Optimal. Leaf size=87 \[ \frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b e \sqrt {a-b} \sqrt {a+b}}-\frac {C \log (a+b \cos (d+e x))}{b e}+\frac {B x}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4377, 2735, 2659, 205, 2668, 31} \[ \frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b e \sqrt {a-b} \sqrt {a+b}}-\frac {C \log (a+b \cos (d+e x))}{b e}+\frac {B x}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 205
Rule 2659
Rule 2668
Rule 2735
Rule 4377
Rubi steps
\begin {align*} \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{a+b \cos (d+e x)} \, dx &=C \int \frac {\sin (d+e x)}{a+b \cos (d+e x)} \, dx+\int \frac {A+B \cos (d+e x)}{a+b \cos (d+e x)} \, dx\\ &=\frac {B x}{b}-\frac {(-A b+a B) \int \frac {1}{a+b \cos (d+e x)} \, dx}{b}-\frac {C \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cos (d+e x)\right )}{b e}\\ &=\frac {B x}{b}-\frac {C \log (a+b \cos (d+e x))}{b e}+\frac {(2 (A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{b e}\\ &=\frac {B x}{b}+\frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b \sqrt {a+b} e}-\frac {C \log (a+b \cos (d+e x))}{b e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 82, normalized size = 0.94 \[ \frac {\frac {2 (a B-A b) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}-C \log (a+b \cos (d+e x))+B (d+e x)}{b e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 326, normalized size = 3.75 \[ \left [\frac {2 \, {\left (B a^{2} - B b^{2}\right )} e x + {\left (B a - A b\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (e x + d\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (e x + d\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (e x + d\right ) + b\right )} \sin \left (e x + d\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}}\right ) - {\left (C a^{2} - C b^{2}\right )} \log \left (b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )} e}, \frac {2 \, {\left (B a^{2} - B b^{2}\right )} e x - 2 \, {\left (B a - A b\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (e x + d\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (e x + d\right )}\right ) - {\left (C a^{2} - C b^{2}\right )} \log \left (b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.45, size = 459, normalized size = 5.28 \[ -{\left (\frac {C {\left (a + b\right )} {\left (a - b\right )}^{2} \log \left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + \frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{2 \, {\left (a - b\right )}}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (\sqrt {a^{2} - b^{2}} B {\left (2 \, a - b\right )} {\left | a - b \right |} - \sqrt {a^{2} - b^{2}} A b {\left | a - b \right |} - \sqrt {a^{2} - b^{2}} A {\left | a - b \right |} {\left | b \right |} + \sqrt {a^{2} - b^{2}} B {\left | a - b \right |} {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {x e + d}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (2 \, B a - A b - B b + A {\left | b \right |} - B {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {x e + d}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{b^{2} - a {\left | b \right |}} + \frac {{\left (C a - C b\right )} \log \left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + \frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{2 \, {\left (a - b\right )}}\right )}{b^{2} - a {\left | b \right |}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.18, size = 226, normalized size = 2.60 \[ -\frac {\ln \left (a \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-\left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right ) b +a +b \right ) a C}{e b \left (a -b \right )}+\frac {\ln \left (a \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-\left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right ) b +a +b \right ) C}{e \left (a -b \right )}+\frac {2 \arctan \left (\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) A}{e \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {2 \arctan \left (\frac {\tan \left (\frac {e x}{2}+\frac {d}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) a B}{e b \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {C \ln \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{e b}+\frac {2 B \arctan \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{e b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.16, size = 886, normalized size = 10.18 \[ -\frac {\ln \left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{b\,e}+\frac {\ln \left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )}{b\,e}-\frac {\ln \left (A^2\,b^3+B^2\,b^3-4\,C^2\,a^3+4\,C^2\,b^3+A^2\,a\,b^2+B^2\,a\,b^2+4\,C^2\,a\,b^2-4\,C^2\,a^2\,b-2\,A\,B\,a\,b^2-2\,A\,B\,a^2\,b+A^2\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}+B^2\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}-4\,C^2\,a^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}+4\,C^2\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}-4\,A\,C\,b^2\,\sqrt {b^2-a^2}+4\,B\,C\,a^2\,\sqrt {b^2-a^2}-4\,A\,C\,b^3\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )-4\,B\,C\,a^3\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )-4\,A\,C\,a\,b\,\sqrt {b^2-a^2}+4\,B\,C\,a\,b\,\sqrt {b^2-a^2}+4\,A\,C\,a^2\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+4\,B\,C\,a\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )-2\,A\,B\,a\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}\right )\,\left (C\,a^2-C\,b^2+A\,b\,\sqrt {b^2-a^2}-B\,a\,\sqrt {b^2-a^2}\right )}{b\,e\,\left (a^2-b^2\right )}-\frac {\ln \left (A^2\,b^3+B^2\,b^3-4\,C^2\,a^3+4\,C^2\,b^3+A^2\,a\,b^2+B^2\,a\,b^2+4\,C^2\,a\,b^2-4\,C^2\,a^2\,b-2\,A\,B\,a\,b^2-2\,A\,B\,a^2\,b-A^2\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}-B^2\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}+4\,C^2\,a^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}-4\,C^2\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}+4\,A\,C\,b^2\,\sqrt {b^2-a^2}-4\,B\,C\,a^2\,\sqrt {b^2-a^2}-4\,A\,C\,b^3\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )-4\,B\,C\,a^3\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+4\,A\,C\,a\,b\,\sqrt {b^2-a^2}-4\,B\,C\,a\,b\,\sqrt {b^2-a^2}+4\,A\,C\,a^2\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+4\,B\,C\,a\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+2\,A\,B\,a\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sqrt {b^2-a^2}\right )\,\left (C\,a^2-C\,b^2-A\,b\,\sqrt {b^2-a^2}+B\,a\,\sqrt {b^2-a^2}\right )}{b\,e\,\left (a^2-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 26.56, size = 672, normalized size = 7.72 \[ \begin {cases} \frac {\tilde {\infty } x \left (A + B \cos {\relax (d )} + C \sin {\relax (d )}\right )}{\cos {\relax (d )}} & \text {for}\: a = 0 \wedge b = 0 \wedge e = 0 \\\frac {A \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{b e} + \frac {B x}{b} - \frac {B \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{b e} + \frac {C \log {\left (\tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{b e} & \text {for}\: a = b \\\frac {A}{b e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}} + \frac {B x}{b} + \frac {B}{b e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}} + \frac {C \log {\left (\tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{b e} - \frac {2 C \log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{b e} & \text {for}\: a = - b \\\frac {A x + \frac {B \sin {\left (d + e x \right )}}{e} - \frac {C \cos {\left (d + e x \right )}}{e}}{a} & \text {for}\: b = 0 \\\frac {x \left (A + B \cos {\relax (d )} + C \sin {\relax (d )}\right )}{a + b \cos {\relax (d )}} & \text {for}\: e = 0 \\- \frac {A b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} + \frac {A b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} + \frac {B a e x}{a b e + b^{2} e} + \frac {B a \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} - \frac {B a \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} + \frac {B b e x}{a b e + b^{2} e} - \frac {C a \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} - \frac {C a \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} + \frac {C a \log {\left (\tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{a b e + b^{2} e} - \frac {C b \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} - \frac {C b \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} + \frac {C b \log {\left (\tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{a b e + b^{2} e} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________