3.5.51 \(\int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [451]

Optimal. Leaf size=243 \[ -\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2} \]

[Out]

-f*arctanh(cosh(d*x+c))/a/d^2-(f*x+e)*csch(d*x+c)/a/d-b*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^2/d+b*(f*x+e)*ln(1+b*ex
p(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d+b*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d-1/2*b*f*polylog(2,e
xp(2*d*x+2*c))/a^2/d^2+b*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2+b*f*polylog(2,-b*exp(d*x+c)/(a
+(a^2+b^2)^(1/2)))/a^2/d^2

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Rubi [A]
time = 0.33, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5706, 5560, 3855, 5688, 3797, 2221, 2317, 2438, 5680} \begin {gather*} \frac {b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d}+\frac {b (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d}-\frac {b f \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-((f*ArcTanh[Cosh[c + d*x]])/(a*d^2)) - ((e + f*x)*Csch[c + d*x])/(a*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))
/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) - (b*(
e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d
^2) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*PolyLog[2, E^(2*(c + d*x))])
/(2*a^2*d^2)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3855

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

Rule 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5688

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[c +
d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 5706

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)*(Coth[c + d*x]^n/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x) \text {csch}(c+d x)}{a d}-\frac {b \int (e+f x) \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f \int \text {csch}(c+d x) \, dx}{a d}\\ &=-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {(2 b) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2}\\ &=-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {(b f) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {(b f) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d}\\ &=-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {(b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}\\ &=-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}\\ \end {align*}

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Mathematica [A]
time = 1.34, size = 416, normalized size = 1.71 \begin {gather*} \frac {-2 b c^2 f-4 b c d f x-2 b d^2 f x^2-a d e \coth \left (\frac {1}{2} (c+d x)\right )-a d f x \coth \left (\frac {1}{2} (c+d x)\right )-2 b c f \log \left (1-e^{-2 (c+d x)}\right )-2 b d f x \log \left (1-e^{-2 (c+d x)}\right )+2 b c f \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 b d f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 b c f \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 b d f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 b d e \log (\sinh (c+d x))+2 b c f \log (\sinh (c+d x))+2 b d e \log (a+b \sinh (c+d x))-2 b c f \log (a+b \sinh (c+d x))+2 a f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+b f \text {PolyLog}\left (2,e^{-2 (c+d x)}\right )+2 b f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 b f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+a d e \tanh \left (\frac {1}{2} (c+d x)\right )+a d f x \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(-2*b*c^2*f - 4*b*c*d*f*x - 2*b*d^2*f*x^2 - a*d*e*Coth[(c + d*x)/2] - a*d*f*x*Coth[(c + d*x)/2] - 2*b*c*f*Log[
1 - E^(-2*(c + d*x))] - 2*b*d*f*x*Log[1 - E^(-2*(c + d*x))] + 2*b*c*f*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])] + 2*b*d*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*b*c*f*Log[1 + (b*E^(c + d*x))/(a + Sqrt[
a^2 + b^2])] + 2*b*d*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*b*d*e*Log[Sinh[c + d*x]] + 2*b*c*f
*Log[Sinh[c + d*x]] + 2*b*d*e*Log[a + b*Sinh[c + d*x]] - 2*b*c*f*Log[a + b*Sinh[c + d*x]] + 2*a*f*Log[Tanh[(c
+ d*x)/2]] + b*f*PolyLog[2, E^(-2*(c + d*x))] + 2*b*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*b
*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + a*d*e*Tanh[(c + d*x)/2] + a*d*f*x*Tanh[(c + d*x)/2])
/(2*a^2*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(229)=458\).
time = 3.19, size = 528, normalized size = 2.17

