\(\int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx\) [252]

Optimal result

Integrand size = 15, antiderivative size = 58 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=-\frac {B \text {arctanh}(\cosh (x))}{a}-\frac {2 (a A-b B) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.33 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {\frac {2 (a A-b B) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+B \left (-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )}{a} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {3042, 3307, 26, 26, 3042, 26, 3480, 26, 3042, 26, 3139, 1083, 219, 4257}

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.

\(\displaystyle \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+i B \csc (i x)}{a-i b \sin (i x)}dx\)

\(\Big \downarrow \) 3307

\(\displaystyle \int -\frac {i \text {csch}(x) (i A \sinh (x)+i B)}{a+b \sinh (x)}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {i \text {csch}(x) (B+A \sinh (x))}{a+b \sinh (x)}dx\)

\(\Big \downarrow \) 26

\(\displaystyle \int \frac {\text {csch}(x) (A \sinh (x)+B)}{a+b \sinh (x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i (B-i A \sin (i x))}{\sin (i x) (a-i b \sin (i x))}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {B-i A \sin (i x)}{\sin (i x) (a-i b \sin (i x))}dx\)

\(\Big \downarrow \) 3480

\(\displaystyle i \left (\frac {B \int -i \text {csch}(x)dx}{a}-\frac {i (a A-b B) \int \frac {1}{a+b \sinh (x)}dx}{a}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (-\frac {i (a A-b B) \int \frac {1}{a+b \sinh (x)}dx}{a}-\frac {i B \int \text {csch}(x)dx}{a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (-\frac {i (a A-b B) \int \frac {1}{a-i b \sin (i x)}dx}{a}-\frac {i B \int i \csc (i x)dx}{a}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {B \int \csc (i x)dx}{a}-\frac {i (a A-b B) \int \frac {1}{a-i b \sin (i x)}dx}{a}\right )\)

\(\Big \downarrow \) 3139

\(\displaystyle i \left (\frac {B \int \csc (i x)dx}{a}-\frac {2 i (a A-b B) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle i \left (\frac {4 i (a A-b B) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}+\frac {B \int \csc (i x)dx}{a}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle i \left (\frac {B \int \csc (i x)dx}{a}+\frac {2 i (a A-b B) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\right )\)

\(\Big \downarrow \) 4257

\(\displaystyle i \left (\frac {2 i (a A-b B) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {i B \text {arctanh}(\cosh (x))}{a}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*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 3307

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)])^(m_.), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*((d + c*Sin[e + 
 f*x])^n/Sin[e + f*x]^n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ 
[n]
 

rule 3480

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b 
- a*B)/(b*c - a*d)   Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ 
(b*c - a*d)   Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f 
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}-\frac {\left (-2 A a +2 b B \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}\)	\(58\)
parts	\(\frac {2 A \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}-\frac {2 B b \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\)	\(86\)
risch	\(\frac {B \ln \left ({\mathrm e}^{x}-1\right )}{a}-\frac {B \ln \left ({\mathrm e}^{x}+1\right )}{a}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right ) A}{\sqrt {a^{2}+b^{2}}}-\frac {B b \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right ) A}{\sqrt {a^{2}+b^{2}}}+\frac {B b \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}\)	\(225\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (54) = 108\).

Time = 0.33 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.97 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (A a - B b\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) + {\left (B a^{2} + B b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (B a^{2} + B b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{3} + a b^{2}} \] Input:

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

Output:

-((A*a - B*b)*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*c 
osh(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)*log(cosh(x) + sinh(x) + 1) 
- (B*a^2 + B*b^2)*log(cosh(x) + sinh(x) - 1))/(a^3 + a*b^2)
 

Sympy [F]

\[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=\int \frac {A + B \operatorname {csch}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \] Input:

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

Output:

Integral((A + B*csch(x))/(a + b*sinh(x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (54) = 108\).

Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.43 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=-B {\left (\frac {b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a}\right )} + \frac {A \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \] Input:

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

Output:

-B*(b*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2) 
))/(sqrt(a^2 + b^2)*a) + log(e^(-x) + 1)/a - log(e^(-x) - 1)/a) + A*log((b 
*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/sqrt(a^2 
+ b^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.55 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=-\frac {B \log \left (e^{x} + 1\right )}{a} + \frac {B \log \left ({\left | e^{x} - 1 \right |}\right )}{a} + \frac {{\left (A a - B b\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a} \] Input:

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

Output:

-B*log(e^x + 1)/a + B*log(abs(e^x - 1))/a + (A*a - B*b)*log(abs(2*b*e^x + 
2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 
 + b^2)*a)
 

Mupad [B] (verification not implemented)

