3.2.31 \(\int \frac {\text {sech}^5(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [131]

Optimal. Leaf size=104 \[ \frac {3 \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {a+b} d}+\frac {\sinh (c+d x)}{4 a d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {3 \sinh (c+d x)}{8 a^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 104, 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 = {3757, 205, 211} \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d \sqrt {a+b}}+\frac {3 \sinh (c+d x)}{8 a^2 d \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sinh (c+d x)}{4 a d \left ((a+b) \sinh ^2(c+d x)+a\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

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Mathematica [A]
time = 0.21, size = 88, normalized size = 0.85 \begin {gather*} \frac {\frac {3 \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} \sqrt {a+b}}+\frac {5 a \sinh (c+d x)+3 (a+b) \sinh ^3(c+d x)}{a \left (a+(a+b) \sinh ^2(c+d x)\right )^2}}{8 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(275\) vs. \(2(90)=180\).
time = 2.70, size = 276, normalized size = 2.65

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 a \,{\mathrm e}^{6 d x +6 c}+3 b \,{\mathrm e}^{6 d x +6 c}+11 a \,{\mathrm e}^{4 d x +4 c}-9 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{2 d x +2 c}+9 b \,{\mathrm e}^{2 d x +2 c}-3 a -3 b \right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} a^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, d \,a^{2}}\)	\(252\)
derivativedivides	\(\frac {\frac {-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 \left (a -4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}-\frac {3 \left (a -4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}+\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {-\frac {3 \left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{8 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {3 \left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{8 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}}{a}}{d}\)	\(276\)
default	\(\frac {\frac {-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 \left (a -4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}-\frac {3 \left (a -4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}+\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {-\frac {3 \left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{8 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {3 \left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{8 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}}{a}}{d}\)	\(276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

[Out]

1/4*(3*(a*e^(7*c) + b*e^(7*c))*e^(7*d*x) + (11*a*e^(5*c) - 9*b*e^(5*c))*e^(5*d*x) - (11*a*e^(3*c) - 9*b*e^(3*c
))*e^(3*d*x) - 3*(a*e^c + b*e^c)*e^(d*x))/(a^4*d + 2*a^3*b*d + a^2*b^2*d + (a^4*d*e^(8*c) + 2*a^3*b*d*e^(8*c)
+ a^2*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^4*d*e^(6*c) - a^2*b^2*d*e^(6*c))*e^(6*d*x) + 2*(3*a^4*d*e^(4*c) - 2*a^3*
b*d*e^(4*c) + 3*a^2*b^2*d*e^(4*c))*e^(4*d*x) + 4*(a^4*d*e^(2*c) - a^2*b^2*d*e^(2*c))*e^(2*d*x)) + 32*integrate
(3/128*(e^(3*d*x + 3*c) + e^(d*x + c))/(a^3 + a^2*b + (a^3*e^(4*c) + a^2*b*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c)
- a^2*b*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. 2712 vs. \(2 (90) = 180\).
time = 0.42, size = 5077, normalized size = 48.82 \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)^5/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(12*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^7 + 84*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 +
12*(a^3 + 2*a^2*b + a*b^2)*sinh(d*x + c)^7 + 4*(11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c)^5 + 4*(11*a^3 + 2*a
^2*b - 9*a*b^2 + 63*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(21*(a^3 + 2*a^2*b + a*b^2)*
cosh(d*x + c)^3 + (11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(11*a^3 + 2*a^2*b - 9*a*b^2)
*cosh(d*x + c)^3 + 4*(105*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 11*a^3 - 2*a^2*b + 9*a*b^2 + 10*(11*a^3 +
2*a^2*b - 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(63*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 10*(11*a
^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c)^3 - 3*(11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*((
a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b + b^2)
*sinh(d*x + c)^8 + 4*(a^2 - b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(
d*x + c)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^
2 - 2*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 30*(a^2 - b^2)*cosh(d*x + c)^
2 + 3*a^2 - 2*a*b + 3*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 10*(a^2 - b^2)*cosh(d*
x + c)^3 + (3*a^2 - 2*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^2 - b^2)*cosh(d*x + c)^2 + 4*(7*(a^2
+ 2*a*b + b^2)*cosh(d*x + c)^6 + 15*(a^2 - b^2)*cosh(d*x + c)^4 + 3*(3*a^2 - 2*a*b + 3*b^2)*cosh(d*x + c)^2 +
a^2 - b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 3*(a^2 - b^2)*cosh(d
*x + c)^5 + (3*a^2 - 2*a*b + 3*b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*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)*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 + 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)) - 12*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c) + 4*(21*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^6 + 5*(11*a^
3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c)^4 - 3*a^3 - 6*a^2*b - 3*a*b^2 - 3*(11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x +
c)^2)*sinh(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^8 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2
+ a^3*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^8 + 4*(a^6
+ a^5*b - a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2
+ (a^6 + a^5*b - a^4*b^2 - a^3*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^6 + a^5*b + a^4*b^2 + 3*a^3*b^3)*d*cosh(d*x +
c)^4 + 8*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 + 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*d*co
sh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^4 + 30*(a^6 + a^5*b
- a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^2 + (3*a^6 + a^5*b + a^4*b^2 + 3*a^3*b^3)*d)*sinh(d*x + c)^4 + 4*(a^6 +
a^5*b - a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^5 +
10*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^3 + (3*a^6 + a^5*b + a^4*b^2 + 3*a^3*b^3)*d*cosh(d*x + c)
)*sinh(d*x + c)^3 + 4*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^6 + 15*(a^6 + a^5*b - a^4*b^2 -
a^3*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^6 + a^5*b + a^4*b^2 + 3*a^3*b^3)*d*cosh(d*x + c)^2 + (a^6 + a^5*b - a^4*b
^2 - a^3*b^3)*d)*sinh(d*x + c)^2 + (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d + 8*((a^6 + 3*a^5*b + 3*a^4*b^2 + a
^3*b^3)*d*cosh(d*x + c)^7 + 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*d*cosh(d*x + c)^5 + (3*a^6 + a^5*b + a^4*b^2 +
3*a^3*b^3)*d*cosh(d*x + c)^3 + (a^6 + a^5*b - a^4*b^2 - a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(6*(a^3
+ 2*a^2*b + a*b^2)*cosh(d*x + c)^7 + 42*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 6*(a^3 + 2*a^
2*b + a*b^2)*sinh(d*x + c)^7 + 2*(11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c)^5 + 2*(11*a^3 + 2*a^2*b - 9*a*b^2
+ 63*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(21*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^3
+ (11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c)^
3 + 2*(105*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 11*a^3 - 2*a^2*b + 9*a*b^2 + 10*(11*a^3 + 2*a^2*b - 9*a*b
^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(63*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 10*(11*a^3 + 2*a^2*b -
9*a*b^2)*cosh(d*x + c)^3 - 3*(11*a^3 + 2*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((a^2 + 2*a*b + b
^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cos...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{5}{\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)**5/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral(sech(c + d*x)**5/(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)^5/(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 )}^5\,{\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)^5*(a + b*tanh(c + d*x)^2)^3),x)

[Out]

int(1/(cosh(c + d*x)^5*(a + b*tanh(c + d*x)^2)^3), x)

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