3.501 \(\int \frac{\text{csch}^3(c+d x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)
Optimal. Leaf size=211 \[ -\frac{b^6 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac{\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}+\frac{b \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac{a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac{b \text{csch}(c+d x)}{a^2 d}-\frac{\text{csch}^2(c+d x)}{2 a d} \]
[Out]
(b*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) + (b*(a^2 + 2*b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (b*C
sch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + (a*(2*a^2 + 3*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (
(2*a^2 - b^2)*Log[Sinh[c + d*x]])/(a^3*d) - (b^6*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (Sech[c + d
*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)
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Rubi [A] time = 0.366715, antiderivative size = 211, normalized size of antiderivative = 1.,
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} \[ -\frac{b^6 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac{\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}+\frac{b \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac{a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac{b \text{csch}(c+d x)}{a^2 d}-\frac{\text{csch}^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
Int[(Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
[Out]
(b*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) + (b*(a^2 + 2*b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (b*C
sch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + (a*(2*a^2 + 3*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (
(2*a^2 - b^2)*Log[Sinh[c + d*x]])/(a^3*d) - (b^6*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (Sech[c + d
*x]^2*(a - b*Sinh[c + d*x]))/(2*(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 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 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 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 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}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{b^3}{x^3 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{b^6 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{b^6 \operatorname{Subst}\left (\int \left (\frac{1}{a b^4 x^3}-\frac{1}{a^2 b^4 x^2}+\frac{-2 a^2+b^2}{a^3 b^6 x}-\frac{1}{a^3 \left (a^2+b^2\right )^2 (a+x)}+\frac{b^2+a x}{b^4 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac{b^2 \left (a^2+2 b^2\right )+a \left (2 a^2+3 b^2\right ) x}{b^6 \left (a^2+b^2\right )^2 \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 (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac{b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}+\frac{\operatorname{Subst}\left (\int \frac{b^2 \left (a^2+2 b^2\right )+a \left (2 a^2+3 b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{b^2+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{b \text{csch}(c+d x)}{a^2 d}-\frac{\text{csch}^2(c+d x)}{2 a d}-\frac{\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac{b^6 \log (a+b \sinh (c+d x))}{a^3 \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 \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^2 \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{\left (a \left (2 a^2+3 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{b \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac{b \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{b \text{csch}(c+d x)}{a^2 d}-\frac{\text{csch}^2(c+d x)}{2 a d}+\frac{a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac{b^6 \log (a+b \sinh (c+d x))}{a^3 \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 [C] time = 0.807566, size = 237, normalized size = 1.12 \[ \frac{-\frac{a \text{sech}^2(c+d x)}{a^2+b^2}-\frac{2 b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2}+\frac{(a-i b) \left (2 a^2+i a b+2 b^2\right ) \log (-\sinh (c+d x)+i)}{\left (a^2+b^2\right )^2}-\frac{2 \left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3}+\frac{(a+i b) \left (2 a^2-i a b+2 b^2\right ) \log (\sinh (c+d x)+i)}{\left (a^2+b^2\right )^2}+\frac{b \tan ^{-1}(\sinh (c+d x))}{a^2+b^2}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{a^2+b^2}+\frac{2 b \text{csch}(c+d x)}{a^2}-\frac{\text{csch}^2(c+d x)}{a}}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
[Out]
((b*ArcTan[Sinh[c + d*x]])/(a^2 + b^2) + (2*b*Csch[c + d*x])/a^2 - Csch[c + d*x]^2/a + ((a - I*b)*(2*a^2 + I*a
*b + 2*b^2)*Log[I - Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*(2*a^2 - b^2)*Log[Sinh[c + d*x]])/a^3 + ((a + I*b)*(2*a
^2 - I*a*b + 2*b^2)*Log[I + Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*b^6*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)^
2) - (a*Sech[c + d*x]^2)/(a^2 + b^2) + (b*Sech[c + d*x]*Tanh[c + d*x])/(a^2 + b^2))/(2*d)
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Maple [B] time = 0.003, size = 539, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3*sech(d*x+c)^3/(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)^2-2/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^6/(a^2+b^2)^2/a^3*ln(ta
