3.1.91 \(\int \frac {\text {sech}^6(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [91]

Optimal. Leaf size=101 \[ -\frac {a (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} (a+b)^{3/2} d}+\frac {\tanh (c+d x)}{b^2 d}+\frac {a^2 \tanh (c+d x)}{2 b^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 101, 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 = {4231, 398, 393, 214} \begin {gather*} \frac {a^2 \tanh (c+d x)}{2 b^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} d (a+b)^{3/2}}+\frac {\tanh (c+d x)}{b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(b^(5/2)*(a + b)^(3/2)*d) + Tanh[c + d*x]/(b
^2*d) + (a^2*Tanh[c + d*x])/(2*b^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))

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 393

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 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 4231

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(101)=202\).
time = 2.52, size = 229, normalized size = 2.27 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (-\frac {a (3 a+4 b) \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{(a+b)^{3/2} \sqrt {b (\cosh (c)-\sinh (c))^4}}+2 (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c) \text {sech}(c+d x) \sinh (d x)+\frac {a (a \text {sech}(2 c) \sinh (2 d x)-(a+2 b) \tanh (2 c))}{a+b}\right )}{8 b^2 d \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(-((a*(3*a + 4*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*
((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*
(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/((a + b)^(3/2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])) + 2*(a + 2*b + a*Cosh[2*(c
+ d*x)])*Sech[c]*Sech[c + d*x]*Sinh[d*x] + (a*(a*Sech[2*c]*Sinh[2*d*x] - (a + 2*b)*Tanh[2*c]))/(a + b)))/(8*b
^2*d*(a + b*Sech[c + d*x]^2)^2)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(89)=178\).
time = 1.75, size = 250, normalized size = 2.48

