3.328 \(\int \frac {\text {sech}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=126 \[ \frac {\left (a^2-3 a b+3 b^2\right ) \tanh (c+d x)}{d (a-b)^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a-b)^{7/2}}+\frac {\tanh ^5(c+d x)}{5 d (a-b)}-\frac {(2 a-3 b) \tanh ^3(c+d x)}{3 d (a-b)^2} \]

[Out]

-b^3*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/(a-b)^(7/2)/d/a^(1/2)+(a^2-3*a*b+3*b^2)*tanh(d*x+c)/(a-b)^3/d-1/
3*(2*a-3*b)*tanh(d*x+c)^3/(a-b)^2/d+1/5*tanh(d*x+c)^5/(a-b)/d

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Rubi [A]  time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3191, 390, 208} \[ \frac {\left (a^2-3 a b+3 b^2\right ) \tanh (c+d x)}{d (a-b)^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a-b)^{7/2}}+\frac {\tanh ^5(c+d x)}{5 d (a-b)}-\frac {(2 a-3 b) \tanh ^3(c+d x)}{3 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

-((b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Tanh[
c + d*x])/((a - b)^3*d) - ((2*a - 3*b)*Tanh[c + d*x]^3)/(3*(a - b)^2*d) + Tanh[c + d*x]^5/(5*(a - b)*d)

Rule 208

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3191

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\text {sech}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2-3 a b+3 b^2}{(a-b)^3}-\frac {(2 a-3 b) x^2}{(a-b)^2}+\frac {x^4}{a-b}-\frac {b^3}{(a-b)^3 \left (a-(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\left (a^2-3 a b+3 b^2\right ) \tanh (c+d x)}{(a-b)^3 d}-\frac {(2 a-3 b) \tanh ^3(c+d x)}{3 (a-b)^2 d}+\frac {\tanh ^5(c+d x)}{5 (a-b) d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{(a-b)^3 d}\\ &=-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{7/2} d}+\frac {\left (a^2-3 a b+3 b^2\right ) \tanh (c+d x)}{(a-b)^3 d}-\frac {(2 a-3 b) \tanh ^3(c+d x)}{3 (a-b)^2 d}+\frac {\tanh ^5(c+d x)}{5 (a-b) d}\\ \end {align*}

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Mathematica [A]  time = 0.93, size = 119, normalized size = 0.94 \[ \frac {\frac {\tanh (c+d x) \left (\left (4 a^2-13 a b+9 b^2\right ) \text {sech}^2(c+d x)+8 a^2+3 (a-b)^2 \text {sech}^4(c+d x)-26 a b+33 b^2\right )}{(a-b)^3}-\frac {15 b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{7/2}}}{15 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

((-15*b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2)) + ((8*a^2 - 26*a*b + 33*b^2 +
(4*a^2 - 13*a*b + 9*b^2)*Sech[c + d*x]^2 + 3*(a - b)^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(a - b)^3)/(15*d)

