3.474 \(\int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=180 \[ -\frac {a \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {a \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {\text {sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 d \left (a^2+b^2\right )^2}-\frac {b \log (\sinh (c+d x))}{a^2 d}-\frac {\text {csch}(c+d x)}{a d} \]

[Out]

-1/2*a*arctan(sinh(d*x+c))/(a^2+b^2)/d-a*(a^2+2*b^2)*arctan(sinh(d*x+c))/(a^2+b^2)^2/d-csch(d*x+c)/a/d+b*(a^2+
2*b^2)*ln(cosh(d*x+c))/(a^2+b^2)^2/d-b*ln(sinh(d*x+c))/a^2/d+b^5*ln(a+b*sinh(d*x+c))/a^2/(a^2+b^2)^2/d-1/2*sec
h(d*x+c)^2*(b+a*sinh(d*x+c))/(a^2+b^2)/d

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Rubi [A]  time = 0.26, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2837, 12, 894, 639, 203, 635, 260} \[ \frac {b^5 \log (a+b \sinh (c+d x))}{a^2 d \left (a^2+b^2\right )^2}-\frac {a \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {a \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {\text {sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )}-\frac {b \log (\sinh (c+d x))}{a^2 d}-\frac {\text {csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-(a*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) - (a*(a^2 + 2*b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) - Csc
h[c + d*x]/(a*d) + (b*(a^2 + 2*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (
b^5*Log[a + b*Sinh[c + d*x]])/(a^2*(a^2 + b^2)^2*d) - (Sech[c + d*x]^2*(b + a*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 203

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

Rule 260

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 635

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 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 894

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

Rule 2837

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*x)/b)^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 {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {b^2}{x^2 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^5 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^5 \operatorname {Subst}\left (\int \left (\frac {1}{a b^4 x^2}-\frac {1}{a^2 b^4 x}+\frac {1}{a^2 \left (a^2+b^2\right )^2 (a+x)}+\frac {-a+x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}-\frac {\left (a^2+2 b^2\right ) (a-x)}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {-a+x}{\left (b^2+x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {\left (b \left (a^2+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {a-x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\left (b \left (a^2+2 b^2\right )\right ) \operatorname {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 (a b \left (a^2+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {a \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {csch}(c+d x)}{a d}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [C]  time = 0.94, size = 227, normalized size = 1.26 \[ -\frac {\text {csch}(c+d x) (a+b \sinh (c+d x)) \left (\frac {b \text {sech}^2(c+d x)}{a^2+b^2}-\frac {(b+i a) \left (a^2+2 b^2\right ) \log (-\sinh (c+d x)+i)}{\left (a^2+b^2\right )^2}+\frac {(-b+i a) \left (a^2+2 b^2\right ) \log (\sinh (c+d x)+i)}{\left (a^2+b^2\right )^2}+\frac {a \tan ^{-1}(\sinh (c+d x))}{a^2+b^2}+\frac {a \tanh (c+d x) \text {sech}(c+d x)}{a^2+b^2}-\frac {2 b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {2 b \log (\sinh (c+d x))}{a^2}+\frac {2 \text {csch}(c+d x)}{a}\right )}{2 d (a \text {csch}(c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-1/2*(Csch[c + d*x]*(a + b*Sinh[c + d*x])*((a*ArcTan[Sinh[c + d*x]])/(a^2 + b^2) + (2*Csch[c + d*x])/a - ((I*a
+ b)*(a^2 + 2*b^2)*Log[I - Sinh[c + d*x]])/(a^2 + b^2)^2 + (2*b*Log[Sinh[c + d*x]])/a^2 + ((I*a - b)*(a^2 + 2
*b^2)*Log[I + Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*b^5*Log[a + b*Sinh[c + d*x]])/(a^2*(a^2 + b^2)^2) + (b*Sech[c
+ d*x]^2)/(a^2 + b^2) + (a*Sech[c + d*x]*Tanh[c + d*x])/(a^2 + b^2)))/(d*(b + a*Csch[c + d*x]))