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b +2 a}+\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 b +2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a +4 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2 b +2 a}\right )}{b^{2}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(250\)
default	\(\frac {\frac {2 a \left (\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b +2 a}+\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 b +2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a +4 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2 b +2 a}\right )}{b^{2}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(250\)
risch	\(-\frac {3 a^{2} {\mathrm e}^{4 d x +4 c}+4 a b \,{\mathrm e}^{4 d x +4 c}+6 a^{2} {\mathrm e}^{2 d x +2 c}+14 a b \,{\mathrm e}^{2 d x +2 c}+8 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}+2 a b}{b^{2} d \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{\sqrt {a b +b^{2}}\, \left (a +b \right ) d b}-\frac {3 a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, \left (a +b \right ) d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{\sqrt {a b +b^{2}}\, \left (a +b \right ) d b}\)	\(468\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*a/b^2*((1/2/(a+b)*a*tanh(1/2*d*x+1/2*c)^3+1/2/(a+b)*a*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^4+b*t
anh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)+1/2*(3*a+4*b)/(a+b)*(-1/4/b^(1/2
)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b
)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))))+2/b^2*tanh(1/2*d*x+1
/2*c)/(tanh(1/2*d*x+1/2*c)^2+1))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (92) = 184\).
time = 0.58, size = 244, normalized size = 2.42 \begin {gather*} \frac {{\left (3 \, a + 4 \, b\right )} a \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, {\left (a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {3 \, a^{2} + 2 \, a b + 2 \, {\left (3 \, a^{2} + 7 \, a b + 4 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (3 \, a^{2} + 4 \, a b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (a^{2} b^{2} + a b^{3} + {\left (3 \, a^{2} b^{2} + 7 \, a b^{3} + 4 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (3 \, a^{2} b^{2} + 7 \, a b^{3} + 4 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{2} b^{2} + a b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*a + 4*b)*a*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqr
t((a + b)*b)))/((a*b^2 + b^3)*sqrt((a + b)*b)*d) + (3*a^2 + 2*a*b + 2*(3*a^2 + 7*a*b + 4*b^2)*e^(-2*d*x - 2*c)
+ (3*a^2 + 4*a*b)*e^(-4*d*x - 4*c))/((a^2*b^2 + a*b^3 + (3*a^2*b^2 + 7*a*b^3 + 4*b^4)*e^(-2*d*x - 2*c) + (3*a
^2*b^2 + 7*a*b^3 + 4*b^4)*e^(-4*d*x - 4*c) + (a^2*b^2 + a*b^3)*e^(-6*d*x - 6*c))*d)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1358 vs. \(2 (92) = 184\).
time = 0.43, size = 2958, normalized size = 29.29 \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)^6/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/4*(4*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^4 + 16*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)*si
nh(d*x + c)^3 + 4*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*sinh(d*x + c)^4 + 12*a^3*b + 20*a^2*b^2 + 8*a*b^3 + 8*(3*a^3
*b + 10*a^2*b^2 + 11*a*b^3 + 4*b^4)*cosh(d*x + c)^2 + 8*(3*a^3*b + 10*a^2*b^2 + 11*a*b^3 + 4*b^4 + 3*(3*a^3*b
+ 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^3 + 4*a^2*b)*cosh(d*x + c)^6 + 6*(3*a^3 + 4*a^
2*b)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^3 + 4*a^2*b)*sinh(d*x + c)^6 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*
x + c)^4 + (9*a^3 + 24*a^2*b + 16*a*b^2 + 15*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(3*a^3
+ 4*a^2*b)*cosh(d*x + c)^3 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a^3 + 4*a^2*b +
(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^2 + (15*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^4 + 9*a^3 + 24*a^2*b + 16*
a*b^2 + 6*(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(3*a^3 + 4*a^2*b)*cosh(d*x + c
)^5 + 2*(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^3 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c))*sinh(d*x
+ c))*sqrt(a*b + b^2)*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2
*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2
+ 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x +
c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(a*b + b^2))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(
d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)
^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 16*((3*a^3*b + 7*a^2*b^2 + 4*a*b^3)
*cosh(d*x + c)^3 + (3*a^3*b + 10*a^2*b^2 + 11*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^3*b^3 + 2*a^2*b
^4 + a*b^5)*d*cosh(d*x + c)^6 + 6*(a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3*b^3 + 2
*a^2*b^4 + a*b^5)*d*sinh(d*x + c)^6 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c)^4 + (15*(a^3
*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x + c)^2 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d)*sinh(d*x + c)^4 +
(3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c)^2 + 4*(5*(a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x
+ c)^3 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^3*b^3 + 2*a^2*
b^4 + a*b^5)*d*cosh(d*x + c)^4 + 6*(3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c)^2 + (3*a^3*b^3
+ 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d)*sinh(d*x + c)^2 + (a^3*b^3 + 2*a^2*b^4 + a*b^5)*d + 2*(3*(a^3*b^3 + 2*a^2*
b^4 + a*b^5)*d*cosh(d*x + c)^5 + 2*(3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c)^3 + (3*a^3*b^3
+ 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh
(d*x + c)^4 + 8*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(3*a^3*b + 7*a^2*b^2 + 4*a*b
^3)*sinh(d*x + c)^4 + 6*a^3*b + 10*a^2*b^2 + 4*a*b^3 + 4*(3*a^3*b + 10*a^2*b^2 + 11*a*b^3 + 4*b^4)*cosh(d*x +
c)^2 + 4*(3*a^3*b + 10*a^2*b^2 + 11*a*b^3 + 4*b^4 + 3*(3*a^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + ((3*a^3 + 4*a^2*b)*cosh(d*x + c)^6 + 6*(3*a^3 + 4*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^3 + 4
*a^2*b)*sinh(d*x + c)^6 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^4 + (9*a^3 + 24*a^2*b + 16*a*b^2 + 15*(3
*a^3 + 4*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^3 + (9*a^3 + 24*a^2*b
+ 16*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a^3 + 4*a^2*b + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c)^2 +
(15*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^4 + 9*a^3 + 24*a^2*b + 16*a*b^2 + 6*(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 + 2*(3*(3*a^3 + 4*a^2*b)*cosh(d*x + c)^5 + 2*(9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x
+ c)^3 + (9*a^3 + 24*a^2*b + 16*a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b - b^2)*arctan(1/2*(a*cosh(d*x +
c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-a*b - b^2)/(a*b + b^2)) + 8*((3*a
^3*b + 7*a^2*b^2 + 4*a*b^3)*cosh(d*x + c)^3 + (3*a^3*b + 10*a^2*b^2 + 11*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*
x + c))/((a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x + c)^6 + 6*(a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x + c)*sin
h(d*x + c)^5 + (a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*sinh(d*x + c)^6 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d
*cosh(d*x + c)^4 + (15*(a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x + c)^2 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 +
4*b^6)*d)*sinh(d*x + c)^4 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c)^2 + 4*(5*(a^3*b^3 + 2*
a^2*b^4 + a*b^5)*d*cosh(d*x + c)^3 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*cosh(d*x + c))*sinh(d*x + c
)^3 + (15*(a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(d*x + c)^4 + 6*(3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d*co
sh(d*x + c)^2 + (3*a^3*b^3 + 10*a^2*b^4 + 11*a*b^5 + 4*b^6)*d)*sinh(d*x + c)^2 + (a^3*b^3 + 2*a^2*b^4 + a*b^5)
*d + 2*(3*(a^3*b^3 + 2*a^2*b^4 + a*b^5)*d*cosh(...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{6}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (92) = 184\).
time = 0.60, size = 225, normalized size = 2.23 \begin {gather*} -\frac {\frac {{\left (3 \, a^{2} + 4 \, a b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} + 2 \, a b\right )}}{{\left (a b^{2} + b^{3}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((3*a^2 + 4*a*b)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a*b^2 + b^3)*sqrt(-a*b - b^
2)) + 2*(3*a^2*e^(4*d*x + 4*c) + 4*a*b*e^(4*d*x + 4*c) + 6*a^2*e^(2*d*x + 2*c) + 14*a*b*e^(2*d*x + 2*c) + 8*b^
2*e^(2*d*x + 2*c) + 3*a^2 + 2*a*b)/((a*b^2 + b^3)*(a*e^(6*d*x + 6*c) + 3*a*e^(4*d*x + 4*c) + 4*b*e^(4*d*x + 4*
c) + 3*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)))/d

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________