Integrand size = 14, antiderivative size = 214 \[ \int \frac {x}{a+b \cos (c+d x)} \, dx=-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}+\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2} \]
-I*x*ln(1+b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/d/(a^2-b^2)^(1/2)+I*x*ln(1 +b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/d/(a^2-b^2)^(1/2)-polylog(2,-b*exp( I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/d^2/(a^2-b^2)^(1/2)+polylog(2,-b*exp(I*(d* x+c))/(a+(a^2-b^2)^(1/2)))/d^2/(a^2-b^2)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(756\) vs. \(2(214)=428\).
Time = 0.94 (sec) , antiderivative size = 756, normalized size of antiderivative = 3.53 \[ \int \frac {x}{a+b \cos (c+d x)} \, dx=\frac {2 (c+d x) \text {arctanh}\left (\frac {(a+b) \cot \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )-2 \left (c+\arccos \left (-\frac {a}{b}\right )\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )+\left (\arccos \left (-\frac {a}{b}\right )-2 i \text {arctanh}\left (\frac {(a+b) \cot \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )+2 i \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )\right ) \log \left (\frac {\sqrt {-a^2+b^2} e^{-\frac {1}{2} i (c+d x)}}{\sqrt {2} \sqrt {b} \sqrt {a+b \cos (c+d x)}}\right )+\left (\arccos \left (-\frac {a}{b}\right )+2 i \left (\text {arctanh}\left (\frac {(a+b) \cot \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )-\text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-a^2+b^2} e^{\frac {1}{2} i (c+d x)}}{\sqrt {2} \sqrt {b} \sqrt {a+b \cos (c+d x)}}\right )-\left (\arccos \left (-\frac {a}{b}\right )-2 i \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )\right ) \log \left (\frac {(a+b) \left (-a+b-i \sqrt {-a^2+b^2}\right ) \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b \left (a+b+\sqrt {-a^2+b^2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )-\left (\arccos \left (-\frac {a}{b}\right )+2 i \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )\right ) \log \left (\frac {(a+b) \left (i a-i b+\sqrt {-a^2+b^2}\right ) \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b \left (a+b+\sqrt {-a^2+b^2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (a-i \sqrt {-a^2+b^2}\right ) \left (a+b-\sqrt {-a^2+b^2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b \left (a+b+\sqrt {-a^2+b^2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (a+i \sqrt {-a^2+b^2}\right ) \left (a+b-\sqrt {-a^2+b^2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b \left (a+b+\sqrt {-a^2+b^2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )\right )}{\sqrt {-a^2+b^2} d^2} \]
(2*(c + d*x)*ArcTanh[((a + b)*Cot[(c + d*x)/2])/Sqrt[-a^2 + b^2]] - 2*(c + ArcCos[-(a/b)])*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + ( ArcCos[-(a/b)] - (2*I)*ArcTanh[((a + b)*Cot[(c + d*x)/2])/Sqrt[-a^2 + b^2] ] + (2*I)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])*Log[Sqrt[ -a^2 + b^2]/(Sqrt[2]*Sqrt[b]*E^((I/2)*(c + d*x))*Sqrt[a + b*Cos[c + d*x]]) ] + (ArcCos[-(a/b)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x)/2])/Sqrt[-a^2 + b^2]] - ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]))*Log[(Sqr t[-a^2 + b^2]*E^((I/2)*(c + d*x)))/(Sqrt[2]*Sqrt[b]*Sqrt[a + b*Cos[c + d*x ]])] - (ArcCos[-(a/b)] - (2*I)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a ^2 + b^2]])*Log[((a + b)*(-a + b - I*Sqrt[-a^2 + b^2])*(1 + I*Tan[(c + d*x )/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))] - (ArcCos[-(a/b)] + (2*I)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])*Log[((a + b )*(I*a - I*b + Sqrt[-a^2 + b^2])*(I + Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[ -a^2 + b^2]*Tan[(c + d*x)/2]))] + I*(PolyLog[2, ((a - I*Sqrt[-a^2 + b^2])* (a + b - Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]* Tan[(c + d*x)/2]))] - PolyLog[2, ((a + I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[- a^2 + b^2]*Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2 ]))]))/(Sqrt[-a^2 + b^2]*d^2)