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fricas [B]  time = 0.83, size = 6046, normalized size = 47.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/30*(60*(a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 480*(a^2*b^2 - a*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + 60*(a^2*b
^2 - a*b^3)*sinh(d*x + c)^8 - 120*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^6 - 120*(a^3*b - 4*a^2*b^2 + 3*a
*b^3 - 14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 240*(14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^3 - 3*(
a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 40*(8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a*b^3)*co
sh(d*x + c)^4 + 40*(105*(a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a*b^3 - 45*(a^3
*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*a^4 - 136*a^3*b + 236*a^2*b^2 - 132*a*b^3 + 16
0*(21*(a^2*b^2 - a*b^3)*cosh(d*x + c)^5 - 15*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^3 + (8*a^4 - 31*a^3*b
+ 47*a^2*b^2 - 24*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 40*(4*a^4 - 17*a^3*b + 28*a^2*b^2 - 15*a*b^3)*cosh(
d*x + c)^2 + 40*(42*(a^2*b^2 - a*b^3)*cosh(d*x + c)^6 - 45*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 + 4*a
^4 - 17*a^3*b + 28*a^2*b^2 - 15*a*b^3 + 6*(8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 15*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 + 5*b^3*cosh(
d*x + c)^8 + 10*b^3*cosh(d*x + c)^6 + 5*(9*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^8 + 40*(3*b^3*cosh(d*x + c
)^3 + b^3*cosh(d*x + c))*sinh(d*x + c)^7 + 10*b^3*cosh(d*x + c)^4 + 10*(21*b^3*cosh(d*x + c)^4 + 14*b^3*cosh(d
*x + c)^2 + b^3)*sinh(d*x + c)^6 + 4*(63*b^3*cosh(d*x + c)^5 + 70*b^3*cosh(d*x + c)^3 + 15*b^3*cosh(d*x + c))*
sinh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^2 + 10*(21*b^3*cosh(d*x + c)^6 + 35*b^3*cosh(d*x + c)^4 + 15*b^3*cosh(d*
x + c)^2 + b^3)*sinh(d*x + c)^4 + 40*(3*b^3*cosh(d*x + c)^7 + 7*b^3*cosh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^3 +
b^3*cosh(d*x + c))*sinh(d*x + c)^3 + b^3 + 5*(9*b^3*cosh(d*x + c)^8 + 28*b^3*cosh(d*x + c)^6 + 30*b^3*cosh(d*x
+ c)^4 + 12*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^2 + 10*(b^3*cosh(d*x + c)^9 + 4*b^3*cosh(d*x + c)^7 + 6*
b^3*cosh(d*x + c)^5 + 4*b^3*cosh(d*x + c)^3 + b^3*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(
d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(
3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b -
b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^
2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(
2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b
)*cosh(d*x + c))*sinh(d*x + c) + b)) + 80*(6*(a^2*b^2 - a*b^3)*cosh(d*x + c)^7 - 9*(a^3*b - 4*a^2*b^2 + 3*a*b^
3)*cosh(d*x + c)^5 + 2*(8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a*b^3)*cosh(d*x + c)^3 + (4*a^4 - 17*a^3*b + 28*a^2
*b^2 - 15*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c
)^10 + 10*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^5 - 4*a^4*b + 6
*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*sinh(d*x + c)^10 + 5*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*
x + c)^8 + 5*(9*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2
- 4*a^2*b^3 + a*b^4)*d)*sinh(d*x + c)^8 + 10*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^
6 + 40*(3*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a
^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 10*(21*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*co
sh(d*x + c)^4 + 14*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*
b^2 - 4*a^2*b^3 + a*b^4)*d)*sinh(d*x + c)^6 + 10*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x +
c)^4 + 4*(63*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 70*(a^5 - 4*a^4*b + 6*a^3*b^2
- 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + 15*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*
sinh(d*x + c)^5 + 10*(21*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^6 + 35*(a^5 - 4*a^4*b
+ 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 15*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cos
h(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sinh(d*x + c)^4 + 5*(a^5 - 4*a^4*b + 6*a^3*b
^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + 40*(3*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x +
c)^7 + 7*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 5*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4
*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*
x + c)^3 + 5*(9*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^8 + 28*(a^5 - 4*a^4*b + 6*a^3*
b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^6 + 30*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c
)^4 + 12*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^
2*b^3 + a*b^4)*d)*sinh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d + 10*((a^5 - 4*a^4*b + 6
*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^9 + 4*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x
+ c)^7 + 6*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 4*(a^5 - 4*a^4*b + 6*a^3*b^2 -
4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(
