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

Optimal. Leaf size=327 \[ -\frac {(2 a+b) \text {Li}_2\left (-\frac {b e^{2 i (c+d x)}}{2 a-2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 a^{3/2} d^2 (a+b)^{3/2}}+\frac {(2 a+b) \text {Li}_2\left (-\frac {b e^{2 i (c+d x)}}{2 a+2 \sqrt {a+b} \sqrt {a}+b}\right )}{8 a^{3/2} d^2 (a+b)^{3/2}}-\frac {i x (2 a+b) \log \left (1+\frac {b e^{2 i (c+d x)}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 a^{3/2} d (a+b)^{3/2}}+\frac {i x (2 a+b) \log \left (1+\frac {b e^{2 i (c+d x)}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 a^{3/2} d (a+b)^{3/2}}-\frac {\log (2 a+b \cos (2 c+2 d x)+b)}{4 a d^2 (a+b)}-\frac {b x \sin (2 c+2 d x)}{2 a d (a+b) (2 a+b \cos (2 c+2 d x)+b)} \]

[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))

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Rubi [A]  time = 0.59, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4586, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac {(2 a+b) \text {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 a^{3/2} d^2 (a+b)^{3/2}}+\frac {(2 a+b) \text {PolyLog}\left (2,-\frac {b e^{2 i (c+d x)}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{8 a^{3/2} d^2 (a+b)^{3/2}}-\frac {i x (2 a+b) \log \left (1+\frac {b e^{2 i (c+d x)}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 a^{3/2} d (a+b)^{3/2}}+\frac {i x (2 a+b) \log \left (1+\frac {b e^{2 i (c+d x)}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}\right )}{4 a^{3/2} d (a+b)^{3/2}}-\frac {\log (2 a+b \cos (2 c+2 d x)+b)}{4 a d^2 (a+b)}-\frac {b x \sin (2 c+2 d x)}{2 a d (a+b) (2 a+b \cos (2 c+2 d x)+b)} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Cos[c + d*x]^2)^2,x]

[Out]

((-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) + ((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 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 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 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]

Rule 2668

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 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 3324

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 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]))

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b \cos ^2(c+d x)\right )^2} \, dx &=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 {\operatorname {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) \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^{i (2 c+2 d x)}\right )}{8 a^{3/2} (a+b)^{3/2} d^2}-\frac {(2 a+b) \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^{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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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*}

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Mathematica [B]  time = 13.60, size = 825, normalized size = 2.52 \[ \frac {\left (-\frac {4 (2 a+b) c \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b}}+2 \log \left (\sec ^2(c+d x)\right )-2 \log \left (a \tan ^2(c+d x)+a+b\right )-\frac {i (2 a+b) \left (\log (1-i \tan (c+d x)) \log \left (\frac {\sqrt {a} \tan (c+d x)+\sqrt {-a-b}}{\sqrt {-a-b}-i \sqrt {a}}\right )+\text {Li}_2\left (\frac {\sqrt {a} (1-i \tan (c+d x))}{\sqrt {a}+i \sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}+\frac {i (2 a+b) \left (\log (i \tan (c+d x)+1) \log \left (\frac {\sqrt {a} \tan (c+d x)+\sqrt {-a-b}}{i \sqrt {a}+\sqrt {-a-b}}\right )+\text {Li}_2\left (\frac {\sqrt {a} (i \tan (c+d x)+1)}{\sqrt {a}-i \sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}-\frac {i (2 a+b) \left (\log (i \tan (c+d x)+1) \log \left (\frac {\sqrt {-a-b}-\sqrt {a} \tan (c+d x)}{\sqrt {-a-b}-i \sqrt {a}}\right )+\text {Li}_2\left (\frac {\sqrt {a} (i \tan (c+d x)+1)}{\sqrt {a}+i \sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}+\frac {i (2 a+b) \left (\log (1-i \tan (c+d x)) \log \left (\frac {\sqrt {-a-b}-\sqrt {a} \tan (c+d x)}{i \sqrt {a}+\sqrt {-a-b}}\right )+\text {Li}_2\left (\frac {\sqrt {a} (\tan (c+d x)+i)}{i \sqrt {a}+\sqrt {-a-b}}\right )\right )}{\sqrt {a} \sqrt {-a-b}}\right ) (2 (2 a+b) d x+b \sin (2 (c+d x))) \left (\sqrt {a} \tan (c+d x)-\sqrt {-a-b}\right ) \left (\sqrt {a} \tan (c+d x)+\sqrt {-a-b}\right ) \cos ^2(c+d x)}{4 a (a+b) d^2 (2 a+b+b \cos (2 (c+d x))) (b \sin (2 (c+d x))-(2 a+b) (2 c-i \log (1-i \tan (c+d x))+i \log (i \tan (c+d x)+1)))}+\frac {b c \sin (2 (c+d x))-b (c+d x) \sin (2 (c+d x))}{2 a (a+b) d^2 (2 a+b+b \cos (2 (c+d x)))} \]

