\(\int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx\) [199]

Optimal result

Integrand size = 15, antiderivative size = 56 \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\frac {2 A \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (a+b \cosh (x))}{b} \]

[Out]

B*ln(a+b*cosh(x))/b+2*A*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(1/2)/(a+b)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4486, 2738, 214, 2747, 31} \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\frac {2 A \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (a+b \cosh (x))}{b} \]

[In]

Int[(A + B*Sinh[x])/(a + b*Cosh[x]),x]

[Out]

(2*A*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[a + b*Cosh[x]])/b

Rule 31

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

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 4486

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{a+b \cosh (x)}+\frac {B \sinh (x)}{a+b \cosh (x)}\right ) \, dx \\ & = A \int \frac {1}{a+b \cosh (x)} \, dx+B \int \frac {\sinh (x)}{a+b \cosh (x)} \, dx \\ & = (2 A) \text {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )+\frac {B \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{b} \\ & = \frac {2 A \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (a+b \cosh (x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=-\frac {2 A \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {B \log (a+b \cosh (x))}{b} \]

[In]

Integrate[(A + B*Sinh[x])/(a + b*Cosh[x]),x]

[Out]

(-2*A*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (B*Log[a + b*Cosh[x]])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(46)=92\).

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00

method	result	size
default	\(-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\frac {2 \left (B a -B b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b -a -b \right )}{2 a -2 b}+\frac {2 b A \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}\)	\(112\)
risch	\(\frac {B x}{b}+\frac {2 x B \,a^{2} b}{-a^{2} b^{2}+b^{4}}-\frac {2 x B \,b^{3}}{-a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b A a -\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{b^{2} A}\right ) a^{2} B}{\left (a^{2}-b^{2}\right ) b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {b A a -\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{b^{2} A}\right ) B}{a^{2}-b^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b A a -\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{b^{2} A}\right ) \sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{\left (a^{2}-b^{2}\right ) b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b A a +\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{b^{2} A}\right ) a^{2} B}{\left (a^{2}-b^{2}\right ) b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {b A a +\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{b^{2} A}\right ) B}{a^{2}-b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {b A a +\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{b^{2} A}\right ) \sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}}}{\left (a^{2}-b^{2}\right ) b}\)	\(420\)

[In]

int((A+B*sinh(x))/(a+b*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

-B/b*ln(tanh(1/2*x)-1)+2/b*(1/2*(B*a-B*b)/(a-b)*ln(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b-a-b)+b*A/((a+b)*(a-b))^(1/2
)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2)))-B/b*ln(tanh(1/2*x)+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (46) = 92\).

Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 5.20 \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\left [\frac {\sqrt {a^{2} - b^{2}} A b \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - {\left (B a^{2} - B b^{2}\right )} x + {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} A b \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (B a^{2} - B b^{2}\right )} x - {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}\right ] \]

[In]

integrate((A+B*sinh(x))/(a+b*cosh(x)),x, algorithm="fricas")

[Out]

[(sqrt(a^2 - b^2)*A*b*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(x) + a*b)
*sinh(x) - 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh
(x) + a)*sinh(x) + b)) - (B*a^2 - B*b^2)*x + (B*a^2 - B*b^2)*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))))/(a^2*
b - b^3), -(2*sqrt(-a^2 + b^2)*A*b*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 - b^2)) + (B*a^2
- B*b^2)*x - (B*a^2 - B*b^2)*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))))/(a^2*b - b^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (48) = 96\).

