3.1.99 \(\int \frac {\text {sech}^5(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [99]

Optimal. Leaf size=106 \[ \frac {3 \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 \sqrt {a} (a+b)^{5/2} d}+\frac {\sinh (c+d x)}{4 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {3 \sinh (c+d x)}{8 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 106, 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 = {4232, 205, 211} \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 \sqrt {a} d (a+b)^{5/2}}+\frac {3 \sinh (c+d x)}{8 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\sinh (c+d x)}{4 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]^5/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

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

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Mathematica [A]
time = 0.21, size = 125, normalized size = 1.18 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \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 )}{64 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((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)))/(6
4*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. \(239\) vs. \(2(92)=184\).
time = 1.97, size = 240, normalized size = 2.26

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 a \,{\mathrm e}^{6 d x +6 c}+11 a \,{\mathrm e}^{4 d x +4 c}+20 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{2 d x +2 c}-20 b \,{\mathrm e}^{2 d x +2 c}-3 a \right )}{4 \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}\)	\(235\)
derivativedivides	\(\frac {\frac {-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )}+\frac {3 \left (a +5 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2}}-\frac {3 \left (a +5 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2}}+\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 \left (a +b \right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\frac {3 \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{8 \sqrt {a +b}\, \sqrt {a}}+\frac {3 \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{8 \sqrt {a +b}\, \sqrt {a}}}{a^{2}+2 a b +b^{2}}}{d}\)	\(240\)
default	\(\frac {\frac {-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )}+\frac {3 \left (a +5 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2}}-\frac {3 \left (a +5 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2}}+\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 \left (a +b \right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\frac {3 \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{8 \sqrt {a +b}\, \sqrt {a}}+\frac {3 \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{8 \sqrt {a +b}\, \sqrt {a}}}{a^{2}+2 a b +b^{2}}}{d}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/4*((11*a*e^(5*c) + 20*b*e^(5*c))*e^(5*d*x) - (11*a*e^(3*c) + 20*b*e^(3*c))*e^(3*d*x) + 3*a*e^(7*d*x + 7*c) -
3*a*e^(d*x + c))/(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) + 4*a^3*b*d*e^(6*c) + 5*a^2*b^2*d*e^(6*c) + 2*a*b^3*d*e^(6*c))*e^(6*d*x) + 2*(3*a^4*
d*e^(4*c) + 14*a^3*b*d*e^(4*c) + 27*a^2*b^2*d*e^(4*c) + 24*a*b^3*d*e^(4*c) + 8*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a
^4*d*e^(2*c) + 4*a^3*b*d*e^(2*c) + 5*a^2*b^2*d*e^(2*c) + 2*a*b^3*d*e^(2*c))*e^(2*d*x)) + 32*integrate(3/128*(e
^(3*d*x + 3*c) + e^(d*x + c))/(a^3 + 2*a^2*b + a*b^2 + (a^3*e^(4*c) + 2*a^2*b*e^(4*c) + a*b^2*e^(4*c))*e^(4*d*
x) + 2*(a^3*e^(2*c) + 4*a^2*b*e^(2*c) + 5*a*b^2*e^(2*c) + 2*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. 2638 vs. \(2 (92) = 184\).
time = 0.40, size = 5006, normalized size = 47.23 \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*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(12*(a^3 + a^2*b)*cosh(d*x + c)^7 + 84*(a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^6 + 12*(a^3 + a^2*b)*si
nh(d*x + c)^7 + 4*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^5 + 4*(11*a^3 + 31*a^2*b + 20*a*b^2 + 63*(a^3 +
a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(21*(a^3 + a^2*b)*cosh(d*x + c)^3 + (11*a^3 + 31*a^2*b + 20*a*b^
