Integrand size = 31, antiderivative size = 86 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {B x}{b}+\frac {2 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b \sqrt {a+b} e}+\frac {C \log (a+b \cosh (d+e x))}{b e} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4462, 2814, 2738, 211, 2747, 31} \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {2 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b e \sqrt {a-b} \sqrt {a+b}}+\frac {C \log (a+b \cosh (d+e x))}{b e}+\frac {B x}{b} \]
[In]
[Out]
Rule 31
Rule 211
Rule 2738
Rule 2747
Rule 2814
Rule 4462
Rubi steps \begin{align*} \text {integral}& = C \int \frac {\sinh (d+e x)}{a+b \cosh (d+e x)} \, dx+\int \frac {A+B \cosh (d+e x)}{a+b \cosh (d+e x)} \, dx \\ & = \frac {B x}{b}-\frac {(-A b+a B) \int \frac {1}{a+b \cosh (d+e x)} \, dx}{b}+\frac {C \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (d+e x)\right )}{b e} \\ & = \frac {B x}{b}+\frac {C \log (a+b \cosh (d+e x))}{b e}-\frac {(2 i (A b-a B)) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{b e} \\ & = \frac {B x}{b}+\frac {2 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \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 \cosh (d+e x))}{b e} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {B (d+e x)+\frac {2 (-A b+a B) \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+C \log (a+b \cosh (d+e x))}{b e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(155\) vs. \(2(77)=154\).
Time = 0.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.81
method | result | size |
derivativedivides | \(\frac {\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{b}+\frac {\left (B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{b}+\frac {\frac {2 \left (a C -b C \right ) \ln \left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )}{2 a -2 b}-\frac {2 \left (-b A +B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}}{e}\) | \(156\) |
default | \(\frac {\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{b}+\frac {\left (B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{b}+\frac {\frac {2 \left (a C -b C \right ) \ln \left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )}{2 a -2 b}-\frac {2 \left (-b A +B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}}{e}\) | \(156\) |
risch | \(\frac {B x}{b}+\frac {x C}{b}+\frac {2 C \,a^{2} b \,e^{2} x}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}-\frac {2 C \,b^{3} e^{2} x}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}+\frac {2 C \,a^{2} b d e}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}-\frac {2 C \,b^{3} d e}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {b A a -a^{2} B -\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (b A -B a \right )}\right ) C \,a^{2}}{\left (a^{2}-b^{2}\right ) e b}-\frac {b \ln \left ({\mathrm e}^{e x +d}+\frac {b A a -a^{2} B -\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (b A -B a \right )}\right ) C}{\left (a^{2}-b^{2}\right ) e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {b A a -a^{2} B -\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (b A -B a \right )}\right ) \sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{\left (a^{2}-b^{2}\right ) e b}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {b A a -a^{2} B +\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (b A -B a \right )}\right ) C \,a^{2}}{\left (a^{2}-b^{2}\right ) e b}-\frac {b \ln \left ({\mathrm e}^{e x +d}+\frac {b A a -a^{2} B +\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (b A -B a \right )}\right ) C}{\left (a^{2}-b^{2}\right ) e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {b A a -a^{2} B +\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (b A -B a \right )}\right ) \sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{\left (a^{2}-b^{2}\right ) e b}\) | \(897\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 405, normalized size of antiderivative = 4.71 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\left [\frac {{\left ({\left (B - C\right )} a^{2} - {\left (B - C\right )} b^{2}\right )} e x - {\left (B a - A b\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (e x + d\right )^{2} + b^{2} \sinh \left (e x + d\right )^{2} + 2 \, a b \cosh \left (e x + d\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (e x + d\right ) + a b\right )} \sinh \left (e x + d\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (e x + d\right ) + b \sinh \left (e x + d\right ) + a\right )}}{b \cosh \left (e x + d\right )^{2} + b \sinh \left (e x + d\right )^{2} + 2 \, a \cosh \left (e x + d\right ) + 2 \, {\left (b \cosh \left (e x + d\right ) + a\right )} \sinh \left (e x + d\right ) + b}\right ) + {\left (C a^{2} - C b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (e x + d\right ) + a\right )}}{\cosh \left (e x + d\right ) - \sinh \left (e x + d\right )}\right )}{{\left (a^{2} b - b^{3}\right )} e}, \frac {{\left ({\left (B - C\right )} a^{2} - {\left (B - C\right )} b^{2}\right )} e x + 2 \, {\left (B a - A b\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (e x + d\right ) + b \sinh \left (e x + d\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (C a^{2} - C b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (e x + d\right ) + a\right )}}{\cosh \left (e x + d\right ) - \sinh \left (e x + d\right )}\right )}{{\left (a^{2} b - b^{3}\right )} e}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (73) = 146\).