Time = 2.16 (sec) , antiderivative size = 539, normalized size of antiderivative = 9.29 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {B\,\ln \left ({\mathrm {e}}^x-1\right )}{a}-\frac {B\,\ln \left ({\mathrm {e}}^x+1\right )}{a}-\frac {\ln \left (\frac {\left (A\,a-B\,b\right )\,\left (\frac {32\,\left (A^2\,a^2\,b-2\,A\,B\,a\,b^2-4\,{\mathrm {e}}^x\,B^2\,a^3+2\,B^2\,a^2\,b-3\,{\mathrm {e}}^x\,B^2\,a\,b^2+2\,B^2\,b^3\right )}{b^5}-\frac {\left (A\,a-B\,b\right )\,\left (\frac {32\,a^2\,\left (2\,B\,b^2+4\,A\,a^2\,{\mathrm {e}}^x+A\,b^2\,{\mathrm {e}}^x-2\,A\,a\,b-3\,B\,a\,b\,{\mathrm {e}}^x\right )}{b^5}+\frac {32\,a\,\left (A\,a-B\,b\right )\,\left (-4\,{\mathrm {e}}^x\,a^3+3\,a^2\,b-3\,{\mathrm {e}}^x\,a\,b^2+2\,b^3\right )}{b^5\,\sqrt {a^2+b^2}}\right )}{a\,\sqrt {a^2+b^2}}\right )}{a\,\sqrt {a^2+b^2}}+\frac {32\,B\,\left (A\,a-B\,b\right )\,\left (A\,b\,{\mathrm {e}}^x-2\,B\,b+4\,B\,a\,{\mathrm {e}}^x\right )}{b^5}\right )\,\left (A\,a-B\,b\right )\,\sqrt {a^2+b^2}}{a^3+a\,b^2}+\frac {\ln \left (\frac {32\,B\,\left (A\,a-B\,b\right )\,\left (A\,b\,{\mathrm {e}}^x-2\,B\,b+4\,B\,a\,{\mathrm {e}}^x\right )}{b^5}-\frac {\left (A\,a-B\,b\right )\,\left (\frac {32\,\left (A^2\,a^2\,b-2\,A\,B\,a\,b^2-4\,{\mathrm {e}}^x\,B^2\,a^3+2\,B^2\,a^2\,b-3\,{\mathrm {e}}^x\,B^2\,a\,b^2+2\,B^2\,b^3\right )}{b^5}+\frac {\left (A\,a-B\,b\right )\,\left (\frac {32\,a^2\,\left (2\,B\,b^2+4\,A\,a^2\,{\mathrm {e}}^x+A\,b^2\,{\mathrm {e}}^x-2\,A\,a\,b-3\,B\,a\,b\,{\mathrm {e}}^x\right )}{b^5}-\frac {32\,a\,\left (A\,a-B\,b\right )\,\left (-4\,{\mathrm {e}}^x\,a^3+3\,a^2\,b-3\,{\mathrm {e}}^x\,a\,b^2+2\,b^3\right )}{b^5\,\sqrt {a^2+b^2}}\right )}{a\,\sqrt {a^2+b^2}}\right )}{a\,\sqrt {a^2+b^2}}\right )\,\left (A\,a-B\,b\right )\,\sqrt {a^2+b^2}}{a^3+a\,b^2} \] Input:

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

Output:

(B*log(exp(x) - 1))/a - (B*log(exp(x) + 1))/a - (log(((A*a - B*b)*((32*(2* 
B^2*b^3 + A^2*a^2*b + 2*B^2*a^2*b - 4*B^2*a^3*exp(x) - 3*B^2*a*b^2*exp(x) 
- 2*A*B*a*b^2))/b^5 - ((A*a - B*b)*((32*a^2*(2*B*b^2 + 4*A*a^2*exp(x) + A* 
b^2*exp(x) - 2*A*a*b - 3*B*a*b*exp(x)))/b^5 + (32*a*(A*a - B*b)*(3*a^2*b + 
 2*b^3 - 4*a^3*exp(x) - 3*a*b^2*exp(x)))/(b^5*(a^2 + b^2)^(1/2))))/(a*(a^2 
 + b^2)^(1/2))))/(a*(a^2 + b^2)^(1/2)) + (32*B*(A*a - B*b)*(A*b*exp(x) - 2 
*B*b + 4*B*a*exp(x)))/b^5)*(A*a - B*b)*(a^2 + b^2)^(1/2))/(a*b^2 + a^3) + 
(log((32*B*(A*a - B*b)*(A*b*exp(x) - 2*B*b + 4*B*a*exp(x)))/b^5 - ((A*a - 
B*b)*((32*(2*B^2*b^3 + A^2*a^2*b + 2*B^2*a^2*b - 4*B^2*a^3*exp(x) - 3*B^2* 
a*b^2*exp(x) - 2*A*B*a*b^2))/b^5 + ((A*a - B*b)*((32*a^2*(2*B*b^2 + 4*A*a^ 
2*exp(x) + A*b^2*exp(x) - 2*A*a*b - 3*B*a*b*exp(x)))/b^5 - (32*a*(A*a - B* 
b)*(3*a^2*b + 2*b^3 - 4*a^3*exp(x) - 3*a*b^2*exp(x)))/(b^5*(a^2 + b^2)^(1/ 
2))))/(a*(a^2 + b^2)^(1/2))))/(a*(a^2 + b^2)^(1/2)))*(A*a - B*b)*(a^2 + b^ 
2)^(1/2))/(a*b^2 + a^3)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.24 \[ \int \frac {A+B \text {csch}(x)}{a+b \sinh (x)} \, dx=\frac {2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} i -2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{2} i +\mathrm {log}\left (e^{x}-1\right ) a^{2} b +\mathrm {log}\left (e^{x}-1\right ) b^{3}-\mathrm {log}\left (e^{x}+1\right ) a^{2} b -\mathrm {log}\left (e^{x}+1\right ) b^{3}}{a \left (a^{2}+b^{2}\right )} \] Input:

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

Output:

(2*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*i - 2*s 
qrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*b**2*i + log(e** 
x - 1)*a**2*b + log(e**x - 1)*b**3 - log(e**x + 1)*a**2*b - log(e**x + 1)* 
b**3)/(a*(a**2 + b**2))