Warning: Unable to verify antiderivative.

[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)]))

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fricas [B]  time = 1.18, size = 3799, normalized size = 11.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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(-1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) - 4*(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) + 2*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(1/2*((2*(2*a + b)*cos(d*x + c) - (4*I
*a + 2*I*b)*sin(d*x + c) - 4*(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) - 2*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(-1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(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) + 2*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(1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*s
in(d*x + c) - 4*(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) - 2*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(-1/
2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) + 4*(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) + 2*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(1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) + 4*
(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) - 2*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(-1/2*((2*(2*a + b)*c
os(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqr
t((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*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(1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(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) - 2*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)*s
qrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) - 4*(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) + 2*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(-1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*s
in(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*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(1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(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) + 2*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(-1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(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) - 2*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(1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*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) - (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(-
1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) + 4*(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) - 2*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(1/2*((2*
(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(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) + 2*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(-1/2*((2*(2*a
+ b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(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) - 2*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*sq
rt((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^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.28, size = 2270, normalized size = 6.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a+b)/(a*(a+b))^(1/2)*x^2-1/4*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-1/4/d^2/(a+b)/a*b/(a*(a+b))^(1/2)*c^2-1/(a+b)/(-2*(a*(a+b))^(1/
2)-2*a-b)*x^2-1/d^2/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b)*c^2-1/2/d^2/(a+b)/(a*(a+b))^(1/2)*c^2-1/2/(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/(a+b)/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*x^2-1/d/(a+b)/(a*(a+b))^(1/2)*c*x-1/4/(a+b)
/a*b/(a*(a+b))^(1/2)*x^2-2/d/(a+b)/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*c*x-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/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)))-2/d/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*c*x-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))-1/2/d/(a+b)/a*b/(a*(a+b))^(1/2)*
c*x-I*x*(2*a*exp(2*I*(d*x+c))+b*exp(2*I*(d*x+c))+b)/a/(a+b)/d/(b*exp(4*I*(d*x+c))+4*a*exp(2*I*(d*x+c))+2*b*exp
(2*I*(d*x+c))+b)-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*
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-1/4*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-1/2*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/(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-I/d^2/(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))*c-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-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-1/2/d/(a+b)/a*b^2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b
)*c*x-1/4/d^2/(a+b)/a*ln(b*exp(4*I*(d*x+c))+4*a*exp(2*I*(d*x+c))+2*b*exp(2*I*(d*x+c))+b)-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))-1/2/d^2/(a+b)/(-2*(a*(a+b))^(1/2)-2*a-b)*poly
log(2,b*exp(2*I*(d*x+c))/(-2*(a*(a+b))^(1/2)-2*a-b))-1/d^2/(a+b)*a/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*
c^2-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-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-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))-1/4/d^2/(a+b)/a*b/(-2*(a*(a+b))^(1/2)-2*a-b)*p
olylog(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^2-1/2/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/d^2/(a+b)/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2
)-2*a-b)*b*c^2-1/2*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-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^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))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\left (b\,{\cos \left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*cos(c + d*x)^2)^2,x)

[Out]

int(x/(a + b*cos(c + d*x)^2)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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