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

Optimal. Leaf size=157 \[ \frac {b^{3/2} (5 a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^3}+\frac {b (a+b) \sinh (c+d x)}{2 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {(a-5 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^3}+\frac {\tanh (c+d x) \text {sech}(c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

1/2*(a-5*b)*arctan(sinh(d*x+c))/(a-b)^3/d+1/2*(5*a-b)*b^(3/2)*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/a^(3/2)/(a-b
)^3/d+1/2*b*(a+b)*sinh(d*x+c)/a/(a-b)^2/d/(a+b*sinh(d*x+c)^2)+1/2*sech(d*x+c)*tanh(d*x+c)/(a-b)/d/(a+b*sinh(d*
x+c)^2)

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Rubi [A]  time = 0.19, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3190, 414, 527, 522, 203, 205} \[ \frac {b^{3/2} (5 a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^3}+\frac {b (a+b) \sinh (c+d x)}{2 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {(a-5 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^3}+\frac {\tanh (c+d x) \text {sech}(c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - 5*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^3*d) + ((5*a - b)*b^(3/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]
)/(2*a^(3/2)*(a - b)^3*d) + (b*(a + b)*Sinh[c + d*x])/(2*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^2)) + (Sech[c + d*
x]*Tanh[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^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 205

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3190

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

Rubi steps

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

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Mathematica [A]  time = 0.93, size = 230, normalized size = 1.46 \[ \frac {(2 a-b) \left (2 a^{3/2} (a-5 b) \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a^{3/2} (a-b) \tanh (c+d x) \text {sech}(c+d x)+b^{3/2} (b-5 a) \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )\right )+b \cosh (2 (c+d x)) \left (2 a^{3/2} (a-5 b) \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a^{3/2} (a-b) \tanh (c+d x) \text {sech}(c+d x)+b^{3/2} (b-5 a) \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )\right )+2 \sqrt {a} b^2 (a-b) \sinh (c+d x)}{2 a^{3/2} d (a-b)^3 (2 a+b \cosh (2 (c+d x))-b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a]*(a - b)*b^2*Sinh[c + d*x] + (2*a - b)*(b^(3/2)*(-5*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] +
2*a^(3/2)*(a - 5*b)*ArcTan[Tanh[(c + d*x)/2]] + a^(3/2)*(a - b)*Sech[c + d*x]*Tanh[c + d*x]) + b*Cosh[2*(c +
d*x)]*(b^(3/2)*(-5*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*a^(3/2)*(a - 5*b)*ArcTan[Tanh[(c + d*x)/
2]] + a^(3/2)*(a - b)*Sech[c + d*x]*Tanh[c + d*x]))/(2*a^(3/2)*(a - b)^3*d*(2*a - b + b*Cosh[2*(c + d*x)]))

