\(\int \frac {x}{(a+b \cos ^2(c+d x))^2} \, dx\) [9]

Optimal result

Integrand size = 16, antiderivative size = 327 \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=-\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}+\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}-\frac {\log (2 a+b+b \cos (2 c+2 d x))}{4 a (a+b) d^2}-\frac {(2 a+b) \operatorname {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{3/2} d^2}+\frac {(2 a+b) \operatorname {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{3/2} d^2}-\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))} \]

[Out]

-1/4*ln(2*a+b+b*cos(2*d*x+2*c))/a/(a+b)/d^2-1/4*I*(2*a+b)*x*ln(1+b*exp(2*I*(d*x+c))/(2*a+b-2*a^(1/2)*(a+b)^(1/
2)))/a^(3/2)/(a+b)^(3/2)/d+1/4*I*(2*a+b)*x*ln(1+b*exp(2*I*(d*x+c))/(2*a+b+2*a^(1/2)*(a+b)^(1/2)))/a^(3/2)/(a+b
)^(3/2)/d-1/8*(2*a+b)*polylog(2,-b*exp(2*I*(d*x+c))/(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/a^(3/2)/(a+b)^(3/2)/d^2+1/8
*(2*a+b)*polylog(2,-b*exp(2*I*(d*x+c))/(2*a+b+2*a^(1/2)*(a+b)^(1/2)))/a^(3/2)/(a+b)^(3/2)/d^2-1/2*b*x*sin(2*d*
x+2*c)/a/(a+b)/d/(2*a+b+b*cos(2*d*x+2*c))

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4682, 3405, 3402, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=-\frac {(2 a+b) \operatorname {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 a^{3/2} d^2 (a+b)^{3/2}}+\frac {(2 a+b) \operatorname {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 a^{3/2} d^2 (a+b)^{3/2}}-\frac {i x (2 a+b) \log \left (1+\frac {b e^{2 i (c+d x)}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 a^{3/2} d (a+b)^{3/2}}+\frac {i x (2 a+b) \log \left (1+\frac {b e^{2 i (c+d x)}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 a^{3/2} d (a+b)^{3/2}}-\frac {\log (2 a+b \cos (2 c+2 d x)+b)}{4 a d^2 (a+b)}-\frac {b x \sin (2 c+2 d x)}{2 a d (a+b) (2 a+b \cos (2 c+2 d x)+b)} \]

[In]

Int[x/(a + b*Cos[c + d*x]^2)^2,x]

[Out]

((-1/4*I)*(2*a + b)*x*Log[1 + (b*E^((2*I)*(c + d*x)))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(a^(3/2)*(a + b)^(3/
2)*d) + ((I/4)*(2*a + b)*x*Log[1 + (b*E^((2*I)*(c + d*x)))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(a^(3/2)*(a + b
)^(3/2)*d) - Log[2*a + b + b*Cos[2*c + 2*d*x]]/(4*a*(a + b)*d^2) - ((2*a + b)*PolyLog[2, -((b*E^((2*I)*(c + d*
x)))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(8*a^(3/2)*(a + b)^(3/2)*d^2) + ((2*a + b)*PolyLog[2, -((b*E^((2*I)*
(c + d*x)))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(8*a^(3/2)*(a + b)^(3/2)*d^2) - (b*x*Sin[2*c + 2*d*x])/(2*a*(
a + b)*d*(2*a + b + b*Cos[2*c + 2*d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3402

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 3405

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

Rule 4682

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]))

Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {x}{(2 a+b+b \cos (2 c+2 d x))^2} \, dx \\ & = -\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))}+\frac {(2 a+b) \int \frac {x}{2 a+b+b \cos (2 c+2 d x)} \, dx}{a (a+b)}+\frac {b \int \frac {\sin (2 c+2 d x)}{2 a+b+b \cos (2 c+2 d x)} \, dx}{2 a (a+b) d} \\ & = -\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))}+\frac {(2 (2 a+b)) \int \frac {e^{i (2 c+2 d x)} x}{b+2 (2 a+b) e^{i (2 c+2 d x)}+b e^{2 i (2 c+2 d x)}} \, dx}{a (a+b)}-\frac {\text {Subst}\left (\int \frac {1}{2 a+b+x} \, dx,x,b \cos (2 c+2 d x)\right )}{4 a (a+b) d^2} \\ & = -\frac {\log (2 a+b+b \cos (2 c+2 d x))}{4 a (a+b) d^2}-\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))}+\frac {(b (2 a+b)) \int \frac {e^{i (2 c+2 d x)} x}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{i (2 c+2 d x)}} \, dx}{a^{3/2} (a+b)^{3/2}}-\frac {(b (2 a+b)) \int \frac {e^{i (2 c+2 d x)} x}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)+2 b e^{i (2 c+2 d x)}} \, dx}{a^{3/2} (a+b)^{3/2}} \\ & = -\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}+\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}-\frac {\log (2 a+b+b \cos (2 c+2 d x))}{4 a (a+b) d^2}-\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))}+\frac {(i (2 a+b)) \int \log \left (1+\frac {2 b e^{i (2 c+2 d x)}}{-4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{4 a^{3/2} (a+b)^{3/2} d}-\frac {(i (2 a+b)) \int \log \left (1+\frac {2 b e^{i (2 c+2 d x)}}{4 \sqrt {a} \sqrt {a+b}+2 (2 a+b)}\right ) \, dx}{4 a^{3/2} (a+b)^{3/2} d} \\ & = -\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}+\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}-\frac {\log (2 a+b+b \cos (2 c+2 d x))}{4 a (a+b) d^2}-\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))}+\frac {(2 a+b) \text {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^{i (2 c+2 d x)}\right )}{8 a^{3/2} (a+b)^{3/2} d^2}-\frac {(2 a+b) \text {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^{i (2 c+2 d x)}\right )}{8 a^{3/2} (a+b)^{3/2} d^2} \\ & = -\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}+\frac {i (2 a+b) x \log \left (1+\frac {b e^{2 i (c+d x)}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 a^{3/2} (a+b)^{3/2} d}-\frac {\log (2 a+b+b \cos (2 c+2 d x))}{4 a (a+b) d^2}-\frac {(2 a+b) \operatorname {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{3/2} d^2}+\frac {(2 a+b) \operatorname {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{3/2} d^2}-\frac {b x \sin (2 c+2 d x)}{2 a (a+b) d (2 a+b+b \cos (2 c+2 d x))} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(825\) vs. \(2(327)=654\).

Time = 15.03 (sec) , antiderivative size = 825, normalized size of antiderivative = 2.52 \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=\frac {b c \sin (2 (c+d x))-b (c+d x) \sin (2 (c+d x))}{2 a (a+b) d^2 (2 a+b+b \cos (2 (c+d x)))}+\frac {\cos ^2(c+d x) \left (-\frac {4 (2 a+b) c \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b}}+2 \log \left (\sec ^2(c+d x)\right )-2 \log \left (a+b+a \tan ^2(c+d x)\right )-\frac {i (2 a+b) \left (\log (1-i \tan (c+d x)) \log \left (\frac {\sqrt {-a-b}+\sqrt {a} \tan (c+d x)}{-i \sqrt {a}+\sqrt {-a-b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a} (1-i \tan (c+d x))}{\sqrt {a}+i \sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}+\frac {i (2 a+b) \left (\log (1+i \tan (c+d x)) \log \left (\frac {\sqrt {-a-b}+\sqrt {a} \tan (c+d x)}{i \sqrt {a}+\sqrt {-a-b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a} (1+i \tan (c+d x))}{\sqrt {a}-i \sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}-\frac {i (2 a+b) \left (\log (1+i \tan (c+d x)) \log \left (\frac {\sqrt {-a-b}-\sqrt {a} \tan (c+d x)}{-i \sqrt {a}+\sqrt {-a-b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a} (1+i \tan (c+d x))}{\sqrt {a}+i \sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}+\frac {i (2 a+b) \left (\log (1-i \tan (c+d x)) \log \left (\frac {\sqrt {-a-b}-\sqrt {a} \tan (c+d x)}{i \sqrt {a}+\sqrt {-a-b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a} (i+\tan (c+d x))}{i \sqrt {a}+\sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}\right ) (2 (2 a+b) d x+b \sin (2 (c+d x))) \left (-\sqrt {-a-b}+\sqrt {a} \tan (c+d x)\right ) \left (\sqrt {-a-b}+\sqrt {a} \tan (c+d x)\right )}{4 a (a+b) d^2 (2 a+b+b \cos (2 (c+d x))) (-((2 a+b) (2 c-i \log (1-i \tan (c+d x))+i \log (1+i \tan (c+d x))))+b \sin (2 (c+d x)))} \]

[In]

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

[Out]

