3.4.60 \(\int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [360]

Optimal. Leaf size=122 \[ -\frac {b \left (a^2-b^2\right ) \text {ArcTan}(\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}+\frac {a b^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a b^2 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]

[Out]

-1/2*b*(a^2-b^2)*arctan(sinh(d*x+c))/(a^2+b^2)^2/d+a*b^2*ln(cosh(d*x+c))/(a^2+b^2)^2/d-a*b^2*ln(a+b*sinh(d*x+c
))/(a^2+b^2)^2/d-1/2*sech(d*x+c)^2*(a-b*sinh(d*x+c))/(a^2+b^2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2916, 12, 837, 815, 649, 209, 266} \begin {gather*} -\frac {b \left (a^2-b^2\right ) \text {ArcTan}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}-\frac {a b^2 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {a b^2 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-1/2*(b*(a^2 - b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (a*b^2*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) -
(a*b^2*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)^2*d) - (Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

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

Rule 266

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

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 2916

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

Rubi steps

\begin {align*} \int \frac {\text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {b^3 \text {Subst}\left (\int \frac {x}{b (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^2 \text {Subst}\left (\int \frac {x}{(a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {\text {Subst}\left (\int \frac {a b^2-b^2 x}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {\text {Subst}\left (\int \left (-\frac {2 a b^2}{\left (a^2+b^2\right ) (a+x)}+\frac {b^2 \left (-a^2+b^2+2 a x\right )}{\left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac {a b^2 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b^2 \text {Subst}\left (\int \frac {-a^2+b^2+2 a x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {a b^2 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {\left (a b^2\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}-\frac {\left (b^2 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {b \left (a^2-b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}+\frac {a b^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a b^2 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 105, normalized size = 0.86 \begin {gather*} \frac {2 b \left (\left (-a^2+b^2\right ) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a b (\log (\cosh (c+d x))-\log (a+b \sinh (c+d x)))\right )-a \left (a^2+b^2\right ) \text {sech}^2(c+d x)+b \left (a^2+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right )^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(2*b*((-a^2 + b^2)*ArcTan[Tanh[(c + d*x)/2]] + a*b*(Log[Cosh[c + d*x]] - Log[a + b*Sinh[c + d*x]])) - a*(a^2 +
b^2)*Sech[c + d*x]^2 + b*(a^2 + b^2)*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)^2*d)

________________________________________________________________________________________

Maple [A]
time = 1.32, size = 212, normalized size = 1.74

method	result	size
derivativedivides	\(\frac {-\frac {2 a \,b^{2} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}-\frac {2 \left (\frac {\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3}-a \,b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {b \left (-a b \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{d}\)	\(212\)
default	\(\frac {-\frac {2 a \,b^{2} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}-\frac {2 \left (\frac {\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3}-a \,b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {b \left (-a b \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{d}\)	\(212\)
risch	\(-\frac {2 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {2 a \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 a \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {a \,b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(449\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*a*b^2/(2*a^4+4*a^2*b^2+2*b^4)*ln(a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)-a)-2/(a^4+2*a^2*b^2+b
^4)*(((1/2*a^2*b+1/2*b^3)*tanh(1/2*d*x+1/2*c)^3+(-a^3-a*b^2)*tanh(1/2*d*x+1/2*c)^2+(-1/2*a^2*b-1/2*b^3)*tanh(1
/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^2+1)^2+1/2*b*(-a*b*ln(tanh(1/2*d*x+1/2*c)^2+1)+(a^2-b^2)*arctan(tanh(1/2*d
*x+1/2*c)))))

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 218, normalized size = 1.79 \begin {gather*} -\frac {a b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {a b^{2} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (a^{2} b - b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 926 vs. \(2 (119) = 238\).
time = 0.37, size = 926, normalized size = 7.59 \begin {gather*} \frac {{\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{3} - 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (2 \, a^{3} + 2 \, a b^{2} - 3 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} b - b^{3}\right )} \sinh \left (d x + c\right )^{4} + a^{2} b - b^{3} + 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b - b^{3} + 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right ) - {\left (a b^{2} \cosh \left (d x + c\right )^{4} + 4 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a b^{2} \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} \cosh \left (d x + c\right )^{2} + a b^{2} + 2 \, {\left (3 \, a b^{2} \cosh \left (d x + c\right )^{2} + a b^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} \cosh \left (d x + c\right )^{3} + a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left (a b^{2} \cosh \left (d x + c\right )^{4} + 4 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a b^{2} \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} \cosh \left (d x + c\right )^{2} + a b^{2} + 2 \, {\left (3 \, a b^{2} \cosh \left (d x + c\right )^{2} + a b^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} \cosh \left (d x + c\right )^{3} + a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - {\left (a^{2} b + b^{3} - 3 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d + 4 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