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fricas [B]  time = 0.77, size = 6548, normalized size = 41.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^2*b - b^3)*cosh(d*x + c)^7 + 28*(a^2*b - b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(a^2*b - b^3)*sinh(
d*x + c)^7 + 4*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^5 + 4*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3 + 21*(a^
2*b - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(a^2*b - b^3)*cosh(d*x + c)^3 + (4*a^3 - 7*a^2*b + 4*a*b^2
- b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^3 + 4*(35*(a^2*b -
b^3)*cosh(d*x + c)^4 - 4*a^3 + 7*a^2*b - 4*a*b^2 + b^3 + 10*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^3 + 4*(21*(a^2*b - b^3)*cosh(d*x + c)^5 + 10*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^3
- 3*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((5*a*b^2 - b^3)*cosh(d*x + c)^8 + 8*(5
*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (5*a*b^2 - b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - a*b^2)*cosh(d*x +
c)^6 + 4*(5*a^2*b - a*b^2 + 7*(5*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - b^3)*cosh(d*
x + c)^3 + 3*(5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^4 +
2*(35*(5*a*b^2 - b^3)*cosh(d*x + c)^4 + 20*a^2*b - 9*a*b^2 + b^3 + 30*(5*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh
(d*x + c)^4 + 8*(7*(5*a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b - a*b^2)*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^
2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a*b^2 - b^3 + 4*(5*a^2*b - a*b^2)*cosh(d*x + c)^2 + 4*(7*(5*a*b^2
- b^3)*cosh(d*x + c)^6 + 15*(5*a^2*b - a*b^2)*cosh(d*x + c)^4 + 5*a^2*b - a*b^2 + 3*(20*a^2*b - 9*a*b^2 + b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((5*a*b^2 - b^3)*cosh(d*x + c)^7 + 3*(5*a^2*b - a*b^2)*cosh(d*x + c)^5 +
(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^3 + (5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log((
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)
+ 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d
*x + c)^2 - a)*sinh(d*x + c))*sqrt(-b/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)) + 4*((a^2*b - 5*a*b^2)*cosh(d*x + c)^8 + 8*(a^2*b - 5*a*b
^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b - 5*a*b^2)*sinh(d*x + c)^8 + 4*(a^3 - 5*a^2*b)*cosh(d*x + c)^6 + 4*
(a^3 - 5*a^2*b + 7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^3
+ 3*(a^3 - 5*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(4*a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c)^4 + 2*(35*(
a^2*b - 5*a*b^2)*cosh(d*x + c)^4 + 4*a^3 - 21*a^2*b + 5*a*b^2 + 30*(a^3 - 5*a^2*b)*cosh(d*x + c)^2)*sinh(d*x +
c)^4 + 8*(7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^5 + 10*(a^3 - 5*a^2*b)*cosh(d*x + c)^3 + (4*a^3 - 21*a^2*b + 5*a*
b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*b - 5*a*b^2 + 4*(a^3 - 5*a^2*b)*cosh(d*x + c)^2 + 4*(7*(a^2*b - 5*a*
b^2)*cosh(d*x + c)^6 + 15*(a^3 - 5*a^2*b)*cosh(d*x + c)^4 + a^3 - 5*a^2*b + 3*(4*a^3 - 21*a^2*b + 5*a*b^2)*cos
h(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^2*b - 5*a*b^2)*cosh(d*x + c)^7 + 3*(a^3 - 5*a^2*b)*cosh(d*x + c)^5 + (4*
a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c)^3 + (a^3 - 5*a^2*b)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c)
+ sinh(d*x + c)) - 4*(a^2*b - b^3)*cosh(d*x + c) + 4*(7*(a^2*b - b^3)*cosh(d*x + c)^6 + 5*(4*a^3 - 7*a^2*b +
4*a*b^2 - b^3)*cosh(d*x + c)^4 - a^2*b + b^3 - 3*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x +
c))/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^8 + 8*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*
cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*sinh(d*x + c)^8 + 4*(a^5 - 3*a^4*b +
3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^2 + (a^
5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d)*sinh(d*x + c)^6 + 2*(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*
d*cosh(d*x + c)^4 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^3 + 3*(a^5 - 3*a^4*b + 3*a^3*
b^2 - a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c
)^4 + 30*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3
+ a*b^4)*d)*sinh(d*x + c)^4 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b - 3*a^3*
b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^5 + 10*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^3 + (4*a
^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b - 3*a^3*b^2 + 3
*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^6 + 15*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^4 + 3*(4*a^5 -
13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d)*sinh(d
*x + c)^2 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d + 8*((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x
+ c)^7 + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^5 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^
3 + a*b^4)*d*cosh(d*x + c)^3 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*(
a^2*b - b^3)*cosh(d*x + c)^7 + 14*(a^2*b - b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a^2*b - b^3)*sinh(d*x + c)^
7 + 2*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3 + 21*(a^2*b - b^3
)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(a^2*b - b^3)*cosh(d*x + c)^3 + (4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*c
osh(d*x + c))*sinh(d*x + c)^4 - 2*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(35*(a^2*b - b^3)*cosh