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fricas [B]  time = 2.05, size = 2568, normalized size = 14.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^5 + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*sinh(d*x + c)^5 + 2*(a^4*b + a
^2*b^3)*cosh(d*x + c)^4 + (2*a^4*b + 2*a^2*b^3 + 5*(3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^
4 + 2*(a^5 + 3*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^3 + 2*(a^5 + 3*a^3*b^2 + 2*a*b^4 + 5*(3*a^5 + 5*a^3*b^2 + 2*a*
b^4)*cosh(d*x + c)^2 + 4*(a^4*b + a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(a^4*b + a^2*b^3)*cosh(d*x + c)^
2 - 2*(a^4*b + a^2*b^3 - 5*(3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^3 - 6*(a^4*b + a^2*b^3)*cosh(d*x + c)^2
- 3*(a^5 + 3*a^3*b^2 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^6 + 6*(3*
a^5 + 5*a^3*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^5 + 5*a^3*b^2)*sinh(d*x + c)^6 - 3*a^5 - 5*a^3*b^2 + (3*
a^5 + 5*a^3*b^2)*cosh(d*x + c)^4 + (3*a^5 + 5*a^3*b^2 + 15*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^
4 + 4*(5*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^3 + (3*a^5 + 5*a^3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (3*a^5 + 5
*a^3*b^2)*cosh(d*x + c)^2 - (3*a^5 + 5*a^3*b^2 - 15*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^4 - 6*(3*a^5 + 5*a^3*b^2
)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(3*a^5 + 5*a^3*b^2)*cosh(d*x + c)^5 + 2*(3*a^5 + 5*a^3*b^2)*cosh(d*x
+ c)^3 - (3*a^5 + 5*a^3*b^2)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + (3*a^5 + 5
*a^3*b^2 + 2*a*b^4)*cosh(d*x + c) - (b^5*cosh(d*x + c)^6 + 6*b^5*cosh(d*x + c)*sinh(d*x + c)^5 + b^5*sinh(d*x
+ c)^6 + b^5*cosh(d*x + c)^4 - b^5*cosh(d*x + c)^2 - b^5 + (15*b^5*cosh(d*x + c)^2 + b^5)*sinh(d*x + c)^4 + 4*
(5*b^5*cosh(d*x + c)^3 + b^5*cosh(d*x + c))*sinh(d*x + c)^3 + (15*b^5*cosh(d*x + c)^4 + 6*b^5*cosh(d*x + c)^2
- b^5)*sinh(d*x + c)^2 + 2*(3*b^5*cosh(d*x + c)^5 + 2*b^5*cosh(d*x + c)^3 - b^5*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^4*b + 2*a^2*b^3)*cosh(d*x + c)^6 + 6*(a^4*b
+ 2*a^2*b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4*b + 2*a^2*b^3)*sinh(d*x + c)^6 - a^4*b - 2*a^2*b^3 + (a^4*b
+ 2*a^2*b^3)*cosh(d*x + c)^4 + (a^4*b + 2*a^2*b^3 + 15*(a^4*b + 2*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
4*(5*(a^4*b + 2*a^2*b^3)*cosh(d*x + c)^3 + (a^4*b + 2*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^4*b + 2*a^
2*b^3)*cosh(d*x + c)^2 - (a^4*b + 2*a^2*b^3 - 15*(a^4*b + 2*a^2*b^3)*cosh(d*x + c)^4 - 6*(a^4*b + 2*a^2*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(a^4*b + 2*a^2*b^3)*cosh(d*x + c)^5 + 2*(a^4*b + 2*a^2*b^3)*cosh(d*x +
c)^3 - (a^4*b + 2*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c)))
+ ((a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c)^6 + 6*(a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^
4*b + 2*a^2*b^3 + b^5)*sinh(d*x + c)^6 - a^4*b - 2*a^2*b^3 - b^5 + (a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c)^4 +
(a^4*b + 2*a^2*b^3 + b^5 + 15*(a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^4*b + 2*a^
2*b^3 + b^5)*cosh(d*x + c)^3 + (a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^4*b + 2*a^2*b^3 +
b^5)*cosh(d*x + c)^2 - (a^4*b + 2*a^2*b^3 + b^5 - 15*(a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c)^4 - 6*(a^4*b + 2
*a^2*b^3 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c)^5 + 2*(a^4*b +
2*a^2*b^3 + b^5)*cosh(d*x + c)^3 - (a^4*b + 2*a^2*b^3 + b^5)*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c
)/(cosh(d*x + c) - sinh(d*x + c))) + (3*a^5 + 5*a^3*b^2 + 2*a*b^4 + 5*(3*a^5 + 5*a^3*b^2 + 2*a*b^4)*cosh(d*x +
c)^4 + 8*(a^4*b + a^2*b^3)*cosh(d*x + c)^3 + 6*(a^5 + 3*a^3*b^2 + 2*a*b^4)*cosh(d*x + c)^2 - 4*(a^4*b + a^2*b
^3)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c)^6 + 6*(a^6 + 2*a^4*b^2 + a^2*b^
4)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d*sinh(d*x + c)^6 + (a^6 + 2*a^4*b^2 + a^2*b^
4)*d*cosh(d*x + c)^4 + (15*(a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c)^2 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*sinh
(d*x + c)^4 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c)^2 + 4*(5*(a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c)
^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x
+ c)^4 + 6*(a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c)^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*sinh(d*x + c)^2 - (
a^6 + 2*a^4*b^2 + a^2*b^4)*d + 2*(3*(a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c)^5 + 2*(a^6 + 2*a^4*b^2 + a^2*b
^4)*d*cosh(d*x + c)^3 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c))