Time = 0.80 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3802, 2694, 27, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x}{a+b \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {x}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3802 |
| \(\displaystyle 2 \int \frac {e^{i (c+d x)} x}{2 e^{i (c+d x)} a+b e^{2 i (c+d x)}+b}dx\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle 2 \left (\frac {b \int \frac {e^{i (c+d x)} x}{2 \left (a+b e^{i (c+d x)}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {b \int \frac {e^{i (c+d x)} x}{2 \left (a+b e^{i (c+d x)}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {b \int \frac {e^{i (c+d x)} x}{a+b e^{i (c+d x)}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {e^{i (c+d x)} x}{a+b e^{i (c+d x)}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 2 \left (\frac {b \left (\frac {i \int \log \left (\frac {e^{i (c+d x)} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b d}-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {i \int \log \left (\frac {e^{i (c+d x)} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b d}-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 2 \left (\frac {b \left (\frac {\int e^{-i (c+d x)} \log \left (\frac {e^{i (c+d x)} b}{a-\sqrt {a^2-b^2}}+1\right )de^{i (c+d x)}}{b d^2}-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {\int e^{-i (c+d x)} \log \left (\frac {e^{i (c+d x)} b}{a+\sqrt {a^2-b^2}}+1\right )de^{i (c+d x)}}{b d^2}-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 \left (\frac {b \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {i x \log \left (1+\frac {b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )\) |
2*((b*(((-I)*x*Log[1 + (b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) - PolyLog[2, -((b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]))]/(b*d^2)))/(2*Sqr t[a^2 - b^2]) - (b*(((-I)*x*Log[1 + (b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^ 2])])/(b*d) - PolyLog[2, -((b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2]))]/(b* d^2)))/(2*Sqrt[a^2 - b^2]))
3.2.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
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]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*( x_)]), x_Symbol] :> Simp[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (188 ) = 376\).
Time = 0.84 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.93
| method | result | size |
| risch | \(-\frac {i \ln \left (\frac {-b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}-a}{\sqrt {a^{2}-b^{2}}-a}\right ) x}{d \sqrt {a^{2}-b^{2}}}+\frac {i \ln \left (\frac {b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}+a}{\sqrt {a^{2}-b^{2}}+a}\right ) x}{d \sqrt {a^{2}-b^{2}}}-\frac {i \ln \left (\frac {-b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}-a}{\sqrt {a^{2}-b^{2}}-a}\right ) c}{d^{2} \sqrt {a^{2}-b^{2}}}+\frac {i \ln \left (\frac {b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}+a}{\sqrt {a^{2}-b^{2}}+a}\right ) c}{d^{2} \sqrt {a^{2}-b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}-a}{\sqrt {a^{2}-b^{2}}-a}\right )}{d^{2} \sqrt {a^{2}-b^{2}}}+\frac {\operatorname {dilog}\left (\frac {b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}+a}{\sqrt {a^{2}-b^{2}}+a}\right )}{d^{2} \sqrt {a^{2}-b^{2}}}+\frac {2 i c \arctan \left (\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{d^{2} \sqrt {-a^{2}+b^{2}}}\) | \(414\) |
-I/d/(a^2-b^2)^(1/2)*ln((-b*exp(I*(d*x+c))+(a^2-b^2)^(1/2)-a)/((a^2-b^2)^( 1/2)-a))*x+I/d/(a^2-b^2)^(1/2)*ln((b*exp(I*(d*x+c))+(a^2-b^2)^(1/2)+a)/((a ^2-b^2)^(1/2)+a))*x-I/d^2/(a^2-b^2)^(1/2)*ln((-b*exp(I*(d*x+c))+(a^2-b^2)^ (1/2)-a)/((a^2-b^2)^(1/2)-a))*c+I/d^2/(a^2-b^2)^(1/2)*ln((b*exp(I*(d*x+c)) +(a^2-b^2)^(1/2)+a)/((a^2-b^2)^(1/2)+a))*c-1/d^2/(a^2-b^2)^(1/2)*dilog((-b *exp(I*(d*x+c))+(a^2-b^2)^(1/2)-a)/((a^2-b^2)^(1/2)-a))+1/d^2/(a^2-b^2)^(1 /2)*dilog((b*exp(I*(d*x+c))+(a^2-b^2)^(1/2)+a)/((a^2-b^2)^(1/2)+a))+2*I/d^ 2*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*exp(I*(d*x+c))+2*a)/(-a^2+b^2)^(1/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (184) = 368\).