(b*c*Sin[2*(c + d*x)] - b*(c + d*x)*Sin[2*(c + d*x)])/(2*a*(a + b)*d^2*(2*a + b + b*Cos[2*(c + d*x)])) + (Cos[
c + d*x]^2*((-4*(2*a + b)*c*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*Sqrt[a + b]) + 2*Log[Sec[c +
d*x]^2] - 2*Log[a + b + a*Tan[c + d*x]^2] - (I*(2*a + b)*(Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[-a - b] + Sqrt[a]*
Tan[c + d*x])/((-I)*Sqrt[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(1 - I*Tan[c + d*x]))/(Sqrt[a] + I*Sqrt[-a
- b])]))/(Sqrt[a]*Sqrt[-a - b]) + (I*(2*a + b)*(Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[-a - b] + Sqrt[a]*Tan[c + d*
x])/(I*Sqrt[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(1 + I*Tan[c + d*x]))/(Sqrt[a] - I*Sqrt[-a - b])]))/(Sqr
t[a]*Sqrt[-a - b]) - (I*(2*a + b)*(Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[-a - b] - Sqrt[a]*Tan[c + d*x])/((-I)*Sqr
t[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(1 + I*Tan[c + d*x]))/(Sqrt[a] + I*Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[
-a - b]) + (I*(2*a + b)*(Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[-a - b] - Sqrt[a]*Tan[c + d*x])/(I*Sqrt[a] + Sqrt[-
a - b])] + PolyLog[2, (Sqrt[a]*(I + Tan[c + d*x]))/(I*Sqrt[a] + Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]))*(2*(2
*a + b)*d*x + b*Sin[2*(c + d*x)])*(-Sqrt[-a - b] + Sqrt[a]*Tan[c + d*x])*(Sqrt[-a - b] + Sqrt[a]*Tan[c + d*x])
)/(4*a*(a + b)*d^2*(2*a + b + b*Cos[2*(c + d*x)])*(-((2*a + b)*(2*c - I*Log[1 - I*Tan[c + d*x]] + I*Log[1 + I*
Tan[c + d*x]])) + b*Sin[2*(c + d*x)]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2269 vs. \(2 (273 ) = 546\).

Time = 1.41 (sec) , antiderivative size = 2270, normalized size of antiderivative = 6.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(2270\)

[In]

int(x/(a+cos(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4/d^2/(a+b)/a*ln(b*exp(4*I*(d*x+c))+4*exp(2*I*(d*x+c))*a+2*b*exp(2*I*(d*x+c))+b)-1/4*I/d^2/(a+b)/a*b^2/(a*(
a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*c-1/4*I/d/(a+b)/a*b
^2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*x-1/2/d/(a+b
)/a*b^2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*c*x-I*x*(2*exp(2*I*(d*x+c))*a+b*exp(2*I*(d*x+c))+b)/a/(a+b)
/d/(b*exp(4*I*(d*x+c))+4*exp(2*I*(d*x+c))*a+2*b*exp(2*I*(d*x+c))+b)-1/2/d^2/(a+b)/(a*(a+b))^(1/2)*c^2-1/4/d^2/
(a+b)/a*b^2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*c^2-1/8/d^2/(a+b)/a*b^2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(
1/2)-2*a-b)*polylog(2,b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))-2/d/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(a+b))
^(1/2)-2*a-b)*c*x-2/d/(a+b)/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*c*x*b-1/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b
)*x^2-1/d/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b)*c*x+1/d^2/(a+b)/a*ln(exp(I*(d*x+c)))-1/(a+b)/(a*(a+b))^(1/2)/(-
2*(a*(a+b))^(1/2)-2*a-b)*x^2*b-1/2/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b)*x^2-1/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(
a+b))^(1/2)-2*a-b)*x^2-2/d/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b)*c*x-1/2*I/d^2/(a+b)/a*b*c/(a^2+a*b)^(1/2)*arctanh(
1/4*(2*b*exp(2*I*(d*x+c))+4*a+2*b)/(a^2+a*b)^(1/2))-I/d^2/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*l
n(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*c-1/2/(a+b)/(a*(a+b))^(1/2)*x^2-1/4/d^2/(a+b)/(a*(a+b))^(1/
2)*polylog(2,b*exp(2*I*(d*x+c))/(2*(a*(a+b))^(1/2)-2*a-b))-1/d^2/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b)*c^2-1/2/d^2/
(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))-I/d/(a+b)/(a*(a+b))^
(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*b*x-I/d^2/(a+b)/(a*(a+b))
^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*b*c-1/4*I/d/(a+b)/a*b/(a
*(a+b))^(1/2)*ln(1-b*exp(2*I*(d*x+c))/(2*(a*(a+b))^(1/2)-2*a-b))*x-1/2*I/d/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b
)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*x-1/d/(a+b)/(a*(a+b))^(1/2)*c*x-1/4/(a+b)/a*b/(a*(a+b))^
(1/2)*x^2-1/4*I/d^2/(a+b)/a*b/(a*(a+b))^(1/2)*ln(1-b*exp(2*I*(d*x+c))/(2*(a*(a+b))^(1/2)-2*a-b))*c-1/2/d/(a+b)
/a*b/(a*(a+b))^(1/2)*c*x-I/d/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a
*(a+b))^(1/2)-2*a-b))*x-1/2*I/d^2/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(
1/2)-2*a-b))*c-1/d^2/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*c^2-1/2/d^2/(a+b)*a/(a*(a+b))^(1/2)/(-
2*(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))-1/8/d^2/(a+b)/a*b/(a*(a+b))^
(1/2)*polylog(2,b*exp(2*I*(d*x+c))/(2*(a*(a+b))^(1/2)-2*a-b))-1/2/d^2/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b)*c^2
-1/4/d^2/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))-1/2*I/d
/(a+b)/(a*(a+b))^(1/2)*ln(1-b*exp(2*I*(d*x+c))/(2*(a*(a+b))^(1/2)-2*a-b))*x-I/d/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-
b)*ln(1-b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*x-I/d^2/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*I
*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*c-1/2*I/d^2/(a+b)/(a*(a+b))^(1/2)*ln(1-b*exp(2*I*(d*x+c))/(2*(a*(a+b))^(
1/2)-2*a-b))*c-I/d^2/(a+b)*c/(a^2+a*b)^(1/2)*arctanh(1/4*(2*b*exp(2*I*(d*x+c))+4*a+2*b)/(a^2+a*b)^(1/2))-1/d^2
/(a+b)/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*c^2*b-1/2/d^2/(a+b)/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-
b)*polylog(2,b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))*b-1/4/(a+b)/a*b^2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1
/2)-2*a-b)*x^2-1/4/d^2/(a+b)/a*b/(a*(a+b))^(1/2)*c^2