nh(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)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*
c)^3*a^2*b-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*b^3+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*a^3+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*a*b
^2+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^2*b+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)*b^3+2/d/(a^2+b^2)^2*ln(tanh(1/2*d*x+1/2*c)^2+1)*a^3+3/d/(a^2+b^2)^2*ln(tanh(1/2*d*x
+1/2*c)^2+1)*a*b^2+3/d/(a^2+b^2)^2*arctan(tanh(1/2*d*x+1/2*c))*a^2*b+5/d/(a^2+b^2)^2*arctan(tanh(1/2*d*x+1/2*c
))*b^3
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Maxima [B] time = 1.7642, size = 564, normalized size = 2.67 \begin{align*} -\frac{b^{6} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d} - \frac{{\left (3 \, a^{2} b + 5 \, 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{{\left (2 \, a^{3} + 3 \, a b^{2}\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{4 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )} -{\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-d x - c\right )} + 2 \,{\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} -{\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )} + 2 \,{\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} +{\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{{\left (a^{4} + a^{2} b^{2} - 2 \,{\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} +{\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} - \frac{{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac{{\left (2 \, 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)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-b^6*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^7 + 2*a^5*b^2 + a^3*b^4)*d) - (3*a^2*b + 5*b^3)*arcta
n(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (2*a^3 + 3*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 +
b^4)*d) - (4*a*b^2*e^(-4*d*x - 4*c) - (3*a^2*b + 2*b^3)*e^(-d*x - c) + 2*(2*a^3 + a*b^2)*e^(-2*d*x - 2*c) + (a
^2*b - 2*b^3)*e^(-3*d*x - 3*c) - (a^2*b - 2*b^3)*e^(-5*d*x - 5*c) + 2*(2*a^3 + a*b^2)*e^(-6*d*x - 6*c) + (3*a^
2*b + 2*b^3)*e^(-7*d*x - 7*c))/((a^4 + a^2*b^2 - 2*(a^4 + a^2*b^2)*e^(-4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-8*d*
x - 8*c))*d) - (2*a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) - (2*a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d)
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Fricas [B] time = 7.08528, size = 7170, normalized size = 33.98 \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)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
[Out]
((3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c)^7 + (3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*sinh(d*x + c)^7 - 2*(2*a^6
+ 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^6 - (4*a^6 + 6*a^4*b^2 + 2*a^2*b^4 - 7*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*co
sh(d*x + c))*sinh(d*x + c)^6 - (a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^5 - (a^5*b - a^3*b^3 - 2*a*b^5 - 21*(
3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c)^2 + 12*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)
^5 - 4*(a^4*b^2 + a^2*b^4)*cosh(d*x + c)^4 - (4*a^4*b^2 + 4*a^2*b^4 - 35*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(
d*x + c)^3 + 30*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^2 + 5*(a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c))*s
inh(d*x + c)^4 + (a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^3 + (a^5*b - a^3*b^3 - 2*a*b^5 + 35*(3*a^5*b + 5*a^
3*b^3 + 2*a*b^5)*cosh(d*x + c)^4 - 40*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^3 - 10*(a^5*b - a^3*b^3 - 2*
a*b^5)*cosh(d*x + c)^2 - 16*(a^4*b^2 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a^6 + 3*a^4*b^2 + a^2*b^
4)*cosh(d*x + c)^2 - (4*a^6 + 6*a^4*b^2 + 2*a^2*b^4 - 21*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c)^5 + 30*
(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^4 + 10*(a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^3 + 24*(a^4*b^2 +
a^2*b^4)*cosh(d*x + c)^2 - 3*(a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^5*b + 5*a^3*b
^3)*cosh(d*x + c)^8 + 56*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(3*a^5*b + 5*a^3*b^3)*cosh
(d*x + c)^2*sinh(d*x + c)^6 + 8*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^5*b + 5*a^3*b^3)*si
nh(d*x + c)^8 + 3*a^5*b + 5*a^3*b^3 - 2*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^4 - 2*(3*a^5*b + 5*a^3*b^3 - 35*(3
*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^4)*sinh(d*x + c)^4 + 8*(7*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^5 - (3*a^5*b +
5*a^3*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^6 - 3*(3*a^5*b + 5*a^3*b
^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^7 - (3*a^5*b + 5*a^3*b^3)*cosh(d
*x + c)^3)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c
) - (b^6*cosh(d*x + c)^8 + 56*b^6*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*b^6*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8