Time = 0.42 (sec) , antiderivative size = 915, normalized size of antiderivative = 4.28 \[ \int \frac {x}{a+b \cos (c+d x)} \, dx=\frac {-i \, b c \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) + i \, b c \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) - i \, b c \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} - 2 \, a\right ) + i \, b c \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} - 2 \, a\right ) - b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) + b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {a \cos \left (d x + c\right ) - i \, a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) + b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {a \cos \left (d x + c\right ) - i \, a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - {\left (i \, b d x + i \, b c\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (\frac {a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - {\left (-i \, b d x - i \, b c\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (\frac {a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - {\left (-i \, b d x - i \, b c\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (\frac {a \cos \left (d x + c\right ) - i \, a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - {\left (i \, b d x + i \, b c\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (\frac {a \cos \left (d x + c\right ) - i \, a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right )}{2 \, {\left (a^{2} - b^{2}\right )} d^{2}} \]
1/2*(-I*b*c*sqrt((a^2 - b^2)/b^2)*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c ) + 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) + I*b*c*sqrt((a^2 - b^2)/b^2)*log(2*b *cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) - I* b*c*sqrt((a^2 - b^2)/b^2)*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b *sqrt((a^2 - b^2)/b^2) - 2*a) + I*b*c*sqrt((a^2 - b^2)/b^2)*log(-2*b*cos(d *x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt((a^2 - b^2)/b^2) - 2*a) - b*sqrt(( a^2 - b^2)/b^2)*dilog(-(a*cos(d*x + c) + I*a*sin(d*x + c) + (b*cos(d*x + c ) + I*b*sin(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + b*sqrt((a^2 - b^ 2)/b^2)*dilog(-(a*cos(d*x + c) + I*a*sin(d*x + c) - (b*cos(d*x + c) + I*b* sin(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) - b*sqrt((a^2 - b^2)/b^2)* dilog(-(a*cos(d*x + c) - I*a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + b*sqrt((a^2 - b^2)/b^2)*dilog(-( a*cos(d*x + c) - I*a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sq rt((a^2 - b^2)/b^2) + b)/b + 1) - (I*b*d*x + I*b*c)*sqrt((a^2 - b^2)/b^2)* log((a*cos(d*x + c) + I*a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c ))*sqrt((a^2 - b^2)/b^2) + b)/b) - (-I*b*d*x - I*b*c)*sqrt((a^2 - b^2)/b^2 )*log((a*cos(d*x + c) + I*a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) - (-I*b*d*x - I*b*c)*sqrt((a^2 - b^2)/b ^2)*log((a*cos(d*x + c) - I*a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) - (I*b*d*x + I*b*c)*sqrt((a^2 - b^...
\[ \int \frac {x}{a+b \cos (c+d x)} \, dx=\int \frac {x}{a + b \cos {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {x}{a+b \cos (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
\[ \int \frac {x}{a+b \cos (c+d x)} \, dx=\int { \frac {x}{b \cos \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x}{a+b \cos (c+d x)} \, dx=\int \frac {x}{a+b\,\cos \left (c+d\,x\right )} \,d x \]