3.4.26 \(\int \frac {\text {sech}^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [326]

Optimal. Leaf size=88 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{5/2} d}+\frac {(a-2 b) \tanh (c+d x)}{(a-b)^2 d}-\frac {\tanh ^3(c+d x)}{3 (a-b) d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 88, 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 = {3270, 398, 214} \begin {gather*} \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a-b)^{5/2}}-\frac {\tanh ^3(c+d x)}{3 d (a-b)}+\frac {(a-2 b) \tanh (c+d x)}{d (a-b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 214

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

Rule 398

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 3270

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

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Mathematica [A]
time = 0.51, size = 84, normalized size = 0.95 \begin {gather*} \frac {\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{5/2}}+\frac {\left (2 a-5 b+(a-b) \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{(a-b)^2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(267\) vs. \(2(78)=156\).
time = 1.84, size = 268, normalized size = 3.05

method	result	size
risch	\(-\frac {2 \left (-3 b \,{\mathrm e}^{4 d x +4 c}+6 a \,{\mathrm e}^{2 d x +2 c}-12 b \,{\mathrm e}^{2 d x +2 c}+2 a -5 b \right )}{3 d \left (a -b \right )^{2} \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}\)	\(242\)
derivativedivides	\(\frac {-\frac {2 b^{2} a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{\left (a -b \right )^{2}}-\frac {2 \left (\left (2 b -a \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {8 b}{3}-\frac {2 a}{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 b -a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{d}\)	\(268\)
default	\(\frac {-\frac {2 b^{2} a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{\left (a -b \right )^{2}}-\frac {2 \left (\left (2 b -a \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {8 b}{3}-\frac {2 a}{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 b -a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{d}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^4/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*b^2/(a-b)^2*a*(-1/2*((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arcta
nh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/2*((-b*(a-b))^(1/2)+b)/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)))-2/(a-b)^
2*((2*b-a)*tanh(1/2*d*x+1/2*c)^5+(8/3*b-2/3*a)*tanh(1/2*d*x+1/2*c)^3+(2*b-a)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*
x+1/2*c)^2+1)^3)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4/(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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1094 vs. \(2 (78) = 156\).
time = 0.40, size = 2444, normalized size = 27.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/6*(12*(a^2*b - a*b^2)*cosh(d*x + c)^4 + 48*(a^2*b - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 12*(a^2*b - a*b^
2)*sinh(d*x + c)^4 - 8*a^3 + 28*a^2*b - 20*a*b^2 - 24*(a^3 - 3*a^2*b + 2*a*b^2)*cosh(d*x + c)^2 - 24*(a^3 - 3*
a^2*b + 2*a*b^2 - 3*(a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x
+ c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x
+ c)^4 + 3*b^2*cosh(d*x + c)^2 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^2*c
osh(d*x + c)^4 + 6*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 + b^2 + 6*(b^2*cosh(d*x + c)^5 + 2*b^2*cosh(d*x
+ c)^3 + b^2*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*cos
h(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))
+ 48*((a^2*b - a*b^2)*cosh(d*x + c)^3 - (a^3 - 3*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 - 3*a^3*
b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^6 + 6*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)
^5 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*sinh(d*x + c)^6 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x
+ c)^4 + 3*(5*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d)*s
inh(d*x + c)^4 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a^4 - 3*a^3*b + 3*a^2*b^2 - a
*b^3)*d*cosh(d*x + c)^3 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*(a^4 -
3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^4 + 6*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + (a
^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d + 6*((a^4 - 3*a^3
*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^5 + 2*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^3 + (a^4 - 3
*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/3*(6*(a^2*b - a*b^2)*cosh(d*x + c)^4 + 24*(a^2*
b - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 6*(a^2*b - a*b^2)*sinh(d*x + c)^4 - 4*a^3 + 14*a^2*b - 10*a*b^2 - 1
2*(a^3 - 3*a^2*b + 2*a*b^2)*cosh(d*x + c)^2 - 12*(a^3 - 3*a^2*b + 2*a*b^2 - 3*(a^2*b - a*b^2)*cosh(d*x + c)^2)
*sinh(d*x + c)^2 - 3*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*
cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^4 + 3*b^2*cosh(d*x + c)^2 + 4*(5*b^2*cosh(d*x
+ c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^2*cosh(d*x + c)^4 + 6*b^2*cosh(d*x + c)^2 + b^2)*sinh(d
*x + c)^2 + b^2 + 6*(b^2*cosh(d*x + c)^5 + 2*b^2*cosh(d*x + c)^3 + b^2*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)) + 24*((a^2*b - a*b^2)*cosh(d*x + c)^3 - (a^3 - 3*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(
d*x + c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^6 + 6*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos
h(d*x + c)*sinh(d*x + c)^5 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*sinh(d*x + c)^6 + 3*(a^4 - 3*a^3*b + 3*a^2*
b^2 - a*b^3)*d*cosh(d*x + c)^4 + 3*(5*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + (a^4 - 3*a^3*b +
3*a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^4 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a^4 -
3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^3 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(
d*x + c)^3 + 3*(5*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^4 + 6*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3
)*d*cosh(d*x + c)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*
b^3)*d + 6*((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^5 + 2*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*co
sh(d*x + c)^3 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{a + b \sinh ^{2}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**4/(a+b*sinh(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**4/(a + b*sinh(c + d*x)**2), x)