*b^6*cosh(d*x + c)*sinh(d*x + c)^7 + b^6*sinh(d*x + c)^8 - 2*b^6*cosh(d*x + c)^4 + b^6 + 2*(35*b^6*cosh(d*x +
c)^4 - b^6)*sinh(d*x + c)^4 + 8*(7*b^6*cosh(d*x + c)^5 - b^6*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^6*cosh(d*
x + c)^6 - 3*b^6*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(b^6*cosh(d*x + c)^7 - b^6*cosh(d*x + c)^3)*sinh(d*x + c
))*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + ((2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^8 + 56*(2
*a^6 + 3*a^4*b^2)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8
*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (2*a^6 + 3*a^4*b^2)*sinh(d*x + c)^8 + 2*a^6 + 3*a^4*b^2 -
2*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^4 - 2*(2*a^6 + 3*a^4*b^2 - 35*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^4)*sinh(d
*x + c)^4 + 8*(7*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^5 - (2*a^6 + 3*a^4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*
(7*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^6 - 3*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((2*a^6 +
3*a^4*b^2)*cosh(d*x + c)^7 - (2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x
+ c) - sinh(d*x + c))) - ((2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^8 + 56*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x +
c)^3*sinh(d*x + c)^5 + 28*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(2*a^6 + 3*a^4*b^2 -
b^6)*cosh(d*x + c)*sinh(d*x + c)^7 + (2*a^6 + 3*a^4*b^2 - b^6)*sinh(d*x + c)^8 + 2*a^6 + 3*a^4*b^2 - b^6 - 2*(
2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^4 - 2*(2*a^6 + 3*a^4*b^2 - b^6 - 35*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x
+ c)^4)*sinh(d*x + c)^4 + 8*(7*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^5 - (2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x
+ c))*sinh(d*x + c)^3 + 4*(7*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^6 - 3*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 8*((2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^7 - (2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x +
c)^3)*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + (7*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)
*cosh(d*x + c)^6 - 3*a^5*b - 5*a^3*b^3 - 2*a*b^5 - 12*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^5 - 5*(a^5*b
- a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^4 - 16*(a^4*b^2 + a^2*b^4)*cosh(d*x + c)^3 + 3*(a^5*b - a^3*b^3 - 2*a*b^5)
*cosh(d*x + c)^2 - 4*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 2*a^5*b^2 + a^3*b^4)*
d*cosh(d*x + c)^8 + 56*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(a^7 + 2*a^5*b^2 + a
^3*b^4)*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a
^7 + 2*a^5*b^2 + a^3*b^4)*d*sinh(d*x + c)^8 - 2*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^4 + 2*(35*(a^7 + 2
*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^4 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d)*sinh(d*x + c)^4 + 8*(7*(a^7 + 2*a^5*b^2
+ a^3*b^4)*d*cosh(d*x + c)^5 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 2*a
^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^6 - 3*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + (a^7
+ 2*a^5*b^2 + a^3*b^4)*d + 8*((a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^7 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d*co
sh(d*x + c)^3)*sinh(d*x + c))
________________________________________________________________________________________
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)**3/(a+b*sinh(d*x+c)),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.77478, size = 652, normalized size = 3.09 \begin{align*} -\frac{b^{7} \log \left ({\left | b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{7} b d + 2 \, a^{5} b^{3} d + a^{3} b^{5} 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 )}{\left (3 \, a^{2} b + 5 \, b^{3}\right )}}{4 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac{{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{2 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac{2 \, a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 3 \, 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 )} + 12 \, a^{3} + 16 \, a b^{2}}{2 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}} - \frac{{\left (2 \, 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{6 \, 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)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
[Out]
-b^7*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^7*b*d + 2*a^5*b^3*d + a^3*b^5*d) + 1/4*(pi + 2*arctan(1
/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a^2*b + 5*b^3)/(a^4*d + 2*a^2*b^2*d + b^4*d) + 1/2*(2*a^3 + 3*a*b^2
)*log((e^(d*x + c) - e^(-d*x - c))^2 + 4)/(a^4*d + 2*a^2*b^2*d + b^4*d) - 1/2*(2*a^3*(e^(d*x + c) - e^(-d*x -
c))^2 + 3*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)) + 12*a^3 + 16*a*b^2)/((a^4*d + 2*a^2*b^2*d + b^4*d)*((e^(d*x + c) - e^(-d*x - c))^2 + 4)) - (2*a^
2 - b^2)*log(abs(e^(d*x + c) - e^(-d*x - c)))/(a^3*d) + 1/2*(6*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
)