Time = 12.98 (sec) , antiderivative size = 741, normalized size of antiderivative = 13.23 \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\begin {cases} \tilde {\infty } \left (2 A \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x - 2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {A \tanh {\left (\frac {x}{2} \right )}}{b} + \frac {B x}{b} - \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b} & \text {for}\: a = b \\- \frac {A}{b \tanh {\left (\frac {x}{2} \right )}} + \frac {B x}{b} - \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b} + \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{b} & \text {for}\: a = - b \\\frac {A x + B \cosh {\left (x \right )}}{a} & \text {for}\: b = 0 \\- \frac {A b \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {A b \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {B a x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \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 {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \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 {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {2 B a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {2 B b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} & \text {otherwise} \end {cases} \]

[In]

integrate((A+B*sinh(x))/(a+b*cosh(x)),x)

[Out]

Piecewise((zoo*(2*A*atan(tanh(x/2)) + B*x - 2*B*log(tanh(x/2) + 1) + B*log(tanh(x/2)**2 + 1)), Eq(a, 0) & Eq(b
, 0)), (A*tanh(x/2)/b + B*x/b - 2*B*log(tanh(x/2) + 1)/b, Eq(a, b)), (-A/(b*tanh(x/2)) + B*x/b - 2*B*log(tanh(
x/2) + 1)/b + 2*B*log(tanh(x/2))/b, Eq(a, -b)), ((A*x + B*cosh(x))/a, Eq(b, 0)), (-A*b*log(-sqrt(a/(a - b) + b
/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) + A*b*log(sqrt(a/(
a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) + B*a*x*
sqrt(a/(a - b) + b/(a - b))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) + B*a*sqrt(a/
(a - b) + b/(a - b))*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqr
t(a/(a - b) + b/(a - b))) + B*a*sqrt(a/(a - b) + b/(a - b))*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*
sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) - 2*B*a*sqrt(a/(a - b) + b/(a - b))*log(tanh(x
/2) + 1)/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) - B*b*x*sqrt(a/(a - b) + b/(a -
b))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) - B*b*sqrt(a/(a - b) + b/(a - b))*log
(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))
) - B*b*sqrt(a/(a - b) + b/(a - b))*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a -
b)) - b**2*sqrt(a/(a - b) + b/(a - b))) + 2*B*b*sqrt(a/(a - b) + b/(a - b))*log(tanh(x/2) + 1)/(a*b*sqrt(a/(a
- b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((A+B*sinh(x))/(a+b*cosh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\frac {2 \, A \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} - \frac {B x}{b} + \frac {B \log \left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}{b} \]

[In]

integrate((A+B*sinh(x))/(a+b*cosh(x)),x, algorithm="giac")

[Out]

2*A*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/sqrt(-a^2 + b^2) - B*x/b + B*log(b*e^(2*x) + 2*a*e^x + b)/b

Mupad [B] (verification not implemented)

Time = 3.86 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.52 \[ \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {A^2\,b^2\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}}{\left (A\,b^3-A\,a^2\,b\right )\,\sqrt {A^2}}+\frac {A^2\,a\,b\,\sqrt {b^2-a^2}}{\left (A\,b^3-A\,a^2\,b\right )\,\sqrt {A^2}}\right )\,\sqrt {A^2}}{\sqrt {b^2-a^2}}-\frac {B\,x}{b}+\frac {B\,b^3\,\ln \left (4\,A^2\,b+8\,A^2\,a\,{\mathrm {e}}^x+4\,A^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^4-a^2\,b^2}-\frac {B\,a^2\,b\,\ln \left (4\,A^2\,b+8\,A^2\,a\,{\mathrm {e}}^x+4\,A^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^4-a^2\,b^2} \]

[In]

int((A + B*sinh(x))/(a + b*cosh(x)),x)

[Out]

(2*atan((A^2*b^2*exp(x)*(b^2 - a^2)^(1/2))/((A*b^3 - A*a^2*b)*(A^2)^(1/2)) + (A^2*a*b*(b^2 - a^2)^(1/2))/((A*b
^3 - A*a^2*b)*(A^2)^(1/2)))*(A^2)^(1/2))/(b^2 - a^2)^(1/2) - (B*x)/b + (B*b^3*log(4*A^2*b + 8*A^2*a*exp(x) + 4
*A^2*b*exp(2*x)))/(b^4 - a^2*b^2) - (B*a^2*b*log(4*A^2*b + 8*A^2*a*exp(x) + 4*A^2*b*exp(2*x)))/(b^4 - a^2*b^2)