d*x + c)), -1/15*(30*(a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 240*(a^2*b^2 - a*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 +
30*(a^2*b^2 - a*b^3)*sinh(d*x + c)^8 - 60*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^6 - 60*(a^3*b - 4*a^2*b
^2 + 3*a*b^3 - 14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 120*(14*(a^2*b^2 - a*b^3)*cosh(d*x + c)
^3 - 3*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 20*(8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a
*b^3)*cosh(d*x + c)^4 + 20*(105*(a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a*b^3 -
45*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*a^4 - 68*a^3*b + 118*a^2*b^2 - 66*a*b^
3 + 80*(21*(a^2*b^2 - a*b^3)*cosh(d*x + c)^5 - 15*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^3 + (8*a^4 - 31*
a^3*b + 47*a^2*b^2 - 24*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 20*(4*a^4 - 17*a^3*b + 28*a^2*b^2 - 15*a*b^3)*
cosh(d*x + c)^2 + 20*(42*(a^2*b^2 - a*b^3)*cosh(d*x + c)^6 - 45*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4
+ 4*a^4 - 17*a^3*b + 28*a^2*b^2 - 15*a*b^3 + 6*(8*a^4 - 31*a^3*b + 47*a^2*b^2 - 24*a*b^3)*cosh(d*x + c)^2)*sin
h(d*x + c)^2 - 15*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 + 5*b^3*
cosh(d*x + c)^8 + 10*b^3*cosh(d*x + c)^6 + 5*(9*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^8 + 40*(3*b^3*cosh(d*
x + c)^3 + b^3*cosh(d*x + c))*sinh(d*x + c)^7 + 10*b^3*cosh(d*x + c)^4 + 10*(21*b^3*cosh(d*x + c)^4 + 14*b^3*c
osh(d*x + c)^2 + b^3)*sinh(d*x + c)^6 + 4*(63*b^3*cosh(d*x + c)^5 + 70*b^3*cosh(d*x + c)^3 + 15*b^3*cosh(d*x +
c))*sinh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^2 + 10*(21*b^3*cosh(d*x + c)^6 + 35*b^3*cosh(d*x + c)^4 + 15*b^3*co
sh(d*x + c)^2 + b^3)*sinh(d*x + c)^4 + 40*(3*b^3*cosh(d*x + c)^7 + 7*b^3*cosh(d*x + c)^5 + 5*b^3*cosh(d*x + c)
^3 + b^3*cosh(d*x + c))*sinh(d*x + c)^3 + b^3 + 5*(9*b^3*cosh(d*x + c)^8 + 28*b^3*cosh(d*x + c)^6 + 30*b^3*cos
h(d*x + c)^4 + 12*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^2 + 10*(b^3*cosh(d*x + c)^9 + 4*b^3*cosh(d*x + c)^7
+ 6*b^3*cosh(d*x + c)^5 + 4*b^3*cosh(d*x + c)^3 + b^3*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-
1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2
- a*b)) + 40*(6*(a^2*b^2 - a*b^3)*cosh(d*x + c)^7 - 9*(a^3*b - 4*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^5 + 2*(8*a^4
- 31*a^3*b + 47*a^2*b^2 - 24*a*b^3)*cosh(d*x + c)^3 + (4*a^4 - 17*a^3*b + 28*a^2*b^2 - 15*a*b^3)*cosh(d*x + c
))*sinh(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^10 + 10*(a^5 - 4*a^4*b + 6*
a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)
*d*sinh(d*x + c)^10 + 5*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^8 + 5*(9*(a^5 - 4*a^4*
b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sinh
(d*x + c)^8 + 10*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^6 + 40*(3*(a^5 - 4*a^4*b + 6*
a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x +
c))*sinh(d*x + c)^7 + 10*(21*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 14*(a^5 - 4*a
^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*s
inh(d*x + c)^6 + 10*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 4*(63*(a^5 - 4*a^4*b +
6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 70*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(
d*x + c)^3 + 15*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*(a^5
- 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^6 + 35*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*
b^4)*d*cosh(d*x + c)^4 + 15*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b
+ 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sinh(d*x + c)^4 + 5*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cos
h(d*x + c)^2 + 40*(3*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^7 + 7*(a^5 - 4*a^4*b + 6*
a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 5*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x
+ c)^3 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(9*(a^5 - 4*a^4*
b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^8 + 28*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*co
sh(d*x + c)^6 + 30*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 12*(a^5 - 4*a^4*b + 6*a
^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sinh(d*x +
c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d + 10*((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)
*d*cosh(d*x + c)^9 + 4*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^7 + 6*(a^5 - 4*a^4*b +
6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 4*(a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*
x + c)^3 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [B]  time = 0.70, size = 253, normalized size = 2.01 \[ -\frac {\frac {15 \, b^{3} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 30 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 90 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 230 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 240 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 130 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 150 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} - 26 \, a b + 33 \, b^{2}\right )}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