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3759 vs. \(2 (277) = 554\).

Time = 1.93 (sec) , antiderivative size = 3759, normalized size of antiderivative = 11.50 \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(4*(a^2*b + a*b^2)*d*x*cos(d*x + c)*sin(d*x + c) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqr
t((a^2 + a*b)/b^2)*dilog((((2*a + b)*cos(d*x + c) + (2*I*a + I*b)*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d
*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b + 1) - (2*a^2*b + a*b^2
+ (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-(((2*a + b)*cos(d*x + c) - (2*I*a + I*b)*sin(d*
x + c) - 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a +
b)/b) + b)/b + 1) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog((((2*a + b
)*cos(d*x + c) + (-2*I*a - I*b)*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sq
rt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b + 1) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*
sqrt((a^2 + a*b)/b^2)*dilog(-(((2*a + b)*cos(d*x + c) - (-2*I*a - I*b)*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*
sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + b)/b + 1) + (2*a^2*b + a
*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog((((2*a + b)*cos(d*x + c) + (2*I*a + I*b)*si
n(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*
a - b)/b) - b)/b + 1) + (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-(((2*a
+ b)*cos(d*x + c) - (2*I*a + I*b)*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))
*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + b)/b + 1) + (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2
)*sqrt((a^2 + a*b)/b^2)*dilog((((2*a + b)*cos(d*x + c) + (-2*I*a - I*b)*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b
*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - b)/b + 1) + (2*a^2*b + a
*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-(((2*a + b)*cos(d*x + c) - (-2*I*a - I*b)*
sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) -
2*a - b)/b) + b)/b + 1) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c
)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(d*x + c) + (2*I*a + I*b)*sin(d*x + c)
- 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)
- b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2
*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log((((2*a + b)*cos(d*x + c) - (2*I*a + I*b)*sin(d*x + c) - 2*(b*cos(d*x
+ c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + b)/b) - (-I*(
2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*
sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(d*x + c) + (-2*I*a - I*b)*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*si
n(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b) - (I*(2*a^2*b + a*b^
2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b
)/b^2)*log((((2*a + b)*cos(d*x + c) - (-2*I*a - I*b)*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt
((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2
*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-((
(2*a + b)*cos(d*x + c) + (2*I*a + I*b)*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b
^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - b)/b) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x
+ I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log((((2*a + b)*cos(d*x
+ c) - (2*I*a + I*b)*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqr
t((a^2 + a*b)/b^2) - 2*a - b)/b) + b)/b) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^
3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*cos(d*x + c) + (-2*I*a -
I*b)*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^
2) - 2*a - b)/b) - b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x
+ c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log((((2*a + b)*cos(d*x + c) - (-2*I*a - I*b)*sin(d*x +
c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b
) + b)/b) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b
+ a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*cos(d*x + c) + 2*I
*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^
2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*cos(d*x + c) -
2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(
2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*cos(d*x +
c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2
- I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*cos(d
*x + c) - 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c
)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*co
s(d*x + c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x
+ c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) +
2*cos(d*x + c) - 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos
(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b
) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*
cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b
)/b) - 2*cos(d*x + c) - 2*I*sin(d*x + c)))/((a^4*b + 2*a^3*b^2 + a^2*b^3)*d^2*cos(d*x + c)^2 + (a^5 + 2*a^4*b
+ a^3*b^2)*d^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F]

\[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=\int { \frac {x}{{\left (b \cos \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx=\int \frac {x}{{\left (b\,{\cos \left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

[In]

int(x/(a + b*cos(c + d*x)^2)^2,x)

[Out]

int(x/(a + b*cos(c + d*x)^2)^2, x)