method	result	size
risch	\(-\frac {2 \left (f x +e \right ) {\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {b f \dilog \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {b f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{2}}+\frac {b f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{2}}+\frac {b f \dilog \left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}-\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {b e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,a^{2}}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{2}}+\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{2}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{2}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{2}}-\frac {b f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} a^{2}}+\frac {b f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}\)	\(528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2/d*(f*x+e)/a*exp(d*x+c)/(exp(2*d*x+2*c)-1)-1/d^2/a^2*b*f*dilog(exp(d*x+c)+1)+1/d^2/a^2*b*f*dilog((-b*exp(d*x
+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2/a^2*b*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)
^(1/2)))+1/d^2/a^2*b*f*dilog(exp(d*x+c))-1/a^2/d*b*e*ln(exp(d*x+c)+1)+1/d/a^2*b*e*ln(b*exp(2*d*x+2*c)+2*a*exp(
d*x+c)-b)-1/a^2/d*b*e*ln(exp(d*x+c)-1)-1/a^2/d*b*f*ln(exp(d*x+c)+1)*x+1/d/a^2*b*f*ln((-b*exp(d*x+c)+(a^2+b^2)^
(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d^2/a^2*b*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/
d/a^2*b*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2/a^2*b*f*ln((b*exp(d*x+c)+(a^2+b^2)^
(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d^2/a^2*b*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/a^2/d^2*b*f*c*ln(exp(d
*x+c)-1)-1/a/d^2*f*ln(exp(d*x+c)+1)+1/a/d^2*f*ln(exp(d*x+c)-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(2*b*d*integrate(1/2*x/(a^2*d*e^(d*x + c) + a^2*d), x) - 2*b*d*integrate(1/2*x/(a^2*d*e^(d*x + c) - a^2*d), x)
+ a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) + 1)/(a^2*d^2)) - a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) - 1)/(a
^2*d^2)) - 2*x*e^(d*x + c)/(a*d*e^(2*d*x + 2*c) - a*d) - 2*integrate((a*b*x*e^(d*x + c) - b^2*x)/(a^2*b*e^(2*d
*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), x))*f + (2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) + b*log(-2*a*e^(-
d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d)
)*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1339 vs. \(2 (230) = 460\).
time = 0.40, size = 1339, normalized size = 5.51 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*(a*d*f*x + a*d*cosh(1) + a*d*sinh(1))*cosh(d*x + c) - (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x
+ c) + b*f*sinh(d*x + c)^2 - b*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c
))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x
+ c)^2 - b*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)
/b^2) - b)/b + 1) + (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 - b*f)*dilo
g(cosh(d*x + c) + sinh(d*x + c)) + (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c
)^2 - b*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - (b*c*f - b*d*cosh(1) - (b*c*f - b*d*cosh(1) - b*d*sinh(1))*
cosh(d*x + c)^2 - b*d*sinh(1) - 2*(b*c*f - b*d*cosh(1) - b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*c*f - b
*d*cosh(1) - b*d*sinh(1))*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^
2) + 2*a) - (b*c*f - b*d*cosh(1) - (b*c*f - b*d*cosh(1) - b*d*sinh(1))*cosh(d*x + c)^2 - b*d*sinh(1) - 2*(b*c*
f - b*d*cosh(1) - b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) - (b*c*f - b*d*cosh(1) - b*d*sinh(1))*sinh(d*x + c)
^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*d*f*x + b*c*f - (b*d*f*x
+ b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x + b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x + b*c*f)*sinh(d*x + c)^
2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b)
+ (b*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x + b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b
*d*f*x + b*c*f)*sinh(d*x + c)^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))
*sqrt((a^2 + b^2)/b^2) - b)/b) - (b*d*f*x + b*d*cosh(1) - (b*d*f*x + b*d*cosh(1) + b*d*sinh(1) + a*f)*cosh(d*x
+ c)^2 + b*d*sinh(1) - 2*(b*d*f*x + b*d*cosh(1) + b*d*sinh(1) + a*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x +
b*d*cosh(1) + b*d*sinh(1) + a*f)*sinh(d*x + c)^2 + a*f)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (b*d*cosh(1)
- (b*d*cosh(1) + b*d*sinh(1) - (b*c + a)*f)*cosh(d*x + c)^2 + b*d*sinh(1) - 2*(b*d*cosh(1) + b*d*sinh(1) - (b
*c + a)*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*cosh(1) + b*d*sinh(1) - (b*c + a)*f)*sinh(d*x + c)^2 - (b*c + a)
*f)*log(cosh(d*x + c) + sinh(d*x + c) - 1) - (b*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x
+ b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x + b*c*f)*sinh(d*x + c)^2)*log(-cosh(d*x + c) - sinh(d*x + c)
+ 1) + 2*(a*d*f*x + a*d*cosh(1) + a*d*sinh(1))*sinh(d*x + c))/(a^2*d^2*cosh(d*x + c)^2 + 2*a^2*d^2*cosh(d*x +
c)*sinh(d*x + c) + a^2*d^2*sinh(d*x + c)^2 - a^2*d^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right ) \coth {\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)*coth(c + d*x)*csch(c + d*x)/(a + b*sinh(c + d*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(c + d*x)*(e + f*x))/(sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

int((coth(c + d*x)*(e + f*x))/(sinh(c + d*x)*(a + b*sinh(c + d*x))), x)

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