-1/15*(15*b^3*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sqrt
(-a^2 + a*b)) + 2*(15*b^2*e^(8*d*x + 8*c) - 30*a*b*e^(6*d*x + 6*c) + 90*b^2*e^(6*d*x + 6*c) + 80*a^2*e^(4*d*x
+ 4*c) - 230*a*b*e^(4*d*x + 4*c) + 240*b^2*e^(4*d*x + 4*c) + 40*a^2*e^(2*d*x + 2*c) - 130*a*b*e^(2*d*x + 2*c)
+ 150*b^2*e^(2*d*x + 2*c) + 8*a^2 - 26*a*b + 33*b^2)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(e^(2*d*x + 2*c) + 1)^5)
)/d

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maple [B]  time = 0.17, size = 907, normalized size = 7.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^6/(a+b*sinh(d*x+c)^2),x)

[Out]

1/d*b^3/(a-b)^3/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*
a)^(1/2))+1/d*b^4/(a-b)^3/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/(
(2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/d*b^3/(a-b)^3/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d
*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/d*b^4/(a-b)^3/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a
)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+2/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2
+1)^5*tanh(1/2*d*x+1/2*c)^9*a^2-6/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^9*a*b+6/d/(a-b)^3/
(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^9*b^2+8/3/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1
/2*c)^7*a^2-32/3/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^7*a*b+16/d/(a-b)^3/(tanh(1/2*d*x+1/
2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^7*b^2+116/15/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^5*a^2-3
32/15/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^5*a*b+132/5/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1
)^5*tanh(1/2*d*x+1/2*c)^5*b^2+8/3/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^3*a^2-32/3/d/(a-b)
^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)^3*a*b+16/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x
+1/2*c)^3*b^2+2/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)*a^2-6/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)
^2+1)^5*tanh(1/2*d*x+1/2*c)*a*b+6/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^5*tanh(1/2*d*x+1/2*c)*b^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is b-a positive or negative?

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mupad [B]  time = 2.89, size = 1152, normalized size = 9.14 \[ \frac {16}{\left (a\,d-b\,d\right )\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32}{5\,\left (a\,d-b\,d\right )\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {\mathrm {atan}\left (\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {4\,b}{d\,{\left (a-b\right )}^3\,\sqrt {b^6}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}+\frac {\left (2\,a-b\right )\,\left (2\,a^4\,d\,\sqrt {b^6}+b^4\,d\,\sqrt {b^6}-5\,a\,b^3\,d\,\sqrt {b^6}-7\,a^3\,b\,d\,\sqrt {b^6}+9\,a^2\,b^2\,d\,\sqrt {b^6}\right )}{b^5\,\sqrt {-a\,d^2\,{\left (a-b\right )}^7}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\,\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}\right )-\frac {\left (2\,a-b\right )\,\left (b^4\,d\,\sqrt {b^6}-3\,a\,b^3\,d\,\sqrt {b^6}-a^3\,b\,d\,\sqrt {b^6}+3\,a^2\,b^2\,d\,\sqrt {b^6}\right )}{b^5\,\sqrt {-a\,d^2\,{\left (a-b\right )}^7}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\,\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}\right )\,\left (\frac {b^4\,\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}{2}+\frac {3\,a^2\,b^2\,\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}{2}-\frac {3\,a\,b^3\,\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}{2}-\frac {a^3\,b\,\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}{2}\right )\right )\,\sqrt {b^6}}{\sqrt {-a^8\,d^2+7\,a^7\,b\,d^2-21\,a^6\,b^2\,d^2+35\,a^5\,b^3\,d^2-35\,a^4\,b^4\,d^2+21\,a^3\,b^5\,d^2-7\,a^2\,b^6\,d^2+a\,b^7\,d^2}}-\frac {2\,b^2}{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a-b\right )}^2\,\left (a\,d-b\,d\right )}+\frac {4\,\left (a\,b-b^2\right )}{{\left (a-b\right )}^2\,\left (a\,d-b\,d\right )\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {8\,\left (4\,a-3\,b\right )}{3\,\left (a-b\right )\,\left (a\,d-b\,d\right )\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^6*(a + b*sinh(c + d*x)^2)),x)

