\(\int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [25]

Optimal result

Integrand size = 21, antiderivative size = 95 \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {\left (a^2+2 b^2\right ) x}{2 a^3}+\frac {2 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d} \]

[Out]

1/2*(a^2+2*b^2)*x/a^3-1/2*cosh(d*x+c)*(2*b-a*sinh(d*x+c))/a^2/d+2*b*arctanh((a-b*tanh(1/2*d*x+1/2*c))/(a^2+b^2
)^(1/2))*(a^2+b^2)^(1/2)/a^3/d

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2944, 2814, 2739, 632, 212} \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d}+\frac {2 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {x \left (a^2+2 b^2\right )}{2 a^3} \]

[In]

Int[Cosh[c + d*x]^2/(a + b*Csch[c + d*x]),x]

[Out]

((a^2 + 2*b^2)*x)/(2*a^3) + (2*b*Sqrt[a^2 + b^2]*ArcTanh[(a - b*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^3*d) -
(Cosh[c + d*x]*(2*b - a*Sinh[c + d*x]))/(2*a^2*d)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[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 2814

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

Rule 2944

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) -
a*d*p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(m + p)*(m + p +
1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{i b+i a \sinh (c+d x)} \, dx \\ & = -\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d}+\frac {\int \frac {-i a b+i \left (a^2+2 b^2\right ) \sinh (c+d x)}{i b+i a \sinh (c+d x)} \, dx}{2 a^2} \\ & = \frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d}-\frac {\left (i b \left (a^2+b^2\right )\right ) \int \frac {1}{i b+i a \sinh (c+d x)} \, dx}{a^3} \\ & = \frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{i b+2 a x+i b x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 d} \\ & = \frac {\left (a^2+2 b^2\right ) x}{2 a^3}-\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d}+\frac {\left (4 b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 a+2 i b \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 d} \\ & = \frac {\left (a^2+2 b^2\right ) x}{2 a^3}+\frac {2 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\cosh (c+d x) (2 b-a \sinh (c+d x))}{2 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15 \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {2 a^2 c+4 b^2 c+2 a^2 d x+4 b^2 d x+8 b \sqrt {-a^2-b^2} \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )-4 a b \cosh (c+d x)+a^2 \sinh (2 (c+d x))}{4 a^3 d} \]

[In]

Integrate[Cosh[c + d*x]^2/(a + b*Csch[c + d*x]),x]

[Out]

(2*a^2*c + 4*b^2*c + 2*a^2*d*x + 4*b^2*d*x + 8*b*Sqrt[-a^2 - b^2]*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 -
b^2]] - 4*a*b*Cosh[c + d*x] + a^2*Sinh[2*(c + d*x)])/(4*a^3*d)

Maple [A] (verified)

Time = 4.00 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.76

method	result	size
risch	\(\frac {x}{2 a}+\frac {x \,b^{2}}{a^{3}}+\frac {{\mathrm e}^{2 d x +2 c}}{8 d a}-\frac {b \,{\mathrm e}^{d x +c}}{2 d \,a^{2}}-\frac {b \,{\mathrm e}^{-d x -c}}{2 d \,a^{2}}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 d a}+\frac {\sqrt {a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{d x +c}+\frac {b +\sqrt {a^{2}+b^{2}}}{a}\right )}{d \,a^{3}}-\frac {\sqrt {a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{d x +c}-\frac {-b +\sqrt {a^{2}+b^{2}}}{a}\right )}{d \,a^{3}}\)	\(167\)
derivativedivides	\(\frac {-\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {-a +2 b}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3}}+\frac {2 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -2 b}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{3}}}{d}\)	\(189\)
default	\(\frac {-\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {-a +2 b}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3}}+\frac {2 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -2 b}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{3}}}{d}\)	\(189\)

[In]

int(cosh(d*x+c)^2/(a+b*csch(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/2*x/a+x/a^3*b^2+1/8/d/a*exp(2*d*x+2*c)-1/2*b/d/a^2*exp(d*x+c)-1/2*b/d/a^2*exp(-d*x-c)-1/8/d/a*exp(-2*d*x-2*c
)+(a^2+b^2)^(1/2)*b/d/a^3*ln(exp(d*x+c)+(b+(a^2+b^2)^(1/2))/a)-(a^2+b^2)^(1/2)*b/d/a^3*ln(exp(d*x+c)-(-b+(a^2+
b^2)^(1/2))/a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (87) = 174\).

