3.2.29 \(\int \frac {\text {sech}^3(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [129]

Optimal. Leaf size=129 \[ \frac {(4 a+3 b) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{3/2} d}+\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]

[Out]

1/8*(4*a+3*b)*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/a^(5/2)/(a+b)^(3/2)/d+1/4*b*sinh(d*x+c)/a/(a+b)/d/(a+(a+
b)*sinh(d*x+c)^2)^2+1/8*(4*a+3*b)*sinh(d*x+c)/a^2/(a+b)/d/(a+(a+b)*sinh(d*x+c)^2)

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Rubi [A]
time = 0.08, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3757, 393, 205, 211} \begin {gather*} \frac {(4 a+3 b) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^{3/2}}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {b \sinh (c+d x)}{4 a d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a + 3*b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(3/2)*d) + (b*Sinh[c + d*x])/(4*a
*(a + b)*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + ((4*a + 3*b)*Sinh[c + d*x])/(8*a^2*(a + b)*d*(a + (a + b)*Sinh[c
+ d*x]^2))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[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 3757

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

Rubi steps

\begin {align*} \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{\left (a+(a+b) x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {\left (\frac {3}{a}+\frac {1}{a+b}\right ) \text {Subst}\left (\int \frac {1}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {(4 a+3 b) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a+b) d}\\ &=\frac {(4 a+3 b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{3/2} d}+\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 123, normalized size = 0.95 \begin {gather*} \frac {-\frac {8 \sinh (c+d x)}{\left (a+(a+b) \sinh ^2(c+d x)\right )^2}+(4 a+3 b) \left (\frac {3 \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {5 a \sinh (c+d x)+3 (a+b) \sinh ^3(c+d x)}{a^2 \left (a+(a+b) \sinh ^2(c+d x)\right )^2}\right )}{24 (a+b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*Sinh[c + d*x])/(a + (a + b)*Sinh[c + d*x]^2)^2 + (4*a + 3*b)*((3*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[
a]])/(a^(5/2)*Sqrt[a + b]) + (5*a*Sinh[c + d*x] + 3*(a + b)*Sinh[c + d*x]^3)/(a^2*(a + (a + b)*Sinh[c + d*x]^2
)^2)))/(24*(a + b)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(341\) vs. \(2(115)=230\).
time = 2.95, size = 342, normalized size = 2.65

