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

Optimal. Leaf size=114 \[ -\frac {b (4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^{5/2}}+\frac {b^2 \tanh (c+d x)}{2 a d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\tanh (c+d x)}{d (a-b)^2} \]

[Out]

-1/2*(4*a-b)*b*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(3/2)/(a-b)^(5/2)/d+tanh(d*x+c)/(a-b)^2/d+1/2*b^2*ta
nh(d*x+c)/a/(a-b)^2/d/(a-(a-b)*tanh(d*x+c)^2)

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Rubi [A]  time = 0.18, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3191, 390, 385, 208} \[ -\frac {b (4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^{5/2}}+\frac {b^2 \tanh (c+d x)}{2 a d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\tanh (c+d x)}{d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 385

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

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

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Mathematica [A]  time = 0.92, size = 105, normalized size = 0.92 \[ \frac {\frac {\frac {b^2 \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}+2 \tanh (c+d x)}{(a-b)^2}-\frac {b (4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} (a-b)^{5/2}}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.61, size = 3147, normalized size = 27.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(4*a^3*b - 5*a^2*b^2 + a*b^3)*cosh(d*x + c)^4 + 16*(4*a^3*b - 5*a^2*b^2 + a*b^3)*cosh(d*x + c)*sinh(d
*x + c)^3 + 4*(4*a^3*b - 5*a^2*b^2 + a*b^3)*sinh(d*x + c)^4 + 8*a^3*b - 4*a^2*b^2 - 4*a*b^3 + 8*(4*a^4 - 5*a^3
*b + a^2*b^2)*cosh(d*x + c)^2 + 8*(4*a^4 - 5*a^3*b + a^2*b^2 + 3*(4*a^3*b - 5*a^2*b^2 + a*b^3)*cosh(d*x + c)^2
)*sinh(d*x + c)^2 + ((4*a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(4*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (4*a*
b^2 - b^3)*sinh(d*x + c)^6 + (16*a^2*b - 8*a*b^2 + b^3)*cosh(d*x + c)^4 + (16*a^2*b - 8*a*b^2 + b^3 + 15*(4*a*
b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a*b^2 - b^3)*cosh(d*x + c)^3 + (16*a^2*b - 8*a*b^2 + b^3
)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a*b^2 - b^3 + (16*a^2*b - 8*a*b^2 + b^3)*cosh(d*x + c)^2 + (15*(4*a*b^2 -
b^3)*cosh(d*x + c)^4 + 16*a^2*b - 8*a*b^2 + b^3 + 6*(16*a^2*b - 8*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)
^2 + 2*(3*(4*a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(16*a^2*b - 8*a*b^2 + b^3)*cosh(d*x + c)^3 + (16*a^2*b - 8*a*b^2
+ 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)) + 16*
((4*a^3*b - 5*a^2*b^2 + a*b^3)*cosh(d*x + c)^3 + (4*a^4 - 5*a^3*b + a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a
^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^6 + 6*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh
(d*x + c)*sinh(d*x + c)^5 + (a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*sinh(d*x + c)^6 + (4*a^6 - 13*a^5*b +
15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4 + (15*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x
+ c)^2 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d)*sinh(d*x + c)^4 + (4*a^6 - 13*a^5*b + 15*a^
4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + 4*(5*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c
)^3 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^5*b - 3*
a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^4 + 6*(4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*c
osh(d*x + c)^2 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d)*sinh(d*x + c)^2 + (a^5*b - 3*a^4*b^2
+ 3*a^3*b^3 - a^2*b^4)*d + 2*(3*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^5 + 2*(4*a^6 - 13*a
^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^3 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*
b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(4*a^3*b - 5*a^2*b^2 + a*b^3)*cosh(d*x + c)^4 + 8*(4*a^3*b - 5*a
^2*b^2 + a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(4*a^3*b - 5*a^2*b^2 + a*b^3)*sinh(d*x + c)^4 + 4*a^3*b - 2*
a^2*b^2 - 2*a*b^3 + 4*(4*a^4 - 5*a^3*b + a^2*b^2)*cosh(d*x + c)^2 + 4*(4*a^4 - 5*a^3*b + a^2*b^2 + 3*(4*a^3*b
- 5*a^2*b^2 + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(4*a*b^2 - b^3)*c
osh(d*x + c)*sinh(d*x + c)^5 + (4*a*b^2 - b^3)*sinh(d*x + c)^6 + (16*a^2*b - 8*a*b^2 + b^3)*cosh(d*x + c)^4 +
(16*a^2*b - 8*a*b^2 + b^3 + 15*(4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a*b^2 - b^3)*cosh(d*
x + c)^3 + (16*a^2*b - 8*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a*b^2 - b^3 + (16*a^2*b - 8*a*b^2 + b
^3)*cosh(d*x + c)^2 + (15*(4*a*b^2 - b^3)*cosh(d*x + c)^4 + 16*a^2*b - 8*a*b^2 + b^3 + 6*(16*a^2*b - 8*a*b^2 +
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(4*a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(16*a^2*b - 8*a*b^2 + b^3)*c
osh(d*x + c)^3 + (16*a^2*b - 8*a*b^2 + 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)) + 8
*((4*a^3*b - 5*a^2*b^2 + a*b^3)*cosh(d*x + c)^3 + (4*a^4 - 5*a^3*b + a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((
a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^6 + 6*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos
h(d*x + c)*sinh(d*x + c)^5 + (a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*sinh(d*x + c)^6 + (4*a^6 - 13*a^5*b +
15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4 + (15*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*
x + c)^2 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d)*sinh(d*x + c)^4 + (4*a^6 - 13*a^5*b + 15*a
^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + 4*(5*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x +
c)^3 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^5*b - 3
*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^4 + 6*(4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*
cosh(d*x + c)^2 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d)*sinh(d*x + c)^2 + (a^5*b - 3*a^4*b^
2 + 3*a^3*b^3 - a^2*b^4)*d + 2*(3*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^5 + 2*(4*a^6 - 13*
a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^3 + (4*a^6 - 13*a^5*b + 15*a^4*b^2 - 7*a^3*b^3 + a^2
*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [B]  time = 0.82, size = 221, normalized size = 1.94 \[ -\frac {\frac {{\left (4 \, a b - b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (4 \, a b e^{\left (4 \, d x + 4 \, c\right )} - b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b + b^{2}\right )}}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left (b e^{\left (6 \, d x + 6 \, c\right )} + 4 \, a e^{\left (4 \, d x + 4 \, c\right )} - b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((4*a*b - b^2)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^3 - 2*a^2*b + a*b^2)*sqrt(-
a^2 + a*b)) + 2*(4*a*b*e^(4*d*x + 4*c) - b^2*e^(4*d*x + 4*c) + 8*a^2*e^(2*d*x + 2*c) - 2*a*b*e^(2*d*x + 2*c) +
2*a*b + b^2)/((a^3 - 2*a^2*b + a*b^2)*(b*e^(6*d*x + 6*c) + 4*a*e^(4*d*x + 4*c) - b*e^(4*d*x + 4*c) + 4*a*e^(2
*d*x + 2*c) - b*e^(2*d*x + 2*c) + b)))/d

