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

Optimal. Leaf size=123 \[ -\frac{b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a+b}}\right )}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \tan \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d x^2\right )\right )}+\frac{x^2}{2 a^2} \]

[Out]

x^2/(2*a^2) - (b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^2)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a +
b)^(3/2)*d) + (b^2*Tan[c + d*x^2])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x^2]))

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Rubi [A]  time = 0.254355, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4204, 3785, 3919, 3831, 2659, 208} \[ -\frac{b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a+b}}\right )}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \tan \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a+b \sec \left (c+d x^2\right )\right )}+\frac{x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sec[c + d*x^2])^2,x]

[Out]

x^2/(2*a^2) - (b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^2)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a +
b)^(3/2)*d) + (b^2*Tan[c + d*x^2])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x^2]))

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]

Rule 3785

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

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.600105, size = 153, normalized size = 1.24 \[ \frac{\frac{b \left (\left (a^2-b^2\right ) \left (c+d x^2\right )+a b \sin \left (c+d x^2\right )\right )+a \left (a^2-b^2\right ) \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{a \cos \left (c+d x^2\right )+b}-\frac{2 b \left (b^2-2 a^2\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}}{2 a^2 d (a-b) (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.06, size = 214, normalized size = 1.7 \begin{align*}{\frac{1}{d{a}^{2}}\arctan \left ( \tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{{b}^{2}}{da \left ({a}^{2}-{b}^{2} \right ) }\tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a- \left ( \tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}b-a-b \right ) ^{-1}}-2\,{\frac{b}{d \left ( a+b \right ) \left ( a-b \right ) \sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,d{x}^{2}+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+{\frac{{b}^{3}}{d{a}^{2} \left ( a+b \right ) \left ( a-b \right ) }{\it Artanh} \left ({(a-b)\tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a^2*arctan(tan(1/2*d*x^2+1/2*c))-1/d*b^2/a/(a^2-b^2)*tan(1/2*d*x^2+1/2*c)/(tan(1/2*d*x^2+1/2*c)^2*a-tan(1/
2*d*x^2+1/2*c)^2*b-a-b)-2/d*b/(a+b)/(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*b^3/a^2/(a+b)/(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))

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Maxima [B]  time = 175.746, size = 11976, normalized size = 97.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^6 - 2*a^4*b^2 + a^2*b^4)*d*x^2*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - 2*a^2*b^4 + b^6)*d*x^2*cos(d*x^2 +
c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*d*x^2*sin(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - 2*a^2*b^4 + b^6)*d*x^2*sin(d*x^2
+ c)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(d*x^2 + c) + (a^6 - 2*a^4*b^2 + a^2*b^4)*d*x^2 + (2*a^4*b - a
^2*b^3 + (2*a^4*b - a^2*b^3)*cos(2*d*x^2 + 2*c)^2 + 4*(2*a^2*b^3 - b^5)*cos(d*x^2 + c)^2 + (2*a^4*b - a^2*b^3)
*sin(2*d*x^2 + 2*c)^2 + 4*(2*a^3*b^2 - a*b^4)*sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + 4*(2*a^2*b^3 - b^5)*sin(d*x^
2 + c)^2 + 2*(2*a^4*b - a^2*b^3 + 2*(2*a^3*b^2 - a*b^4)*cos(d*x^2 + c))*cos(2*d*x^2 + 2*c) + 4*(2*a^3*b^2 - a*
b^4)*cos(d*x^2 + c))*sqrt(-a^2 + b^2)*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)*cos(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*sin(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*cos(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)*sin(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)*co
s(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)*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)*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)*c
os(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)), (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)*sin(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)*cos(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)*c
os(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)*co
s(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*cos(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*cos(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*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*c
os(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*si
n(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)*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)*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)*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))) + 2*(2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(d*x^2 + c) + (a^6 - 2*a^4*b^2 + a^2*b^4)*d*x^2 - (a^3*
b^3 - a*b^5)*sin(d*x^2 + c))*cos(2*d*x^2 + 2*c) + 2*(a^4*b^2 - a^2*b^4 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*s
in(d*x^2 + c) + (a^3*b^3 - a*b^5)*cos(d*x^2 + c))*sin(2*d*x^2 + 2*c) + 2*(a^3*b^3 - a*b^5)*sin(d*x^2 + c))/((a
^8 - 2*a^6*b^2 + a^4*b^4)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*d*cos(d*x^2 + c)^2 + (a^8
- 2*a^6*b^2 + a^4*b^4)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*sin(2*d*x^2 + 2*c)*sin(d*x^
2 + c) + 4*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*d*sin(d*x^2 + c)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*cos(d*x^2 +
c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d + 2*(2*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*cos(d*x^2 + c) + (a^8 - 2*a^6*b^2 +
a^4*b^4)*d)*cos(2*d*x^2 + 2*c))

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Fricas [B]  time = 1.98987, size = 1112, normalized size = 9.04 \begin{align*} \left [\frac{2 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cos \left (d x^{2} + c\right ) + 2 \,{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} +{\left (2 \, a^{2} b^{2} - b^{4} +{\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x^{2} + c\right )\right )} \sqrt{a^{2} - b^{2}} \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 ) + 2 \,{\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d x^{2} + c\right )}{4 \,{\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d x^{2} + c\right ) +{\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}, \frac{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cos \left (d x^{2} + c\right ) +{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} -{\left (2 \, a^{2} b^{2} - b^{4} +{\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x^{2} + c\right )\right )} \sqrt{-a^{2} + b^{2}} \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 ) +{\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d x^{2} + c\right )}{2 \,{\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d x^{2} + c\right ) +{\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.24385, size = 263, normalized size = 2.14 \begin{align*} -\frac{b^{2} \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )}{{\left (a^{3} d - a b^{2} d\right )}{\left (a \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )^{2} - a - b\right )}} + \frac{{\left (2 \, a^{2} b - b^{3}\right )}{\left (\pi \left \lfloor \frac{d x^{2} + 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^{2} + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt{-a^{2} + b^{2}}} + \frac{d x^{2} + c}{2 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2*tan(1/2*d*x^2 + 1/2*c)/((a^3*d - a*b^2*d)*(a*tan(1/2*d*x^2 + 1/2*c)^2 - b*tan(1/2*d*x^2 + 1/2*c)^2 - a -
b)) + (2*a^2*b - b^3)*(pi*floor(1/2*(d*x^2 + c)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*d*x^2 + 1/2*c) -
b*tan(1/2*d*x^2 + 1/2*c))/sqrt(-a^2 + b^2)))/((a^4*d - a^2*b^2*d)*sqrt(-a^2 + b^2)) + 1/2*(d*x^2 + c)/(a^2*d)