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Giac [A]
time = 0.71, size = 138, normalized size = 1.57 \begin {gather*} \frac {\frac {3 \, b^{2} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a + 5 \, b\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/3*(3*b^2*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^2 - 2*a*b + b^2)*sqrt(-a^2 + a*b)) +
2*(3*b*e^(4*d*x + 4*c) - 6*a*e^(2*d*x + 2*c) + 12*b*e^(2*d*x + 2*c) - 2*a + 5*b)/((a^2 - 2*a*b + b^2)*(e^(2*d
*x + 2*c) + 1)^3))/d

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Mupad [B]
time = 2.46, size = 710, normalized size = 8.07 \begin {gather*} \frac {8}{3\,\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 )}-\frac {4}{\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 {\mathrm {atan}\left (\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {4}{d\,{\left (a-b\right )}^2\,\sqrt {b^4}\,\left (a^2-2\,a\,b+b^2\right )}+\frac {\left (2\,a-b\right )\,\left (2\,a^3\,d\,\sqrt {b^4}-b^3\,d\,\sqrt {b^4}+4\,a\,b^2\,d\,\sqrt {b^4}-5\,a^2\,b\,d\,\sqrt {b^4}\right )}{b^4\,\left (a^2-2\,a\,b+b^2\right )\,\sqrt {-a\,d^2\,{\left (a-b\right )}^5}\,\sqrt {-a^6\,d^2+5\,a^5\,b\,d^2-10\,a^4\,b^2\,d^2+10\,a^3\,b^3\,d^2-5\,a^2\,b^4\,d^2+a\,b^5\,d^2}}\right )+\frac {\left (2\,a-b\right )\,\left (b^3\,d\,\sqrt {b^4}-2\,a\,b^2\,d\,\sqrt {b^4}+a^2\,b\,d\,\sqrt {b^4}\right )}{b^4\,\left (a^2-2\,a\,b+b^2\right )\,\sqrt {-a\,d^2\,{\left (a-b\right )}^5}\,\sqrt {-a^6\,d^2+5\,a^5\,b\,d^2-10\,a^4\,b^2\,d^2+10\,a^3\,b^3\,d^2-5\,a^2\,b^4\,d^2+a\,b^5\,d^2}}\right )\,\left (\frac {b^3\,\sqrt {-a^6\,d^2+5\,a^5\,b\,d^2-10\,a^4\,b^2\,d^2+10\,a^3\,b^3\,d^2-5\,a^2\,b^4\,d^2+a\,b^5\,d^2}}{2}-a\,b^2\,\sqrt {-a^6\,d^2+5\,a^5\,b\,d^2-10\,a^4\,b^2\,d^2+10\,a^3\,b^3\,d^2-5\,a^2\,b^4\,d^2+a\,b^5\,d^2}+\frac {a^2\,b\,\sqrt {-a^6\,d^2+5\,a^5\,b\,d^2-10\,a^4\,b^2\,d^2+10\,a^3\,b^3\,d^2-5\,a^2\,b^4\,d^2+a\,b^5\,d^2}}{2}\right )\right )\,\sqrt {b^4}}{\sqrt {-a^6\,d^2+5\,a^5\,b\,d^2-10\,a^4\,b^2\,d^2+10\,a^3\,b^3\,d^2-5\,a^2\,b^4\,d^2+a\,b^5\,d^2}}+\frac {2\,b}{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a-b\right )\,\left (a\,d-b\,d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^4*(a + b*sinh(c + d*x)^2)),x)

[Out]

8/(3*(a*d - b*d)*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - 4/((a*d - b*d)*(2*exp(2*c
+ 2*d*x) + exp(4*c + 4*d*x) + 1)) + (atan((exp(2*c)*exp(2*d*x)*(4/(d*(a - b)^2*(b^4)^(1/2)*(a^2 - 2*a*b + b^2
)) + ((2*a - b)*(2*a^3*d*(b^4)^(1/2) - b^3*d*(b^4)^(1/2) + 4*a*b^2*d*(b^4)^(1/2) - 5*a^2*b*d*(b^4)^(1/2)))/(b^
4*(a^2 - 2*a*b + b^2)*(-a*d^2*(a - b)^5)^(1/2)*(a*b^5*d^2 - a^6*d^2 + 5*a^5*b*d^2 - 5*a^2*b^4*d^2 + 10*a^3*b^3
*d^2 - 10*a^4*b^2*d^2)^(1/2))) + ((2*a - b)*(b^3*d*(b^4)^(1/2) - 2*a*b^2*d*(b^4)^(1/2) + a^2*b*d*(b^4)^(1/2)))
/(b^4*(a^2 - 2*a*b + b^2)*(-a*d^2*(a - b)^5)^(1/2)*(a*b^5*d^2 - a^6*d^2 + 5*a^5*b*d^2 - 5*a^2*b^4*d^2 + 10*a^3
*b^3*d^2 - 10*a^4*b^2*d^2)^(1/2)))*((b^3*(a*b^5*d^2 - a^6*d^2 + 5*a^5*b*d^2 - 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 -
10*a^4*b^2*d^2)^(1/2))/2 - a*b^2*(a*b^5*d^2 - a^6*d^2 + 5*a^5*b*d^2 - 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 - 10*a^4
*b^2*d^2)^(1/2) + (a^2*b*(a*b^5*d^2 - a^6*d^2 + 5*a^5*b*d^2 - 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 - 10*a^4*b^2*d^2)
^(1/2))/2))*(b^4)^(1/2))/(a*b^5*d^2 - a^6*d^2 + 5*a^5*b*d^2 - 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 - 10*a^4*b^2*d^2)
^(1/2) + (2*b)/((exp(2*c + 2*d*x) + 1)*(a - b)*(a*d - b*d))

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