3.2.14 \(\int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) [114]

Optimal. Leaf size=75 \[ \frac {(a+b)^2 \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a+2 b) \tanh (c+d x)}{b^2 d}+\frac {\tanh ^3(c+d x)}{3 b d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3756, 398, 211} \begin {gather*} \frac {(a+b)^2 \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a+2 b) \tanh (c+d x)}{b^2 d}+\frac {\tanh ^3(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 3756

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

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

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Mathematica [A]
time = 0.28, size = 71, normalized size = 0.95 \begin {gather*} \frac {(a+b)^2 \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {\left (3 a+5 b+b \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{3 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)^2*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((3*a + 5*b + b*Sech[c + d*x]^2)*Tan
h[c + d*x])/(3*b^2*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs. \(2(65)=130\).
time = 2.49, size = 252, normalized size = 3.36

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\left (a -\sqrt {b \left (a +b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (-a -\sqrt {b \left (a +b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{b^{2}}+\frac {2 \left (-2 b -a \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {8 b}{3}-2 a \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-2 b -a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{d}\)	\(252\)
default	\(\frac {\frac {2 a \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\left (a -\sqrt {b \left (a +b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (-a -\sqrt {b \left (a +b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{b^{2}}+\frac {2 \left (-2 b -a \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {8 b}{3}-2 a \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-2 b -a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{d}\)	\(252\)
risch	\(\frac {2 a \,{\mathrm e}^{4 d x +4 c}+2 b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}+8 b \,{\mathrm e}^{2 d x +2 c}+2 a +\frac {10 b}{3}}{d \,b^{2} \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{\sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{\sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}\)	\(433\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2/b^2*a*(a^2+2*a*b+b^2)*(-1/2*(a-(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*(-a-(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)))+2/b^
2*((-2*b-a)*tanh(1/2*d*x+1/2*c)^5+(-8/3*b-2*a)*tanh(1/2*d*x+1/2*c)^3+(-2*b-a)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d
*x+1/2*c)^2+1)^3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (65) = 130\).
time = 0.51, size = 140, normalized size = 1.87 \begin {gather*} -\frac {2 \, {\left (6 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, a + 5 \, b\right )}}{3 \, {\left (3 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2} e^{\left (-6 \, d x - 6 \, c\right )} + b^{2}\right )} d} - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(6*(a + 2*b)*e^(-2*d*x - 2*c) + 3*(a + b)*e^(-4*d*x - 4*c) + 3*a + 5*b)/((3*b^2*e^(-2*d*x - 2*c) + 3*b^2*
e^(-4*d*x - 4*c) + b^2*e^(-6*d*x - 6*c) + b^2)*d) - (a^2 + 2*a*b + b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) +
a - b)/sqrt(a*b))/(sqrt(a*b)*b^2*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (65) = 130\).
time = 0.38, size = 2032, normalized size = 27.09 \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*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/6*(12*(a^2*b + a*b^2)*cosh(d*x + c)^4 + 48*(a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 12*(a^2*b + a*b^
2)*sinh(d*x + c)^4 + 12*a^2*b + 20*a*b^2 + 24*(a^2*b + 2*a*b^2)*cosh(d*x + c)^2 + 24*(a^2*b + 2*a*b^2 + 3*(a^2
*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b^2)*
cosh(d*x + c)*sinh(d*x + c)^5 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 +
3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(
d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 3*
(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x +
c)^2 + a^2 + 2*a*b + b^2 + 6*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (
a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2
+ 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x +
c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2
*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*((a + b)*cosh(d*x + c)^2 + 2*(a + b
)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(-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)
) + 48*((a^2*b + a*b^2)*cosh(d*x + c)^3 + (a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x +
c)^6 + 6*a*b^3*d*cosh(d*x + c)*sinh(d*x + c)^5 + a*b^3*d*sinh(d*x + c)^6 + 3*a*b^3*d*cosh(d*x + c)^4 + 3*a*b^3
*d*cosh(d*x + c)^2 + a*b^3*d + 3*(5*a*b^3*d*cosh(d*x + c)^2 + a*b^3*d)*sinh(d*x + c)^4 + 4*(5*a*b^3*d*cosh(d*x
+ c)^3 + 3*a*b^3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*a*b^3*d*cosh(d*x + c)^4 + 6*a*b^3*d*cosh(d*x + c)^2
+ a*b^3*d)*sinh(d*x + c)^2 + 6*(a*b^3*d*cosh(d*x + c)^5 + 2*a*b^3*d*cosh(d*x + c)^3 + a*b^3*d*cosh(d*x + c))*s
inh(d*x + c)), 1/3*(6*(a^2*b + a*b^2)*cosh(d*x + c)^4 + 24*(a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 6*(
a^2*b + a*b^2)*sinh(d*x + c)^4 + 6*a^2*b + 10*a*b^2 + 12*(a^2*b + 2*a*b^2)*cosh(d*x + c)^2 + 12*(a^2*b + 2*a*b
^2 + 3*(a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 6*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(d*
x + c)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b +
b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x
+ c)^2 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)
*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 6*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x
+ c)^3 + (a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*
(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(a*b)/(a*b)) + 24*((a^2*b + a*b^2)*
cosh(d*x + c)^3 + (a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^6 + 6*a*b^3*d*cosh(d*
x + c)*sinh(d*x + c)^5 + a*b^3*d*sinh(d*x + c)^6 + 3*a*b^3*d*cosh(d*x + c)^4 + 3*a*b^3*d*cosh(d*x + c)^2 + a*b
^3*d + 3*(5*a*b^3*d*cosh(d*x + c)^2 + a*b^3*d)*sinh(d*x + c)^4 + 4*(5*a*b^3*d*cosh(d*x + c)^3 + 3*a*b^3*d*cosh
(d*x + c))*sinh(d*x + c)^3 + 3*(5*a*b^3*d*cosh(d*x + c)^4 + 6*a*b^3*d*cosh(d*x + c)^2 + a*b^3*d)*sinh(d*x + c)
^2 + 6*(a*b^3*d*cosh(d*x + c)^5 + 2*a*b^3*d*cosh(d*x + c)^3 + a*b^3*d*cosh(d*x + c))*sinh(d*x + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (65) = 130\).
time = 0.63, size = 135, normalized size = 1.80 \begin {gather*} \frac {\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a + 5 \, b\right )}}{b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(a^2 + 2*a*b + b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*b^
2) + 2*(3*a*e^(4*d*x + 4*c) + 3*b*e^(4*d*x + 4*c) + 6*a*e^(2*d*x + 2*c) + 12*b*e^(2*d*x + 2*c) + 3*a + 5*b)/(b
^2*(e^(2*d*x + 2*c) + 1)^3))/d