((a^2*b + b^3)*cosh(d*x + c)^3 + (a^2*b + b^3)*sinh(d*x + c)^3 - 2*(a^3 + a*b^2)*cosh(d*x + c)^2 - (2*a^3 + 2*
a*b^2 - 3*(a^2*b + b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - ((a^2*b - b^3)*cosh(d*x + c)^4 + 4*(a^2*b - b^3)*cosh
(d*x + c)*sinh(d*x + c)^3 + (a^2*b - b^3)*sinh(d*x + c)^4 + a^2*b - b^3 + 2*(a^2*b - b^3)*cosh(d*x + c)^2 + 2*
(a^2*b - b^3 + 3*(a^2*b - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^2*b - b^3)*cosh(d*x + c)^3 + (a^2*b -
b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (a^2*b + b^3)*cosh(d*x + c) - (a*b^
2*cosh(d*x + c)^4 + 4*a*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^2*sinh(d*x + c)^4 + 2*a*b^2*cosh(d*x + c)^2 +
a*b^2 + 2*(3*a*b^2*cosh(d*x + c)^2 + a*b^2)*sinh(d*x + c)^2 + 4*(a*b^2*cosh(d*x + c)^3 + a*b^2*cosh(d*x + c))*
sinh(d*x + c))*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + (a*b^2*cosh(d*x + c)^4 + 4*a*b^2
*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^2*sinh(d*x + c)^4 + 2*a*b^2*cosh(d*x + c)^2 + a*b^2 + 2*(3*a*b^2*cosh(d*x
+ c)^2 + a*b^2)*sinh(d*x + c)^2 + 4*(a*b^2*cosh(d*x + c)^3 + a*b^2*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d
*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - (a^2*b + b^3 - 3*(a^2*b + b^3)*cosh(d*x + c)^2 + 4*(a^3 + a*b^2)*co
sh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x
+ c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2
+ 2*(3*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d)*sinh(d*x + c)^2 + (a^4 + 2*a^2*
b^2 + b^4)*d + 4*((a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c))*sinh(d*
x + c))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (c + d x \right )} \operatorname {sech}^{2}{\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(sech(d*x+c)**2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(tanh(c + d*x)*sech(c + d*x)**2/(a + b*sinh(c + d*x)), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (119) = 238\).
time = 0.45, size = 286, normalized size = 2.34 \begin {gather*} -\frac {\frac {4 \, a b^{3} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {2 \, a b^{2} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{2} b - b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, a^{3} + 8 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

-1/4*(4*a*b^3*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^4*b + 2*a^2*b^3 + b^5) - 2*a*b^2*log((e^(d*x +
c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) + (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(
a^2*b - b^3)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a*b^2*(e^(d*x + c) - e^(-d*x - c))^2 - 2*a^2*b*(e^(d*x + c) - e^(-d*
x - c)) - 2*b^3*(e^(d*x + c) - e^(-d*x - c)) + 4*a^3 + 8*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^(d*x + c) - e^(-d
*x - c))^2 + 4)))/d

________________________________________________________________________________________

Mupad [B]
time = 1.93, size = 337, normalized size = 2.76 \begin {gather*} \frac {\frac {2\,a}{d\,\left (a^2+b^2\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,\left (a^3+a\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+b^3\right )}{d\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}+\frac {b\,\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}{2\,\left (-1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b+1{}\mathrm {i}\,d\,b^2\right )}-\frac {a\,b^2\,\ln \left (b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-14\,a^2\,b^4-a^4\,b^2-b^6+28\,a^3\,b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+14\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a\,b^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+2\,a^5\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^4+2\,d\,a^2\,b^2+d\,b^4}+\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (-d\,a^2+2{}\mathrm {i}\,d\,a\,b+d\,b^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*a)/(d*(a^2 + b^2)) - (2*b*exp(c + d*x))/(d*(a^2 + b^2)))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - ((2
*(a*b^2 + a^3))/(d*(a^2 + b^2)^2) - (exp(c + d*x)*(a^2*b + b^3))/(d*(a^2 + b^2)^2))/(exp(2*c + 2*d*x) + 1) + (
b*log(exp(c + d*x) + 1i)*1i)/(2*(b^2*d - a^2*d + a*b*d*2i)) + (b*log(exp(c + d*x)*1i + 1))/(2*(b^2*d*1i - a^2*
d*1i + 2*a*b*d)) - (a*b^2*log(b^6*exp(2*c)*exp(2*d*x) - 14*a^2*b^4 - a^4*b^2 - b^6 + 28*a^3*b^3*exp(d*x)*exp(c
) + 14*a^2*b^4*exp(2*c)*exp(2*d*x) + a^4*b^2*exp(2*c)*exp(2*d*x) + 2*a*b^5*exp(d*x)*exp(c) + 2*a^5*b*exp(d*x)*
exp(c)))/(a^4*d + b^4*d + 2*a^2*b^2*d)

________________________________________________________________________________________