*a^3*b
^2)*cos(c)*sin(c)^2)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c)^2 + 3*((a^5 - 12*a^3*b^2 + 16*a*b^4)*cos(c)^5 + 2*(a^5
- 12*a^3*b^2 + 16*a*b^4)*cos(c)^3*sin(c)^2 + (a^5 - 12*a^3*b^2 + 16*a*b^4)*cos(c)*sin(c)^4)*cos(d*x^2 + 2*c)
+ 3*(a^5*cos(d*x^2 + 2*c)^4*sin(c) + 8*a^4*b*cos(d*x^2 + 2*c)^3*cos(c)*sin(c) + (a^5 - 12*a^3*b^2 + 16*a*b^4)*
cos(c)^4*sin(c) + 2*(a^5 - 12*a^3*b^2 + 16*a*b^4)*cos(c)^2*sin(c)^3 + (a^5 - 12*a^3*b^2 + 16*a*b^4)*sin(c)^5 -
2*(3*(a^5 - 4*a^3*b^2)*cos(c)^2*sin(c) + (a^5 - 4*a^3*b^2)*sin(c)^3)*cos(d*x^2 + 2*c)^2 - 16*((a^4*b - 2*a^2*
b^3)*cos(c)^3*sin(c) + (a^4*b - 2*a^2*b^3)*cos(c)*sin(c)^3)*cos(d*x^2 + 2*c))*sin(d*x^2 + 2*c))*sqrt(-a^2 + b^
2))))/(sqrt(-a^2 + b^2)*a*d)

________________________________________________________________________________________

mupad [B]  time = 2.24, size = 157, normalized size = 2.38 \[ \frac {x^2}{2\,a}+\frac {b\,\ln \left (2\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}-\frac {b\,x\,\left (a+b\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{\sqrt {a+b}\,\sqrt {a-b}}\right )}{2\,a\,d\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {b\,\ln \left (2\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}+\frac {b\,x\,\left (a+b\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{\sqrt {a+b}\,\sqrt {a-b}}\right )}{2\,a\,d\,\sqrt {a+b}\,\sqrt {a-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/cos(c + d*x^2)),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________