2)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^3 + 4*(105*(a^3 + a^2*b)*co
sh(d*x + c)^4 - 11*a^3 - 31*a^2*b - 20*a*b^2 + 10*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c
)^3 + 4*(63*(a^3 + a^2*b)*cosh(d*x + c)^5 + 10*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^3 - 3*(11*a^3 + 31
*a^2*b + 20*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*(a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)
^7 + a^2*sinh(d*x + c)^8 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x
+ c)^6 + 8*(7*a^2*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 + 8*a*b + 8*b^2)
*cosh(d*x + c)^4 + 2*(35*a^2*cosh(d*x + c)^4 + 30*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(
d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^5 + 10*(a^2 + 2*a*b)*cosh(d*x + c)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 4*(7*a^2*cosh(d*x + c)^6 + 15*(a^2 + 2*a*b)*cosh(d*x +
c)^4 + 3*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*(a^2*cosh(d*x + c)^
7 + 3*(a^2 + 2*a*b)*cosh(d*x + c)^5 + (3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*s
inh(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*c
osh(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)) - 12*(a^3 + a^2*b)*
cosh(d*x + c) + 4*(21*(a^3 + a^2*b)*cosh(d*x + c)^6 + 5*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^4 - 3*a^3
- 3*a^2*b - 3*(11*a^3 + 31*a^2*b + 20*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 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b^4)*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 + 5*a^5*b + 9*a^4*b^2 + 7*a^3
*b^3 + 2*a^2*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^6 + 17*a^5*b + 41*a^4*b^2 + 51*a^3*b^3 + 32*a^2*b^4 + 8*a*b^5)*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 + 5*a^5*b + 9*a^4*b^2
+ 7*a^3*b^3 + 2*a^2*b^4)*d*cosh(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*cos
h(d*x + c)^4 + 30*(a^6 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b^4)*d*cosh(d*x + c)^2 + (3*a^6 + 17*a^5*b +
41*a^4*b^2 + 51*a^3*b^3 + 32*a^2*b^4 + 8*a*b^5)*d)*sinh(d*x + c)^4 + 4*(a^6 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3
+ 2*a^2*b^4)*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 + 5*a^
5*b + 9*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b^4)*d*cosh(d*x + c)^3 + (3*a^6 + 17*a^5*b + 41*a^4*b^2 + 51*a^3*b^3 + 32*
a^2*b^4 + 8*a*b^5)*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 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^6 + 17*a^5*b + 41*a^4
*b^2 + 51*a^3*b^3 + 32*a^2*b^4 + 8*a*b^5)*d*cosh(d*x + c)^2 + (a^6 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b
^4)*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 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b^4)*d*cosh(d*x + c)^5 + (3*a^6 + 17*a^5*b
+ 41*a^4*b^2 + 51*a^3*b^3 + 32*a^2*b^4 + 8*a*b^5)*d*cosh(d*x + c)^3 + (a^6 + 5*a^5*b + 9*a^4*b^2 + 7*a^3*b^3 +
2*a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(6*(a^3 + a^2*b)*cosh(d*x + c)^7 + 42*(a^3 + a^2*b)*cosh(d*x
+ c)*sinh(d*x + c)^6 + 6*(a^3 + a^2*b)*sinh(d*x + c)^7 + 2*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^5 + 2*
(11*a^3 + 31*a^2*b + 20*a*b^2 + 63*(a^3 + a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(21*(a^3 + a^2*b)*cosh(
d*x + c)^3 + (11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(11*a^3 + 31*a^2*b + 20*a*b^2)*
cosh(d*x + c)^3 + 2*(105*(a^3 + a^2*b)*cosh(d*x + c)^4 - 11*a^3 - 31*a^2*b - 20*a*b^2 + 10*(11*a^3 + 31*a^2*b
+ 20*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(63*(a^3 + a^2*b)*cosh(d*x + c)^5 + 10*(11*a^3 + 31*a^2*b + 2
0*a*b^2)*cosh(d*x + c)^3 - 3*(11*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*(a^2*cosh(d*x +
c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a...

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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 \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)**5/(a+b*sech(d*x+c)**2)**3,x)

[Out]

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

[Out]

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

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