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maple [B]  time = 0.14, size = 798, normalized size = 7.00 \[ \frac {b^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a -b \right )^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) a}+\frac {b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a -b \right )^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) a}+\frac {2 b \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{d \left (a -b \right )^{2} \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}+\frac {2 b^{2} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{d \left (a -b \right )^{2} \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {2 b \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{d \left (a -b \right )^{2} \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {2 b^{2} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{d \left (a -b \right )^{2} \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}-\frac {b^{2} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 d \left (a -b \right )^{2} a \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {b^{3} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 d \left (a -b \right )^{2} a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}+\frac {b^{2} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 d \left (a -b \right )^{2} a \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}-\frac {b^{3} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 d \left (a -b \right )^{2} a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a -b \right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*b^2/(a-b)^2/(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)^3+1/d*b^2/(a-b)^2/(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*ta
nh(1/2*d*x+1/2*c)+2/d*b/(a-b)^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))+2/d*b^2/(a-b)^2/(-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))-2/d*b/(a-b)^2/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctan
h(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+2/d*b^2/(a-b)^2/(-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))-1/2/d*b^2/(a-b)^2/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^3/(a-b)^2/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^2/(a-b)^2/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^3/(a-b)^2/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))+2/d/(a-b)^2*tanh(1/2*d*x+1/2*c)/(tanh(
1/2*d*x+1/2*c)^2+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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