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giac [B]  time = 0.70, size = 458, normalized size = 2.54 \[ \frac {\frac {12 \, b^{6} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{3} + 5 \, a b^{2}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {12 \, b \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{2}} + \frac {4 \, {\left (b^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 9 \, a^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 15 \, a^{3} b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 6 \, a b^{4} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 6 \, a^{4} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 6 \, a^{2} b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, b^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, a^{5} - 48 \, a^{3} b^{2} - 24 \, a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}\right )}}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

1/12*(12*b^6*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^6*b + 2*a^4*b^3 + a^2*b^5) - 3*(pi + 2*arctan(1
/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a^3 + 5*a*b^2)/(a^4 + 2*a^2*b^2 + b^4) + 6*(a^2*b + 2*b^3)*log((e^(
d*x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) - 12*b*log(abs(e^(d*x + c) - e^(-d*x - c)))/a^2 + 4*(b
^5*(e^(d*x + c) - e^(-d*x - c))^3 - 9*a^5*(e^(d*x + c) - e^(-d*x - c))^2 - 15*a^3*b^2*(e^(d*x + c) - e^(-d*x -
c))^2 - 6*a*b^4*(e^(d*x + c) - e^(-d*x - c))^2 - 6*a^4*b*(e^(d*x + c) - e^(-d*x - c)) - 6*a^2*b^3*(e^(d*x + c
) - e^(-d*x - c)) + 4*b^5*(e^(d*x + c) - e^(-d*x - c)) - 24*a^5 - 48*a^3*b^2 - 24*a*b^4)/((a^6 + 2*a^4*b^2 + a
^2*b^4)*((e^(d*x + c) - e^(-d*x - c))^3 + 4*e^(d*x + c) - 4*e^(-d*x - c))))/d

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maple [B]  time = 0.00, size = 478, normalized size = 2.66 \[ \frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {b^{5} \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d \left (a^{2}+b^{2}\right )^{2} a^{2}}-\frac {1}{2 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {2 \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {3 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {5 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

