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

Optimal. Leaf size=130 \[ -\frac{b^4 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )}-\frac{\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac{a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}+\frac{b \text{csch}(c+d x)}{a^2 d}-\frac{\text{csch}^2(c+d x)}{2 a d} \]

[Out]

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

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Rubi [A]  time = 0.235471, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2837, 12, 894, 635, 203, 260} \[ -\frac{b^4 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )}-\frac{\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac{a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}+\frac{b \text{csch}(c+d x)}{a^2 d}-\frac{\text{csch}^2(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 12

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

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.36427, size = 164, normalized size = 1.26 \[ \frac{-\frac{2 b^4 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )}+\frac{\left (a-\sqrt{-b^2}\right ) \log \left (\sqrt{-b^2}-b \sinh (c+d x)\right )}{a^2+b^2}+\frac{\left (a+\sqrt{-b^2}\right ) \log \left (\sqrt{-b^2}+b \sinh (c+d x)\right )}{a^2+b^2}+\frac{2 b \text{csch}(c+d x)}{a^2}-\frac{2 (a-b) (a+b) \log (\sinh (c+d x))}{a^3}-\frac{\text{csch}^2(c+d x)}{a}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*Csch[c + d*x])/a^2 - Csch[c + d*x]^2/a - (2*(a - b)*(a + b)*Log[Sinh[c + d*x]])/a^3 + ((a - Sqrt[-b^2])*
Log[Sqrt[-b^2] - b*Sinh[c + d*x]])/(a^2 + b^2) - (2*b^4*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)) + ((a + Sq
rt[-b^2])*Log[Sqrt[-b^2] + b*Sinh[c + d*x]])/(a^2 + b^2))/(2*d)

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Maple [A]  time = 0.003, size = 219, normalized size = 1.7 \begin{align*} -{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{b}{2\,d{a}^{2}}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{{b}^{2}}{d{a}^{3}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{b}{2\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{{b}^{4}}{d{a}^{3} \left ({a}^{2}+{b}^{2} \right ) }\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a-2\,\tanh \left ( 1/2\,dx+c/2 \right ) b-a \right ) }+{\frac{a}{d \left ({a}^{2}+{b}^{2} \right ) }\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) }+2\,{\frac{b\arctan \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/2/d/a^2*tanh(1/2*d*x+1/2*c)*b-1/8/d/a/tanh(1/2*d*x+1/2*c)^2-1/d/a*ln(tanh(1/2
*d*x+1/2*c))+1/d/a^3*ln(tanh(1/2*d*x+1/2*c))*b^2+1/2/d*b/a^2/tanh(1/2*d*x+1/2*c)-1/d*b^4/a^3/(a^2+b^2)*ln(tanh
(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)*b-a)+1/d/(a^2+b^2)*a*ln(tanh(1/2*d*x+1/2*c)^2+1)+2/d/(a^2+b^2)*b*arc
tan(tanh(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.69105, size = 319, normalized size = 2.45 \begin{align*} -\frac{b^{4} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{5} + a^{3} b^{2}\right )} d} - \frac{2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac{2 \,{\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.37535, size = 2503, normalized size = 19.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a^3*b + a*b^3)*cosh(d*x + c)^3 + 2*(a^3*b + a*b^3)*sinh(d*x + c)^3 - 2*(a^4 + a^2*b^2)*cosh(d*x + c)^2 - 2
*(a^4 + a^2*b^2 - 3*(a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 2*(a^3*b*cosh(d*x + c)^4 + 4*a^3*b*cosh(d
*x + c)*sinh(d*x + c)^3 + a^3*b*sinh(d*x + c)^4 - 2*a^3*b*cosh(d*x + c)^2 + a^3*b + 2*(3*a^3*b*cosh(d*x + c)^2
- a^3*b)*sinh(d*x + c)^2 + 4*(a^3*b*cosh(d*x + c)^3 - a^3*b*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c
) + sinh(d*x + c)) - 2*(a^3*b + a*b^3)*cosh(d*x + c) - (b^4*cosh(d*x + c)^4 + 4*b^4*cosh(d*x + c)*sinh(d*x + c
)^3 + b^4*sinh(d*x + c)^4 - 2*b^4*cosh(d*x + c)^2 + b^4 + 2*(3*b^4*cosh(d*x + c)^2 - b^4)*sinh(d*x + c)^2 + 4*
(b^4*cosh(d*x + c)^3 - b^4*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*cosh(d*x + c)^4 + 4*a^4*cosh(d*x + c)*sinh(d*x + c)^3 + a^4*sinh(d*x + c)^4 - 2*a^4*cosh(d*x +
c)^2 + a^4 + 2*(3*a^4*cosh(d*x + c)^2 - a^4)*sinh(d*x + c)^2 + 4*(a^4*cosh(d*x + c)^3 - a^4*cosh(d*x + c))*sin
h(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - ((a^4 - b^4)*cosh(d*x + c)^4 + 4*(a^4 - b^4
)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - b^4)*sinh(d*x + c)^4 + a^4 - b^4 - 2*(a^4 - b^4)*cosh(d*x + c)^2 - 2*
(a^4 - b^4 - 3*(a^4 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^4 - b^4)*cosh(d*x + c)^3 - (a^4 - b^4)*cos
h(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - 2*(a^3*b + a*b^3 - 3*(a^3*b
+ a*b^3)*cosh(d*x + c)^2 + 2*(a^4 + a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + a^3*b^2)*d*cosh(d*x + c)^4
+ 4*(a^5 + a^3*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^5 + a^3*b^2)*d*sinh(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*
cosh(d*x + c)^2 + 2*(3*(a^5 + a^3*b^2)*d*cosh(d*x + c)^2 - (a^5 + a^3*b^2)*d)*sinh(d*x + c)^2 + (a^5 + a^3*b^2
)*d + 4*((a^5 + a^3*b^2)*d*cosh(d*x + c)^3 - (a^5 + a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.346, size = 370, normalized size = 2.85 \begin{align*} -\frac{b^{5} \log \left ({\left | b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{5} b d + a^{3} b^{3} d} + \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 )} b}{2 \,{\left (a^{2} d + b^{2} d\right )}} + \frac{a \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{2 \,{\left (a^{2} d + b^{2} d\right )}} - \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3} d} + \frac{3 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, a^{2}}{2 \, a^{3} d{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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