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (5 b +4 a \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a +b \right )}-\frac {\left (4 a^{2}+13 a b +12 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} \left (a +b \right )}+\frac {\left (4 a^{2}+13 a b +12 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} \left (a +b \right )}+\frac {\left (5 b +4 a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a +b \right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (4 a +3 b \right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4 a \left (a +b \right )}}{d}\)	\(342\)
default	\(\frac {\frac {-\frac {\left (5 b +4 a \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a +b \right )}-\frac {\left (4 a^{2}+13 a b +12 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} \left (a +b \right )}+\frac {\left (4 a^{2}+13 a b +12 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} \left (a +b \right )}+\frac {\left (5 b +4 a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a +b \right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (4 a +3 b \right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4 a \left (a +b \right )}}{d}\)	\(342\)
risch	\(\frac {\left (4 a^{2} {\mathrm e}^{6 d x +6 c}+7 a b \,{\mathrm e}^{6 d x +6 c}+3 b^{2} {\mathrm e}^{6 d x +6 c}+4 a^{2} {\mathrm e}^{4 d x +4 c}-a b \,{\mathrm e}^{4 d x +4 c}-9 b^{2} {\mathrm e}^{4 d x +4 c}-4 a^{2} {\mathrm e}^{2 d x +2 c}+a b \,{\mathrm e}^{2 d x +2 c}+9 b^{2} {\mathrm e}^{2 d x +2 c}-4 a^{2}-7 a b -3 b^{2}\right ) {\mathrm e}^{d x +c}}{4 \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} a^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d a}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right ) b}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d a}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right ) b}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*(-1/8*(5*b+4*a)/a/(a+b)*tanh(1/2*d*x+1/2*c)^7-1/8*(4*a^2+13*a*b+12*b^2)/a^2/(a+b)*tanh(1/2*d*x+1/2*c)^5
+1/8*(4*a^2+13*a*b+12*b^2)/a^2/(a+b)*tanh(1/2*d*x+1/2*c)^3+1/8*(5*b+4*a)/a/(a+b)*tanh(1/2*d*x+1/2*c))/(a*tanh(
1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2+1/4/a*(4*a+3*b)/(a+b)*(-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)*arctanh(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))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((4*a^2*e^(7*c) + 7*a*b*e^(7*c) + 3*b^2*e^(7*c))*e^(7*d*x) + (4*a^2*e^(5*c) - a*b*e^(5*c) - 9*b^2*e^(5*c))
*e^(5*d*x) - (4*a^2*e^(3*c) - a*b*e^(3*c) - 9*b^2*e^(3*c))*e^(3*d*x) - (4*a^2*e^c + 7*a*b*e^c + 3*b^2*e^c)*e^(
d*x))/(a^5*d + 3*a^4*b*d + 3*a^3*b^2*d + a^2*b^3*d + (a^5*d*e^(8*c) + 3*a^4*b*d*e^(8*c) + 3*a^3*b^2*d*e^(8*c)
+ a^2*b^3*d*e^(8*c))*e^(8*d*x) + 4*(a^5*d*e^(6*c) + a^4*b*d*e^(6*c) - a^3*b^2*d*e^(6*c) - a^2*b^3*d*e^(6*c))*e
^(6*d*x) + 2*(3*a^5*d*e^(4*c) + a^4*b*d*e^(4*c) + a^3*b^2*d*e^(4*c) + 3*a^2*b^3*d*e^(4*c))*e^(4*d*x) + 4*(a^5*
d*e^(2*c) + a^4*b*d*e^(2*c) - a^3*b^2*d*e^(2*c) - a^2*b^3*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/32*((4*a*e^(3*
c) + 3*b*e^(3*c))*e^(3*d*x) + (4*a*e^c + 3*b*e^c)*e^(d*x))/(a^4 + 2*a^3*b + a^2*b^2 + (a^4*e^(4*c) + 2*a^3*b*e
^(4*c) + a^2*b^2*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3627 vs. \(2 (115) = 230\).
time = 0.47, size = 6614, normalized size = 51.27 \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)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^7 + 28*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b
^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*sinh(d*x + c)^7 + 4*(4*a^4 + 3
*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^5 + 4*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3 + 21*(4*a^4 + 11*a^
3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)
*cosh(d*x + c)^3 + (4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(4*a^4 + 3*a^3*
b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^3 + 4*(35*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 -
4*a^4 - 3*a^3*b + 10*a^2*b^2 + 9*a*b^3 + 10*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + 4*(21*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^5 + 10*(4*a^4 + 3*a^3*b - 10*a^2*b^2 -
9*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - ((4*a^
3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^8 + 8*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(
d*x + c)^7 + (4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*sinh(d*x + c)^8 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cos
h(d*x + c)^6 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(4*a^3 + 3*a^2*b - 4*a*b^2 -
3*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 +
11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^4 + 12*a^3 + a^2*b + 6*a*b^2 + 9*b^3 + 30*(4*a^3 + 3*a^2*b - 4*a*b
^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 10
*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(
d*x + c)^3 + 4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^2 + 4*(
7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)
^4 + 4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 3*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 8*((4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^7 + 3*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x +
c)^5 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^3 + (4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)
)*sinh(d*x + c))*sqrt(-a^2 - a*b)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a
+ b)*sinh(d*x + c)^4 - 2*(3*a + b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c)^2 +
4*((a + b)*cosh(d*x + c)^3 - (3*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*si
nh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a
+ b)/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*
cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a - b)
*cosh(d*x + c))*sinh(d*x + c) + a + b)) - 4*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c) + 4*(7*(4*
a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^6 + 5*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x +
c)^4 - 4*a^4 - 11*a^3*b - 10*a^2*b^2 - 3*a*b^3 - 3*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^8 + 8*(a^7 + 4*a^6*b + 6*a^5
*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4
)*d*sinh(d*x + c)^8 + 4*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 4*a^6*b + 6*a^5*
b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3
*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^
4*b^3 + a^3*b^4)*d*cosh(d*x + c)^3 + 3*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(35*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 30*(a^7 + 2*a^6*b - 2*a^4*b^3 -
a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(a^7
+ 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d
*cosh(d*x + c)^5 + 10*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 +
4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b
^4)*d*cosh(d*x + c)^6 + 15*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7 + 4*a^6*b + 2*a^
5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^2 +
(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d + 8*((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*c
osh(d*x + c)^7 + 3*(a^7 + 2*a^6*b - 2*a^4*b^3 -...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**3/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral(sech(c + d*x)**3/(a + b*tanh(c + d*x)**2)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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