3.88 \(\int \frac {\text {sech}^3(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\)
Optimal. Leaf size=73 \[ \frac {\sinh (c+d x)}{2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} d (a+b)^{3/2}} \]
[Out]
1/2*sinh(d*x+c)/(a+b)/d/(a+b+a*sinh(d*x+c)^2)+1/2*arctan(sinh(d*x+c)*a^(1/2)/(a+b)^(1/2))/(a+b)^(3/2)/d/a^(1/2
)
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used =
{4147, 199, 205} \[ \frac {\sinh (c+d x)}{2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} d (a+b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]
[Out]
ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(2*Sqrt[a]*(a + b)^(3/2)*d) + Sinh[c + d*x]/(2*(a + b)*d*(a + b +
a*Sinh[c + d*x]^2))
Rule 199
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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 4147
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 )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{2 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{2 (a+b) d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} (a+b)^{3/2} d}+\frac {\sinh (c+d x)}{2 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 108, normalized size = 1.48 \[ \frac {\text {sech}^4(c+d x) (a \cosh (2 c+2 d x)+a+2 b)^2 \left (\frac {\sinh (c+d x)}{(a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{3/2}}\right )}{8 d \left (a+b \text {sech}^2(c+d x)\right )^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]
[Out]
((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*(a +
b)^(3/2)) + Sinh[c + d*x]/((a + b)*(a + b + a*Sinh[c + d*x]^2))))/(8*d*(a + b*Sech[c + d*x]^2)^2)
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fricas [B] time = 0.48, size = 1570, normalized size = 21.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/4*(4*(a^2 + a*b)*cosh(d*x + c)^3 + 12*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^2 + 4*(a^2 + a*b)*sinh(d*x +
c)^3 - (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)*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*co
sh(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*(a^2 + a*b)*cosh
(d*x + c) + 4*(3*(a^2 + a*b)*cosh(d*x + c)^2 - a^2 - a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x
+ c)^4 + 4*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x +
c)^4 + 2*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x
+ c)^2 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d)*sinh(d*x + c)^2 + (a^4 + 2*a^3*b + a^2*b^2)*d + 4*((a^4 + 2*
a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/
2*(2*(a^2 + a*b)*cosh(d*x + c)^3 + 6*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a^2 + a*b)*sinh(d*x + c)^3
+ (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)*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3
+ (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 + a*b)) + (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)*sqrt(
a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a + b)) - 2*(a^2 + a*b)*cosh(d*x + c) +
2*(3*(a^2 + a*b)*cosh(d*x + c)^2 - a^2 - a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^4 + 4
*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*(
a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a
^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d)*sinh(d*x + c)^2 + (a^4 + 2*a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2
*b^2)*d*cosh(d*x + c)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[6,-20]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[89,-63]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[12,-32]Warning, need to ch
oose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [
a,b]=[2,72]Undef/Unsigned Inf encountered in limitEvaluation time: 0.58Limit: Max order reached or unable to m
ake series expansion Error: Bad Argument Value
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maple [B] time = 0.27, size = 241, normalized size = 3.30 \[ -\frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right ) \left (a +b \right )}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right ) \left (a +b \right )}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 d \left (a +b \right )^{\frac {3}{2}} \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 d \left (a +b \right )^{\frac {3}{2}} \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x)
[Out]
-1/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)
/(a+b)*tanh(1/2*d*x+1/2*c)^3+1/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*
tanh(1/2*d*x+1/2*c)^2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)+1/2/d/(a+b)^(3/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tan
h(1/2*d*x+1/2*c)-2*b^(1/2))/a^(1/2))+1/2/d/(a+b)^(3/2)/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 \[ \frac {e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}{a^{2} d + a b d + {\left (a^{2} d e^{\left (4 \, c\right )} + a b d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{2} d e^{\left (2 \, c\right )} + 3 \, a b d e^{\left (2 \, c\right )} + 2 \, b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + 8 \, \int \frac {e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{8 \, {\left (a^{2} + a b + {\left (a^{2} e^{\left (4 \, c\right )} + a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{2} e^{\left (2 \, c\right )} + 3 \, a b e^{\left (2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
(e^(3*d*x + 3*c) - e^(d*x + c))/(a^2*d + a*b*d + (a^2*d*e^(4*c) + a*b*d*e^(4*c))*e^(4*d*x) + 2*(a^2*d*e^(2*c)
+ 3*a*b*d*e^(2*c) + 2*b^2*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/8*(e^(3*d*x + 3*c) + e^(d*x + c))/(a^2 + a*b +
(a^2*e^(4*c) + a*b*e^(4*c))*e^(4*d*x) + 2*(a^2*e^(2*c) + 3*a*b*e^(2*c) + 2*b^2*e^(2*c))*e^(2*d*x)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \]
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)^2),x)
[Out]
int(1/(cosh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)**3/(a+b*sech(d*x+c)**2)**2,x)
[Out]
Integral(sech(c + d*x)**3/(a + b*sech(c + d*x)**2)**2, x)
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