3.1.97 \(\int \frac {\text {sech}^3(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [97]

Optimal. Leaf size=123 \[ \frac {(4 a+b) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{5/2} d}-\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {(4 a+b) \sinh (c+d x)}{8 a (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 123, 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 = {4232, 393, 205, 211} \begin {gather*} \frac {(4 a+b) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{3/2} d (a+b)^{5/2}}+\frac {(4 a+b) \sinh (c+d x)}{8 a d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {b \sinh (c+d x)}{4 a d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(3/2)*(a + b)^(5/2)*d) - (b*Sinh[c + d*x])/(4*a*(
a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + ((4*a + b)*Sinh[c + d*x])/(8*a*(a + b)^2*d*(a + b + a*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 4232

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + 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 \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{\left (a+b+a 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+b+a \sinh ^2(c+d x)\right )^2}+\frac {(4 a+b) \text {Subst}\left (\int \frac {1}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a (a+b) d}\\ &=-\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {(4 a+b) \sinh (c+d x)}{8 a (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac {(4 a+b) \text {Subst}\left (\int \frac {1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{8 a (a+b)^2 d}\\ &=\frac {(4 a+b) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{5/2} d}-\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {(4 a+b) \sinh (c+d x)}{8 a (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 159, normalized size = 1.29 \begin {gather*} -\frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \left (\frac {8 \sinh (c+d x)}{\left (a+b+a \sinh ^2(c+d x)\right )^2}-(4 a+b) \left (\frac {3 \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}+\frac {5 (a+b) \sinh (c+d x)+3 a \sinh ^3(c+d x)}{(a+b)^2 \left (a+b+a \sinh ^2(c+d x)\right )^2}\right )\right )}{192 a d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((8*Sinh[c + d*x])/(a + b + a*Sinh[c + d*x]^2)^2 - (
4*a + b)*((3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (5*(a + b)*Sinh[c + d*x] +
3*a*Sinh[c + d*x]^3)/((a + b)^2*(a + b + a*Sinh[c + d*x]^2)^2))))/(a*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(109)=218\).
time = 2.08, size = 294, normalized size = 2.39