(d*x + c)^4 - 4*a^3 + 7*a^2*b - 4*a*b^2 + b^3 + 10*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + 2*(21*(a^2*b - b^3)*cosh(d*x + c)^5 + 10*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^3 - 3*(4*a^
3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((5*a*b^2 - b^3)*cosh(d*x + c)^8 + 8*(5*a*b^2 -
b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (5*a*b^2 - b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - a*b^2)*cosh(d*x + c)^6 + 4
*(5*a^2*b - a*b^2 + 7*(5*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - b^3)*cosh(d*x + c)^3
+ 3*(5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(5
*a*b^2 - b^3)*cosh(d*x + c)^4 + 20*a^2*b - 9*a*b^2 + b^3 + 30*(5*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 + 8*(7*(5*a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b - a*b^2)*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^2 + b^3)*
cosh(d*x + c))*sinh(d*x + c)^3 + 5*a*b^2 - b^3 + 4*(5*a^2*b - a*b^2)*cosh(d*x + c)^2 + 4*(7*(5*a*b^2 - b^3)*co
sh(d*x + c)^6 + 15*(5*a^2*b - a*b^2)*cosh(d*x + c)^4 + 5*a^2*b - a*b^2 + 3*(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 8*((5*a*b^2 - b^3)*cosh(d*x + c)^7 + 3*(5*a^2*b - a*b^2)*cosh(d*x + c)^5 + (20*a^2*
b - 9*a*b^2 + b^3)*cosh(d*x + c)^3 + (5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*sqrt
(b/a)*(cosh(d*x + c) + sinh(d*x + c))) + ((5*a*b^2 - b^3)*cosh(d*x + c)^8 + 8*(5*a*b^2 - b^3)*cosh(d*x + c)*si
nh(d*x + c)^7 + (5*a*b^2 - b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - a*b^2)*cosh(d*x + c)^6 + 4*(5*a^2*b - a*b^2 + 7
*(5*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - b^3)*cosh(d*x + c)^3 + 3*(5*a^2*b - a*b^2)
*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(5*a*b^2 - b^3)*cosh(d*
x + c)^4 + 20*a^2*b - 9*a*b^2 + b^3 + 30*(5*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(5*a*b^2 -
b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b - a*b^2)*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c))*sinh(d
*x + c)^3 + 5*a*b^2 - b^3 + 4*(5*a^2*b - a*b^2)*cosh(d*x + c)^2 + 4*(7*(5*a*b^2 - b^3)*cosh(d*x + c)^6 + 15*(5
*a^2*b - a*b^2)*cosh(d*x + c)^4 + 5*a^2*b - a*b^2 + 3*(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^2 + 8*((5*a*b^2 - b^3)*cosh(d*x + c)^7 + 3*(5*a^2*b - a*b^2)*cosh(d*x + c)^5 + (20*a^2*b - 9*a*b^2 + b^3)*co
sh(d*x + c)^3 + (5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*
cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - b)*
sinh(d*x + c))*sqrt(b/a)/b) + 2*((a^2*b - 5*a*b^2)*cosh(d*x + c)^8 + 8*(a^2*b - 5*a*b^2)*cosh(d*x + c)*sinh(d*
x + c)^7 + (a^2*b - 5*a*b^2)*sinh(d*x + c)^8 + 4*(a^3 - 5*a^2*b)*cosh(d*x + c)^6 + 4*(a^3 - 5*a^2*b + 7*(a^2*b
- 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^3 + 3*(a^3 - 5*a^2*b)*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(4*a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2*b - 5*a*b^2)*cosh(d*x
+ c)^4 + 4*a^3 - 21*a^2*b + 5*a*b^2 + 30*(a^3 - 5*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^2*b - 5*a
*b^2)*cosh(d*x + c)^5 + 10*(a^3 - 5*a^2*b)*cosh(d*x + c)^3 + (4*a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c))*sinh(
d*x + c)^3 + a^2*b - 5*a*b^2 + 4*(a^3 - 5*a^2*b)*cosh(d*x + c)^2 + 4*(7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^6 + 15
*(a^3 - 5*a^2*b)*cosh(d*x + c)^4 + a^3 - 5*a^2*b + 3*(4*a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*((a^2*b - 5*a*b^2)*cosh(d*x + c)^7 + 3*(a^3 - 5*a^2*b)*cosh(d*x + c)^5 + (4*a^3 - 21*a^2*b + 5*a*b^2)
*cosh(d*x + c)^3 + (a^3 - 5*a^2*b)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(a^
2*b - b^3)*cosh(d*x + c) + 2*(7*(a^2*b - b^3)*cosh(d*x + c)^6 + 5*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x +
c)^4 - a^2*b + b^3 - 3*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^4*b - 3*a^3*b^2
+ 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^8 + 8*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)*sinh(d*x +
c)^7 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*sinh(d*x + c)^8 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*c
osh(d*x + c)^6 + 4*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 -
a^2*b^3)*d)*sinh(d*x + c)^6 + 2*(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 8*(7*
(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^3 + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x
+ c))*sinh(d*x + c)^5 + 2*(35*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^4 + 30*(a^5 - 3*a^4*b +
3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d)*sinh(d*x + c)
^4 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*
d*cosh(d*x + c)^5 + 10*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^3 + (4*a^5 - 13*a^4*b + 15*a^3*b^
2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(
d*x + c)^6 + 15*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^4 + 3*(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7
*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d)*sinh(d*x + c)^2 + (a^4*b - 3*a^
3*b^2 + 3*a^2*b^3 - a*b^4)*d + 8*((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^7 + 3*(a^5 - 3*a^4*b
+ 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^5 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c
)^3 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[6,-20]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[66,-29]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-21,2]Warning, need to cho
ose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a
,b]=[15,2]Undef/Unsigned Inf encountered in limitEvaluation time: 0.74Limit: Max order reached or unable to ma
ke series expansion Error: Bad Argument Value

