3.6 \(\int \frac{x}{a+b \cos ^2(x)} \, dx\)

Optimal. Leaf size=203 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{PolyLog}\left (2,-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x \log \left (1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}} \]

[Out]

((-I/2)*x*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) + ((I/2)*x*Log[1 +
(b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) - PolyLog[2, -((b*E^((2*I)*x))/(2*a
+ b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b]) + PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sq
rt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])

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Rubi [A]  time = 0.333843, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4586, 3321, 2264, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{PolyLog}\left (2,-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x \log \left (1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Cos[x]^2),x]

[Out]

((-I/2)*x*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) + ((I/2)*x*Log[1 +
(b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) - PolyLog[2, -((b*E^((2*I)*x))/(2*a
+ b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b]) + PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sq
rt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])

Rule 4586

Int[(Cos[(c_.) + (d_.)*(x_)]^2*(b_.) + (a_))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/2^n, Int[x^m*(2*a + b + b*Co
s[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1
] || (EqQ[m, 1] && EqQ[n, -2]))

Rule 3321

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c
+ d*x)^m*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(
2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x}{a+b \cos ^2(x)} \, dx &=2 \int \frac{x}{2 a+b+b \cos (2 x)} \, dx\\ &=4 \int \frac{e^{2 i x} x}{b+2 (2 a+b) e^{2 i x}+b e^{4 i x}} \, dx\\ &=\frac{(2 b) \int \frac{e^{2 i x} x}{-4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)+2 b e^{2 i x}} \, dx}{\sqrt{a} \sqrt{a+b}}-\frac{(2 b) \int \frac{e^{2 i x} x}{4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)+2 b e^{2 i x}} \, dx}{\sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i \int \log \left (1+\frac{2 b e^{2 i x}}{-4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \int \log \left (1+\frac{2 b e^{2 i x}}{4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{-4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt{a} \sqrt{a+b}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left (1+\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{\text{Li}_2\left (-\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{Li}_2\left (-\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{4 \sqrt{a} \sqrt{a+b}}\\ \end{align*}

Mathematica [B]  time = 0.547437, size = 532, normalized size = 2.62 \[ \frac{i \left (\text{PolyLog}\left (2,\frac{\left (-2 i \sqrt{-a (a+b)}+2 a+b\right ) \left (-\sqrt{-a (a+b)} \tan (x)+a+b\right )}{b \left (\sqrt{-a (a+b)} \tan (x)+a+b\right )}\right )-\text{PolyLog}\left (2,\frac{\left (2 i \sqrt{-a (a+b)}+2 a+b\right ) \left (-\sqrt{-a (a+b)} \tan (x)+a+b\right )}{b \left (\sqrt{-a (a+b)} \tan (x)+a+b\right )}\right )\right )+4 x \tanh ^{-1}\left (\frac{(a+b) \cot (x)}{\sqrt{-a (a+b)}}\right )+2 \cos ^{-1}\left (-\frac{2 a}{b}-1\right ) \tanh ^{-1}\left (\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right )-\log \left (\frac{2 (a+b) \left (\sqrt{-a (a+b)}-i a\right ) (\tan (x)-i)}{b \left (\sqrt{-a (a+b)} \tan (x)+a+b\right )}\right ) \left (\cos ^{-1}\left (-\frac{2 a}{b}-1\right )+2 i \tanh ^{-1}\left (\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right )\right )-\log \left (\frac{2 (a+b) \left (\sqrt{-a (a+b)}+i a\right ) (\tan (x)+i)}{b \left (\sqrt{-a (a+b)} \tan (x)+a+b\right )}\right ) \left (\cos ^{-1}\left (-\frac{2 a}{b}-1\right )-2 i \tanh ^{-1}\left (\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right )\right )+\log \left (\frac{\sqrt{2} e^{-i x} \sqrt{-a (a+b)}}{\sqrt{b} \sqrt{2 a+b \cos (2 x)+b}}\right ) \left (\cos ^{-1}\left (-\frac{2 a}{b}-1\right )-2 i \left (\tanh ^{-1}\left (\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right )+\tanh ^{-1}\left (\frac{(a+b) \cot (x)}{\sqrt{-a (a+b)}}\right )\right )\right )+\log \left (\frac{\sqrt{2} e^{i x} \sqrt{-a (a+b)}}{\sqrt{b} \sqrt{2 a+b \cos (2 x)+b}}\right ) \left (\cos ^{-1}\left (-\frac{2 a}{b}-1\right )+2 i \left (\tanh ^{-1}\left (\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right )+\tanh ^{-1}\left (\frac{(a+b) \cot (x)}{\sqrt{-a (a+b)}}\right )\right )\right )}{4 \sqrt{-a (a+b)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Cos[x]^2),x]

[Out]