1/2/d/a*tanh(1/2*d*x+1/2*c)+1/d*b^5/(a^2+b^2)^2/a^2*ln(tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)*b-a)-1/2/
d/a/tanh(1/2*d*x+1/2*c)-1/d/a^2*b*ln(tanh(1/2*d*x+1/2*c))+1/d/(a^2+b^2)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2
*d*x+1/2*c)^3*a^3+1/d/(a^2+b^2)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3*a*b^2+2/d/(a^2+b^2)^2/(tan
h(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^2*a^2*b+2/d/(a^2+b^2)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1
/2*c)^2*b^3-1/d/(a^2+b^2)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)*a^3-1/d/(a^2+b^2)^2/(tanh(1/2*d*x+
1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)*a*b^2+1/d/(a^2+b^2)^2*ln(tanh(1/2*d*x+1/2*c)^2+1)*a^2*b+2/d/(a^2+b^2)^2*ln(t
anh(1/2*d*x+1/2*c)^2+1)*b^3-3/d/(a^2+b^2)^2*arctan(tanh(1/2*d*x+1/2*c))*a^3-5/d/(a^2+b^2)^2*arctan(tanh(1/2*d*
x+1/2*c))*a*b^2

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maxima [A]  time = 0.42, size = 350, normalized size = 1.94 \[ \frac {b^{5} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d} + \frac {{\left (3 \, a^{3} + 5 \, a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (a^{2} b + 2 \, b^{3}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {2 \, a b e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a b e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{2} + 2 \, b^{2}\right )} e^{\left (-d x - c\right )} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (3 \, a^{2} + 2 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{{\left (a^{3} + a b^{2} + {\left (a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

b^5*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^6 + 2*a^4*b^2 + a^2*b^4)*d) + (3*a^3 + 5*a*b^2)*arctan
(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (a^2*b + 2*b^3)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4
)*d) - (2*a*b*e^(-2*d*x - 2*c) - 2*a*b*e^(-4*d*x - 4*c) + (3*a^2 + 2*b^2)*e^(-d*x - c) + 2*(a^2 + 2*b^2)*e^(-3
*d*x - 3*c) + (3*a^2 + 2*b^2)*e^(-5*d*x - 5*c))/((a^3 + a*b^2 + (a^3 + a*b^2)*e^(-2*d*x - 2*c) - (a^3 + a*b^2)
*e^(-4*d*x - 4*c) - (a^3 + a*b^2)*e^(-6*d*x - 6*c))*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c)
- 1)/(a^2*d)

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mupad [B]  time = 7.50, size = 398, normalized size = 2.21 \[ \frac {2\,b}{d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^2\,\left (a^2+b^2\right )}-\frac {b\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{a^2\,d}+\frac {2\,b\,\ln \left (1+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,1{}\mathrm {i}\right )}{d\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}-\frac {2\,b^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {2\,b\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+1{}\mathrm {i}\right )}{d\,{\left (b+a\,1{}\mathrm {i}\right )}^2}-\frac {2\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}+\frac {b^5\,\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^2\,\left (a^2+b^2\right )}-\frac {a\,b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {a\,\ln \left (1+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,d\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}+\frac {a\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,d\,{\left (b+a\,1{}\mathrm {i}\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^3*sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)

[Out]

(2*b)/(d*(exp(2*c + 2*d*x) + 1)^2*(a^2 + b^2)) - (b*log(exp(2*c)*exp(2*d*x) - 1))/(a^2*d) - (a*log(exp(d*x)*ex
p(c)*1i + 1)*3i)/(2*d*(a*1i - b)^2) + (2*b*log(exp(d*x)*exp(c)*1i + 1))/(d*(a*1i - b)^2) - (2*b^3)/(d*(exp(2*c
+ 2*d*x) + 1)*(a^2 + b^2)^2) - (2*exp(c + d*x))/(a*d*(exp(2*c + 2*d*x) - 1)) + (a*log(exp(d*x)*exp(c) + 1i)*3
i)/(2*d*(a*1i + b)^2) + (2*b*log(exp(d*x)*exp(c) + 1i))/(d*(a*1i + b)^2) - (2*a^2*b)/(d*(exp(2*c + 2*d*x) + 1)
*(a^2 + b^2)^2) - (a^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)*(a^2 + b^2)^2) + (b^5*log(2*a*exp(d*x)*exp(c) -
b + b*exp(2*c)*exp(2*d*x)))/(a^2*d*(a^2 + b^2)^2) + (2*a*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)^2*(a^2 + b^2
)) - (a*b^2*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)*(a^2 + b^2)^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**2*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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