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Mupad [B]
time = 1.65, size = 252, normalized size = 3.36 \begin {gather*} \frac {4}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {8}{3\,b\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {2\,\left (a+b\right )}{b^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{b^2}-\frac {2\,\left (a+b\right )\,\left (a+b+a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{5/2}}\right )\,{\left (a+b\right )}^2}{2\,\sqrt {-a}\,b^{5/2}\,d}-\frac {\ln \left (\frac {2\,\left (a+b\right )\,\left (a+b+a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{5/2}}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{b^2}\right )\,{\left (a+b\right )}^2}{2\,\sqrt {-a}\,b^{5/2}\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(b*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - 8/(3*b*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(
6*c + 6*d*x) + 1)) + (2*(a + b))/(b^2*d*(exp(2*c + 2*d*x) + 1)) + (log(- (4*exp(2*c + 2*d*x)*(a + b))/b^2 - (2
*(a + b)*(a + b + a*exp(2*c + 2*d*x) - b*exp(2*c + 2*d*x)))/((-a)^(1/2)*b^(5/2)))*(a + b)^2)/(2*(-a)^(1/2)*b^(
5/2)*d) - (log((2*(a + b)*(a + b + a*exp(2*c + 2*d*x) - b*exp(2*c + 2*d*x)))/((-a)^(1/2)*b^(5/2)) - (4*exp(2*c
+ 2*d*x)*(a + b))/b^2)*(a + b)^2)/(2*(-a)^(1/2)*b^(5/2)*d)

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