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

Optimal. Leaf size=55 \[ \frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d}-\frac {\tan ^{-1}(\sinh (c+d x))}{b d} \]

[Out]

-arctan(sinh(d*x+c))/b/d+arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))*(a+b)^(1/2)/b/d/a^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 55, 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 = {3676, 391, 203, 205} \[ \frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b d}-\frac {\tan ^{-1}(\sinh (c+d x))}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTan[Sinh[c + d*x]]/(b*d)) + (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b*d)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

Rule 3676

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

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Mathematica [A]  time = 0.21, size = 55, normalized size = 1.00 \[ -\frac {\frac {\sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a}}+2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((Sqrt[a + b]*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/Sqrt[a] + 2*ArcTan[Tanh[(c + d*x)/2]])/(b*d))

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fricas [B]  time = 0.45, size = 540, normalized size = 9.82 \[ \left [\frac {\sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {a + b}{a}} + a + b}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right ) - 4 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{2 \, b d}, \frac {\sqrt {\frac {a + b}{a}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {a + b}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) + \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )}}\right ) - 2 \, \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(-(a + b)/a)*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*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x
+ c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c))*sqrt(-(a + b)/a) + 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)*c
osh(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*arctan(cosh(d*x + c) + sinh(d*x + c)))/(b*d), (sqrt((a + b)/a)*arct
an(1/2*sqrt((a + b)/a)*(cosh(d*x + c) + sinh(d*x + c))) + sqrt((a + b)/a)*arctan(1/2*((a + b)*cosh(d*x + c)^3
+ 3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cos
h(d*x + c)^2 + 3*a - b)*sinh(d*x + c))*sqrt((a + b)/a)/(a + b)) - 2*arctan(cosh(d*x + c) + sinh(d*x + c)))/(b*
d)]

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giac [B]  time = 0.40, size = 563, normalized size = 10.24 \[ \frac {\frac {{\left (2 \, \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} a b^{2} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} + \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} \sqrt {-a b} {\left (a + b\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} {\left | b \right |} - {\left (a b^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |}\right )} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} + \sqrt {{\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} {\left (a b + b^{2}\right )}}}{a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {-a b} {\left | b \right |}} - \frac {{\left (2 \, \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} a b^{2} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} - \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} \sqrt {-a b} {\left (a + b\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} {\left | b \right |} - {\left (a b^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |}\right )} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} - \sqrt {{\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} {\left (a b + b^{2}\right )}}}{a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {-a b} {\left | b \right |}} - \frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*a*b^2*abs(a*e^(2*c) + b*e^(2*c)) + sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a
+ b))*sqrt(-a*b)*(a + b)*abs(a*e^(2*c) + b*e^(2*c))*abs(b) - (a*b^2 - b^3)*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a +
b))*abs(a*e^(2*c) + b*e^(2*c)))*arctan(e^(d*x)/sqrt((a*b*e^(2*c) - b^2*e^(2*c) + sqrt((a*b*e^(2*c) - b^2*e^(2
*c))^2 - (a*b*e^(4*c) + b^2*e^(4*c))*(a*b + b^2)))/(a*b*e^(4*c) + b^2*e^(4*c))))*e^(-2*c)/((a^3*b + 3*a^2*b^2
+ 3*a*b^3 + b^4)*sqrt(-a*b)*abs(b)) - (2*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))*a*b^2*abs(a*e^(2*c) + b*e^(2*c
)) - sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))*sqrt(-a*b)*(a + b)*abs(a*e^(2*c) + b*e^(2*c))*abs(b) - (a*b^2 - b^
3)*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c)))*arctan(e^(d*x)/sqrt((a*b*e^(2*c) - b^2*e
^(2*c) - sqrt((a*b*e^(2*c) - b^2*e^(2*c))^2 - (a*b*e^(4*c) + b^2*e^(4*c))*(a*b + b^2)))/(a*b*e^(4*c) + b^2*e^(
4*c))))*e^(-2*c)/((a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-a*b)*abs(b)) - 2*arctan(e^(d*x + c))/b)/d