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maple [B]  time = 0.18, size = 1265, normalized size = 8.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*b^2/(a-b)^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+
1/2*c)^3+1/d*b^3/(a-b)^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tan
h(1/2*d*x+1/2*c)^3+1/d*b^2/(a-b)^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*
b+a)*tanh(1/2*d*x+1/2*c)-1/d*b^3/(a-b)^3/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2
*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)-5/2/d*b^2/(a-b)^3*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arc
tan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+5/2/d*b^2/(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))+3/d*b^3/(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))-5/2/d*b
^2/(a-b)^3*a/(-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))-5/2/d*b^2/(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))+3/d*b^3/(a-b)^3/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*ar
ctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/2/d*b^3/(a-b)^3/a/((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))-1/2/d*b^4/(a-b)^3/a/(-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))
+1/2/d*b^3/(a-b)^3/a/((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))-1/2/d*b^4/(a-b)^3/a/(-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))-1/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3*
a+1/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3*b+1/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh
(1/2*d*x+1/2*c)*a-1/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)*b+1/d/(a-b)^3*arctan(tanh(1/2*d*
x+1/2*c))*a-5/d/(a-b)^3*arctan(tanh(1/2*d*x+1/2*c))*b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a e^{c} - 5 \, b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d} + \frac {{\left (a b e^{\left (7 \, c\right )} + b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (4 \, a^{2} e^{\left (5 \, c\right )} - 3 \, a b e^{\left (5 \, c\right )} + b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} - {\left (4 \, a^{2} e^{\left (3 \, c\right )} - 3 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (a b e^{c} + b^{2} e^{c}\right )} e^{\left (d x\right )}}{a^{3} b d - 2 \, a^{2} b^{2} d + a b^{3} d + {\left (a^{3} b d e^{\left (8 \, c\right )} - 2 \, a^{2} b^{2} d e^{\left (8 \, c\right )} + a b^{3} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (a^{4} d e^{\left (6 \, c\right )} - 2 \, a^{3} b d e^{\left (6 \, c\right )} + a^{2} b^{2} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \, {\left (4 \, a^{4} d e^{\left (4 \, c\right )} - 9 \, a^{3} b d e^{\left (4 \, c\right )} + 6 \, a^{2} b^{2} d e^{\left (4 \, c\right )} - a b^{3} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{4} d e^{\left (2 \, c\right )} - 2 \, a^{3} b d e^{\left (2 \, c\right )} + a^{2} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + 8 \, \int \frac {{\left (5 \, a b^{2} e^{\left (3 \, c\right )} - b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (5 \, a b^{2} e^{c} - b^{3} e^{c}\right )} e^{\left (d x\right )}}{8 \, {\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4} + {\left (a^{4} b e^{\left (4 \, c\right )} - 3 \, a^{3} b^{2} e^{\left (4 \, c\right )} + 3 \, a^{2} b^{3} e^{\left (4 \, c\right )} - a b^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (2 \, a^{5} e^{\left (2 \, c\right )} - 7 \, a^{4} b e^{\left (2 \, c\right )} + 9 \, a^{3} b^{2} e^{\left (2 \, c\right )} - 5 \, a^{2} b^{3} e^{\left (2 \, c\right )} + a b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*e^c - 5*b*e^c)*arctan(e^(d*x + c))*e^(-c)/(a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d) + ((a*b*e^(7*c) + b^2*e^(
7*c))*e^(7*d*x) + (4*a^2*e^(5*c) - 3*a*b*e^(5*c) + b^2*e^(5*c))*e^(5*d*x) - (4*a^2*e^(3*c) - 3*a*b*e^(3*c) + b
^2*e^(3*c))*e^(3*d*x) - (a*b*e^c + b^2*e^c)*e^(d*x))/(a^3*b*d - 2*a^2*b^2*d + a*b^3*d + (a^3*b*d*e^(8*c) - 2*a
^2*b^2*d*e^(8*c) + a*b^3*d*e^(8*c))*e^(8*d*x) + 4*(a^4*d*e^(6*c) - 2*a^3*b*d*e^(6*c) + a^2*b^2*d*e^(6*c))*e^(6
*d*x) + 2*(4*a^4*d*e^(4*c) - 9*a^3*b*d*e^(4*c) + 6*a^2*b^2*d*e^(4*c) - a*b^3*d*e^(4*c))*e^(4*d*x) + 4*(a^4*d*e
^(2*c) - 2*a^3*b*d*e^(2*c) + a^2*b^2*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/8*((5*a*b^2*e^(3*c) - b^3*e^(3*c))*
e^(3*d*x) + (5*a*b^2*e^c - b^3*e^c)*e^(d*x))/(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4 + (a^4*b*e^(4*c) - 3*a^3*b
^2*e^(4*c) + 3*a^2*b^3*e^(4*c) - a*b^4*e^(4*c))*e^(4*d*x) + 2*(2*a^5*e^(2*c) - 7*a^4*b*e^(2*c) + 9*a^3*b^2*e^(
2*c) - 5*a^2*b^3*e^(2*c) + a*b^4*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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