Time = 0.28 (sec) , antiderivative size = 446, normalized size of antiderivative = 4.69 \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {a^{2} \cosh \left (d x + c\right )^{4} + a^{2} \sinh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (a^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) - a}\right ) - a^{2} + 4 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (a^{3} d \cosh \left (d x + c\right )^{2} + 2 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{3} d \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="fricas")

[Out]

1/8*(a^2*cosh(d*x + c)^4 + a^2*sinh(d*x + c)^4 + 4*(a^2 + 2*b^2)*d*x*cosh(d*x + c)^2 - 4*a*b*cosh(d*x + c)^3 +
4*(a^2*cosh(d*x + c) - a*b)*sinh(d*x + c)^3 - 4*a*b*cosh(d*x + c) + 2*(3*a^2*cosh(d*x + c)^2 + 2*(a^2 + 2*b^2
)*d*x - 6*a*b*cosh(d*x + c))*sinh(d*x + c)^2 + 8*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh
(d*x + c)^2)*sqrt(a^2 + b^2)*log((a^2*cosh(d*x + c)^2 + a^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + a^2 + 2*b^
2 + 2*(a^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(a*cosh(d*x + c) + a*sinh(d*x + c) + b))/(a*
cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)*sinh(d*x + c) - a)) - a^2 +
4*(a^2*cosh(d*x + c)^3 + 2*(a^2 + 2*b^2)*d*x*cosh(d*x + c) - 3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c))/(a^3*
d*cosh(d*x + c)^2 + 2*a^3*d*cosh(d*x + c)*sinh(d*x + c) + a^3*d*sinh(d*x + c)^2)

Sympy [F]

\[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

[In]

integrate(cosh(d*x+c)**2/(a+b*csch(d*x+c)),x)

[Out]

Integral(cosh(c + d*x)**2/(a + b*csch(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.77 \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {{\left (4 \, b e^{\left (-d x - c\right )} - a\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a^{2} d} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} {\left (d x + c\right )}}{2 \, a^{3} d} - \frac {4 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a^{2} d} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3} d} \]

[In]

integrate(cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="maxima")

[Out]

-1/8*(4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) + 1/2*(a^2 + 2*b^2)*(d*x + c)/(a^3*d) - 1/8*(4*b*e^(-d*x -
c) + a*e^(-2*d*x - 2*c))/(a^2*d) - (a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c) -
b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.63 \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {\frac {4 \, {\left (a^{2} + 2 \, b^{2}\right )} {\left (d x + c\right )}}{a^{3}} + \frac {a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (d x + c\right )}}{a^{2}} - \frac {{\left (4 \, a b e^{\left (d x + c\right )} + a^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{a^{3}} - \frac {8 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}}}{8 \, d} \]

[In]

integrate(cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="giac")

[Out]

1/8*(4*(a^2 + 2*b^2)*(d*x + c)/a^3 + (a*e^(2*d*x + 2*c) - 4*b*e^(d*x + c))/a^2 - (4*a*b*e^(d*x + c) + a^2)*e^(
-2*d*x - 2*c)/a^3 - 8*(a^2*b + b^3)*log(abs(2*a*e^(d*x + c) + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^(d*x + c) + 2
*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3))/d

Mupad [B] (verification not implemented)

Time = 2.44 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.23 \[ \int \frac {\cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,a\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,a\,d}+\frac {x\,\left (a^2+2\,b^2\right )}{2\,a^3}-\frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,a^2\,d}-\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,a^2\,d}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{a^4}-\frac {2\,b\,\sqrt {a^2+b^2}\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^4}\right )\,\sqrt {a^2+b^2}}{a^3\,d}+\frac {b\,\ln \left (\frac {2\,b\,\sqrt {a^2+b^2}\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^4}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{a^4}\right )\,\sqrt {a^2+b^2}}{a^3\,d} \]

[In]

int(cosh(c + d*x)^2/(a + b/sinh(c + d*x)),x)

[Out]

exp(2*c + 2*d*x)/(8*a*d) - exp(- 2*c - 2*d*x)/(8*a*d) + (x*(a^2 + 2*b^2))/(2*a^3) - (b*exp(- c - d*x))/(2*a^2*
d) - (b*exp(c + d*x))/(2*a^2*d) - (b*log((2*b*exp(c + d*x)*(a^2 + b^2))/a^4 - (2*b*(a^2 + b^2)^(1/2)*(a - b*ex
p(c + d*x)))/a^4)*(a^2 + b^2)^(1/2))/(a^3*d) + (b*log((2*b*(a^2 + b^2)^(1/2)*(a - b*exp(c + d*x)))/a^4 + (2*b*
exp(c + d*x)*(a^2 + b^2))/a^4)*(a^2 + b^2)^(1/2))/(a^3*d)