Time = 15.33 (sec) , antiderivative size = 695, normalized size of antiderivative = 8.08 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\begin {cases} \frac {\tilde {\infty } x \left (A + B \cosh {\left (d \right )} + C \sinh {\left (d \right )}\right )}{\cosh {\left (d \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge e = 0 \\\frac {A \tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{b e} + \frac {B x}{b} - \frac {B \tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{b e} + \frac {C x}{b} - \frac {2 C \log {\left (\tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{b e} & \text {for}\: a = b \\- \frac {A}{b e \tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )}} + \frac {B x}{b} - \frac {B}{b e \tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )}} + \frac {C x}{b} - \frac {2 C \log {\left (\tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{b e} + \frac {2 C \log {\left (\tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{b e} & \text {for}\: a = - b \\\frac {A x + \frac {B \sinh {\left (d + e x \right )}}{e} + \frac {C \cosh {\left (d + e x \right )}}{e}}{a} & \text {for}\: b = 0 \\\frac {x \left (A + B \cosh {\left (d \right )} + C \sinh {\left (d \right )}\right )}{a + b \cosh {\left (d \right )}} & \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}} + \tanh {\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}} + \tanh {\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}} + \tanh {\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}} + \tanh {\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 e x}{a b e + b^{2} e} + \frac {C a \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\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}} + \tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} - \frac {2 C a \log {\left (\tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{a b e + b^{2} e} + \frac {C b e x}{a b e + b^{2} e} + \frac {C b \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\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}} + \tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} \right )}}{a b e + b^{2} e} - \frac {2 C b \log {\left (\tanh {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{a b e + b^{2} e} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {\frac {{\left (e x + d\right )} {\left (B - C\right )}}{b} + \frac {C \log \left (b e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} + b\right )}{b} - \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b e^{\left (e x + d\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b}}{e} \]
[In]
[Out]
Time = 2.94 (sec) , antiderivative size = 653, normalized size of antiderivative = 7.59 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{B\,e\,a^3\,b-A\,e\,a^2\,b^2-B\,e\,a\,b^3+A\,e\,b^4}+\frac {a^2\,b^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{B\,e\,a^3\,b^4-A\,e\,a^2\,b^5-B\,e\,a\,b^6+A\,e\,b^7}+\frac {A\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}}{b\,e\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}-\frac {B\,a\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}}{b^2\,e\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}\right )\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{\sqrt {b^4\,e^2-a^2\,b^2\,e^2}}+\frac {B\,x}{b}-\frac {C\,x}{b}+\frac {C\,b^3\,e\,\ln \left (4\,A^2\,b^3+4\,B^2\,a^2\,b-8\,A\,B\,a\,b^2+8\,B^2\,a^3\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,A^2\,b^3\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}+8\,A^2\,a\,b^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,B^2\,a^2\,b\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}-16\,A\,B\,a^2\,b\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d-8\,A\,B\,a\,b^2\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}\right )}{b^4\,e^2-a^2\,b^2\,e^2}-\frac {C\,a^2\,b\,e\,\ln \left (4\,A^2\,b^3+4\,B^2\,a^2\,b-8\,A\,B\,a\,b^2+8\,B^2\,a^3\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,A^2\,b^3\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}+8\,A^2\,a\,b^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,B^2\,a^2\,b\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}-16\,A\,B\,a^2\,b\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d-8\,A\,B\,a\,b^2\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}\right )}{b^4\,e^2-a^2\,b^2\,e^2} \]
[In]
[Out]