[Out]

16/((a*d - b*d)*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - 32/(5
*(a*d - b*d)*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c +
10*d*x) + 1)) + (atan((exp(2*c)*exp(2*d*x)*((4*b)/(d*(a - b)^3*(b^6)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)) +
((2*a - b)*(2*a^4*d*(b^6)^(1/2) + b^4*d*(b^6)^(1/2) - 5*a*b^3*d*(b^6)^(1/2) - 7*a^3*b*d*(b^6)^(1/2) + 9*a^2*b
^2*d*(b^6)^(1/2)))/(b^5*(-a*d^2*(a - b)^7)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)*(a*b^7*d^2 - a^8*d^2 + 7*a^7*
b*d^2 - 7*a^2*b^6*d^2 + 21*a^3*b^5*d^2 - 35*a^4*b^4*d^2 + 35*a^5*b^3*d^2 - 21*a^6*b^2*d^2)^(1/2))) - ((2*a - b
)*(b^4*d*(b^6)^(1/2) - 3*a*b^3*d*(b^6)^(1/2) - a^3*b*d*(b^6)^(1/2) + 3*a^2*b^2*d*(b^6)^(1/2)))/(b^5*(-a*d^2*(a
- b)^7)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)*(a*b^7*d^2 - a^8*d^2 + 7*a^7*b*d^2 - 7*a^2*b^6*d^2 + 21*a^3*b^5
*d^2 - 35*a^4*b^4*d^2 + 35*a^5*b^3*d^2 - 21*a^6*b^2*d^2)^(1/2)))*((b^4*(a*b^7*d^2 - a^8*d^2 + 7*a^7*b*d^2 - 7*
a^2*b^6*d^2 + 21*a^3*b^5*d^2 - 35*a^4*b^4*d^2 + 35*a^5*b^3*d^2 - 21*a^6*b^2*d^2)^(1/2))/2 + (3*a^2*b^2*(a*b^7*
d^2 - a^8*d^2 + 7*a^7*b*d^2 - 7*a^2*b^6*d^2 + 21*a^3*b^5*d^2 - 35*a^4*b^4*d^2 + 35*a^5*b^3*d^2 - 21*a^6*b^2*d^
2)^(1/2))/2 - (3*a*b^3*(a*b^7*d^2 - a^8*d^2 + 7*a^7*b*d^2 - 7*a^2*b^6*d^2 + 21*a^3*b^5*d^2 - 35*a^4*b^4*d^2 +
35*a^5*b^3*d^2 - 21*a^6*b^2*d^2)^(1/2))/2 - (a^3*b*(a*b^7*d^2 - a^8*d^2 + 7*a^7*b*d^2 - 7*a^2*b^6*d^2 + 21*a^3
*b^5*d^2 - 35*a^4*b^4*d^2 + 35*a^5*b^3*d^2 - 21*a^6*b^2*d^2)^(1/2))/2))*(b^6)^(1/2))/(a*b^7*d^2 - a^8*d^2 + 7*
a^7*b*d^2 - 7*a^2*b^6*d^2 + 21*a^3*b^5*d^2 - 35*a^4*b^4*d^2 + 35*a^5*b^3*d^2 - 21*a^6*b^2*d^2)^(1/2) - (2*b^2)
/((exp(2*c + 2*d*x) + 1)*(a - b)^2*(a*d - b*d)) + (4*(a*b - b^2))/((a - b)^2*(a*d - b*d)*(2*exp(2*c + 2*d*x) +
exp(4*c + 4*d*x) + 1)) - (8*(4*a - 3*b))/(3*(a - b)*(a*d - b*d)*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + ex
p(6*c + 6*d*x) + 1))

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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(sech(d*x+c)**6/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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