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (4 a -b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a +b \right )}-\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a +b \right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (4 a +b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{4 a \left (a^{2}+2 a b +b^{2}\right )}}{d}\)	\(294\)
default	\(\frac {\frac {-\frac {\left (4 a -b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a +b \right )}-\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a +b \right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (4 a +b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{4 a \left (a^{2}+2 a b +b^{2}\right )}}{d}\)	\(294\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (4 a^{2} {\mathrm e}^{6 d x +6 c}+a b \,{\mathrm e}^{6 d x +6 c}+4 a^{2} {\mathrm e}^{4 d x +4 c}+9 a b \,{\mathrm e}^{4 d x +4 c}-4 b^{2} {\mathrm e}^{4 d x +4 c}-4 a^{2} {\mathrm e}^{2 d x +2 c}-9 a b \,{\mathrm e}^{2 d x +2 c}+4 b^{2} {\mathrm e}^{2 d x +2 c}-4 a^{2}-a b \right )}{4 a \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}\)	\(414\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*(-1/8*(4*a-b)/a/(a+b)*tanh(1/2*d*x+1/2*c)^7-1/8*(4*a^2-5*a*b+3*b^2)/(a+b)^2/a*tanh(1/2*d*x+1/2*c)^5+1/8
*(4*a^2-5*a*b+3*b^2)/(a+b)^2/a*tanh(1/2*d*x+1/2*c)^3+1/8*(4*a-b)/a/(a+b)*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+
1/2*c)^4+b*tanh(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)^2+1/4*(4*a+b)/a/(a^2
+2*a*b+b^2)*(1/2/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)+2*b^(1/2))/a^(1/2))+1/2/(a+
b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/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*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/4*((4*a^2*e^(7*c) + a*b*e^(7*c))*e^(7*d*x) + (4*a^2*e^(5*c) + 9*a*b*e^(5*c) - 4*b^2*e^(5*c))*e^(5*d*x) - (4*
a^2*e^(3*c) + 9*a*b*e^(3*c) - 4*b^2*e^(3*c))*e^(3*d*x) - (4*a^2*e^c + a*b*e^c)*e^(d*x))/(a^5*d + 2*a^4*b*d + a
^3*b^2*d + (a^5*d*e^(8*c) + 2*a^4*b*d*e^(8*c) + a^3*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^5*d*e^(6*c) + 4*a^4*b*d*e^
(6*c) + 5*a^3*b^2*d*e^(6*c) + 2*a^2*b^3*d*e^(6*c))*e^(6*d*x) + 2*(3*a^5*d*e^(4*c) + 14*a^4*b*d*e^(4*c) + 27*a^
3*b^2*d*e^(4*c) + 24*a^2*b^3*d*e^(4*c) + 8*a*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^5*d*e^(2*c) + 4*a^4*b*d*e^(2*c) +
5*a^3*b^2*d*e^(2*c) + 2*a^2*b^3*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/32*((4*a*e^(3*c) + b*e^(3*c))*e^(3*d*x)
+ (4*a*e^c + 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) + 4*a^3*b*e^(2*c) + 5*a^2*b^2*e^(2*c) + 2*a*b^3*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. 3262 vs. \(2 (109) = 218\).
time = 0.47, size = 6037, normalized size = 49.08 \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*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^7 + 28*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x +
c)^6 + 4*(4*a^4 + 5*a^3*b + a^2*b^2)*sinh(d*x + c)^7 + 4*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x +
c)^5 + 4*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 21*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^5 + 20*(7*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + (4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x +
c))*sinh(d*x + c)^4 - 4*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 + 4*(35*(4*a^4 + 5*a^3*b + a
^2*b^2)*cosh(d*x + c)^4 - 4*a^4 - 13*a^3*b - 5*a^2*b^2 + 4*a*b^3 + 10*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 10*(4*a^4 + 13*a^3*b +
5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c
)^2 - ((4*a^3 + a^2*b)*cosh(d*x + c)^8 + 8*(4*a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + a^2*b)*sin
h(d*x + c)^8 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^6 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2 + 7*(4*a^3 + a^2*b
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + a^2*b)*
cosh(d*x + c)^4 + 12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3 + 30*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(
d*x + c)^4 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^5 + 10*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + (12*a^3 +
35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + a^2*b + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*c
osh(d*x + c)^2 + 4*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^6 + 15*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^4 + 4*a^3
+ 9*a^2*b + 2*a*b^2 + 3*(12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 +
a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + (12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^
3)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 - a*b)*log((a*cosh(d*
x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh
(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) - 4
*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(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)/(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)) - 4*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c) + 4*(7*(4*a^4 + 5*
a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 5*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^4 - 4*a^4 - 5*a^3*
b - a^2*b^2 - 3*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^7 + 3*a^6*b + 3*a
^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^8 + 4*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b
^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b + 9*a^5*
b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 +
8*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + 3*(a^7 + 5*a^6*
b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 3*a^6*b + 3*a^5*b^2 + a
^4*b^3)*d*cosh(d*x + c)^4 + 30*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7
+ 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d)*sinh(d*x + c)^4 + 4*(a^7 + 5*a^6*b + 9*a^5*b
^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^5 +
10*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 5
1*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b
^3)*d*cosh(d*x + c)^6 + 15*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7 +
17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b + 9*a^5*b^2 +
7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d + 8*((a^7 + 3*a^6*b + 3*a^
5*b^2 + a^4*b^3)*d*cosh(d*x + c)^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^5 +
(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^3 + (a^7 + 5*a^6*b + 9*
a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x
+ c)^7 + 14*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(4*a^4 + 5*a^3*b + a^2*b^2)*sinh(d*x
+ c)^7 + 2*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a...

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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 \operatorname {sech}^{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*sech(d*x+c)**2)**3,x)

[Out]

Integral(sech(c + d*x)**3/(a + b*sech(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*sech(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 (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\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/cosh(c + d*x)^2)^3),x)

[Out]

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

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