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maple [B]  time = 0.34, size = 494, normalized size = 8.98 \[ -\frac {a \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 )}{d b \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {a \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 )}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {a \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 )}{d b \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {a \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 )}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {\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 )}{d \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\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 ) b}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\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 )}{d \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {\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 ) b}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*a/b/((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/d*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/d*a/b/((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))+1/d*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))-1/d/((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/d/(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))*b+1/d/((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))+1/d/(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))*b-2/d/b*arctan(tanh(1/2*d*x+1/2*c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b d} + 8 \, \int \frac {{\left (a e^{\left (3 \, c\right )} + b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a e^{c} + b e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a b + b^{2} + {\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(e^(d*x + c))/(b*d) + 8*integrate(1/4*((a*e^(3*c) + b*e^(3*c))*e^(3*d*x) + (a*e^c + b*e^c)*e^(d*x))/(
a*b + b^2 + (a*b*e^(4*c) + b^2*e^(4*c))*e^(4*d*x) + 2*(a*b*e^(2*c) - b^2*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 1.70, size = 449, normalized size = 8.16 \[ \frac {\sqrt {a+b}\,\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a+b}\,\sqrt {a\,b^2\,d^2}}{2\,a\,b\,d}\right )-2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,a\,b^2\,d\,\sqrt {a+b}-6\,a^2\,b\,d\,\sqrt {a+b}\right )}{a^3\,b^3\,d^2\,{\left (a+b\right )}^2\,\left (a^2+2\,a\,b+b^2\right )}+\frac {32\,\left (3\,a^2\,\sqrt {a\,b^2\,d^2}-b^2\,\sqrt {a\,b^2\,d^2}+2\,a\,b\,\sqrt {a\,b^2\,d^2}\right )}{a^3\,b^2\,d\,{\left (a+b\right )}^{3/2}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {a\,b^2\,d^2}}\right )-\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,a^2\,\sqrt {a\,b^2\,d^2}-b^2\,\sqrt {a\,b^2\,d^2}+2\,a\,b\,\sqrt {a\,b^2\,d^2}\right )}{a^3\,b^2\,d\,{\left (a+b\right )}^{3/2}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {a\,b^2\,d^2}}\right )\,\left (a^4\,b\,\left (a+b\right )\,\sqrt {a\,b^2\,d^2}+a^2\,b^3\,\left (a+b\right )\,\sqrt {a\,b^2\,d^2}+2\,a^3\,b^2\,\left (a+b\right )\,\sqrt {a\,b^2\,d^2}\right )}{192\,a-64\,b}\right )\right )}{2\,\sqrt {a\,b^2\,d^2}}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (9\,a^2\,\sqrt {b^2\,d^2}+b^2\,\sqrt {b^2\,d^2}-6\,a\,b\,\sqrt {b^2\,d^2}\right )}{9\,d\,a^2\,b-6\,d\,a\,b^2+d\,b^3}\right )}{\sqrt {b^2\,d^2}} \]

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)),x)

[Out]

((a + b)^(1/2)*(2*atan((exp(d*x)*exp(c)*(a + b)^(1/2)*(a*b^2*d^2)^(1/2))/(2*a*b*d)) - 2*atan(((exp(d*x)*exp(c)
*((64*(2*a*b^2*d*(a + b)^(1/2) - 6*a^2*b*d*(a + b)^(1/2)))/(a^3*b^3*d^2*(a + b)^2*(2*a*b + a^2 + b^2)) + (32*(
3*a^2*(a*b^2*d^2)^(1/2) - b^2*(a*b^2*d^2)^(1/2) + 2*a*b*(a*b^2*d^2)^(1/2)))/(a^3*b^2*d*(a + b)^(3/2)*(2*a*b +
a^2 + b^2)*(a*b^2*d^2)^(1/2))) - (32*exp(3*c)*exp(3*d*x)*(3*a^2*(a*b^2*d^2)^(1/2) - b^2*(a*b^2*d^2)^(1/2) + 2*
a*b*(a*b^2*d^2)^(1/2)))/(a^3*b^2*d*(a + b)^(3/2)*(2*a*b + a^2 + b^2)*(a*b^2*d^2)^(1/2)))*(a^4*b*(a + b)*(a*b^2
*d^2)^(1/2) + a^2*b^3*(a + b)*(a*b^2*d^2)^(1/2) + 2*a^3*b^2*(a + b)*(a*b^2*d^2)^(1/2)))/(192*a - 64*b))))/(2*(
a*b^2*d^2)^(1/2)) - (2*atan((exp(d*x)*exp(c)*(9*a^2*(b^2*d^2)^(1/2) + b^2*(b^2*d^2)^(1/2) - 6*a*b*(b^2*d^2)^(1
/2)))/(b^3*d - 6*a*b^2*d + 9*a^2*b*d)))/(b^2*d^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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