(4*x*ArcTanh[((a + b)*Cot[x])/Sqrt[-(a*(a + b))]] + 2*ArcCos[-1 - (2*a)/b]*ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b)
)]] + (ArcCos[-1 - (2*a)/b] - (2*I)*(ArcTanh[((a + b)*Cot[x])/Sqrt[-(a*(a + b))]] + ArcTanh[(a*Tan[x])/Sqrt[-(
a*(a + b))]]))*Log[(Sqrt[2]*Sqrt[-(a*(a + b))])/(Sqrt[b]*E^(I*x)*Sqrt[2*a + b + b*Cos[2*x]])] + (ArcCos[-1 - (
2*a)/b] + (2*I)*(ArcTanh[((a + b)*Cot[x])/Sqrt[-(a*(a + b))]] + ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]]))*Log[(
Sqrt[2]*Sqrt[-(a*(a + b))]*E^(I*x))/(Sqrt[b]*Sqrt[2*a + b + b*Cos[2*x]])] - (ArcCos[-1 - (2*a)/b] + (2*I)*ArcT
anh[(a*Tan[x])/Sqrt[-(a*(a + b))]])*Log[(2*(a + b)*((-I)*a + Sqrt[-(a*(a + b))])*(-I + Tan[x]))/(b*(a + b + Sq
rt[-(a*(a + b))]*Tan[x]))] - (ArcCos[-1 - (2*a)/b] - (2*I)*ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]])*Log[(2*(a +
b)*(I*a + Sqrt[-(a*(a + b))])*(I + Tan[x]))/(b*(a + b + Sqrt[-(a*(a + b))]*Tan[x]))] + I*(PolyLog[2, ((2*a +
b - (2*I)*Sqrt[-(a*(a + b))])*(a + b - Sqrt[-(a*(a + b))]*Tan[x]))/(b*(a + b + Sqrt[-(a*(a + b))]*Tan[x]))] -
PolyLog[2, ((2*a + b + (2*I)*Sqrt[-(a*(a + b))])*(a + b - Sqrt[-(a*(a + b))]*Tan[x]))/(b*(a + b + Sqrt[-(a*(a
+ b))]*Tan[x]))]))/(4*Sqrt[-(a*(a + b))])

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Maple [B]  time = 0.094, size = 501, normalized size = 2.5 \begin{align*}{-ix\ln \left ( 1-{b{{\rm e}^{2\,ix}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ) \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{iax\ln \left ( 1-{b{{\rm e}^{2\,ix}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{{\frac{i}{2}}bx\ln \left ( 1-{b{{\rm e}^{2\,ix}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{{x}^{2} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{a{x}^{2}{\frac{1}{\sqrt{ \left ( a+b \right ) a}}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{\frac{b{x}^{2}}{2}{\frac{1}{\sqrt{ \left ( a+b \right ) a}}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{\frac{1}{2}{\it polylog} \left ( 2,{b{{\rm e}^{2\,ix}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ) \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{\frac{a}{2}{\it polylog} \left ( 2,{b{{\rm e}^{2\,ix}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{\frac{b}{4}{\it polylog} \left ( 2,{b{{\rm e}^{2\,ix}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}} \left ( -2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}}-{{\frac{i}{2}}x\ln \left ( 1-{b{{\rm e}^{2\,ix}} \left ( 2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}}-{\frac{{x}^{2}}{2}{\frac{1}{\sqrt{ \left ( a+b \right ) a}}}}-{\frac{1}{4}{\it polylog} \left ( 2,{b{{\rm e}^{2\,ix}} \left ( 2\,\sqrt{ \left ( a+b \right ) a}-2\,a-b \right ) ^{-1}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I/(-2*((a+b)*a)^(1/2)-2*a-b)*ln(1-b*exp(2*I*x)/(-2*((a+b)*a)^(1/2)-2*a-b))*x-I/((a+b)*a)^(1/2)/(-2*((a+b)*a)^
(1/2)-2*a-b)*ln(1-b*exp(2*I*x)/(-2*((a+b)*a)^(1/2)-2*a-b))*a*x-1/2*I/((a+b)*a)^(1/2)/(-2*((a+b)*a)^(1/2)-2*a-b
)*ln(1-b*exp(2*I*x)/(-2*((a+b)*a)^(1/2)-2*a-b))*b*x-1/(-2*((a+b)*a)^(1/2)-2*a-b)*x^2-1/((a+b)*a)^(1/2)/(-2*((a
+b)*a)^(1/2)-2*a-b)*a*x^2-1/2/((a+b)*a)^(1/2)/(-2*((a+b)*a)^(1/2)-2*a-b)*b*x^2-1/2/(-2*((a+b)*a)^(1/2)-2*a-b)*
polylog(2,b*exp(2*I*x)/(-2*((a+b)*a)^(1/2)-2*a-b))-1/2/((a+b)*a)^(1/2)/(-2*((a+b)*a)^(1/2)-2*a-b)*polylog(2,b*
exp(2*I*x)/(-2*((a+b)*a)^(1/2)-2*a-b))*a-1/4/((a+b)*a)^(1/2)/(-2*((a+b)*a)^(1/2)-2*a-b)*polylog(2,b*exp(2*I*x)
/(-2*((a+b)*a)^(1/2)-2*a-b))*b-1/2*I/((a+b)*a)^(1/2)*x*ln(1-b*exp(2*I*x)/(2*((a+b)*a)^(1/2)-2*a-b))-1/2/((a+b)
*a)^(1/2)*x^2-1/4/((a+b)*a)^(1/2)*polylog(2,b*exp(2*I*x)/(2*((a+b)*a)^(1/2)-2*a-b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(b*cos(x)^2 + a), x)

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Fricas [B]  time = 0.925609, size = 4247, normalized size = 20.92 \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*cos(x)^2),x, algorithm="fricas")

[Out]

1/16*(4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*
sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x*sqrt((a^2 +
a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/
b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*
a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^
2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a
- 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)
/b) - 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos
(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x*sqr
t((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^
2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2
*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*s
qrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (
-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*
a - b)/b) - 2*b)/b) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(
b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 4
*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*s
qrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)
*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*
sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a +
b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 +
a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2
*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)
+ 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x
) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 4*b*sqrt(
(a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a
^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(
1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*
b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1))/(a^2 + a*b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*cos(x)**2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(b*cos(x)^2 + a), x)