3.99 \(\int \frac{\text{sech}^5(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)
Optimal. Leaf size=106 \[ \frac{3 \sinh (c+d x)}{8 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac{\sinh (c+d x)}{4 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 \sqrt{a} d (a+b)^{5/2}} \]
[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))
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Rubi [A] time = 0.0921015, antiderivative size = 106, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{4147, 199, 205} \[ \frac{3 \sinh (c+d x)}{8 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac{\sinh (c+d x)}{4 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 \sqrt{a} d (a+b)^{5/2}} \]
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 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]
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]
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{\operatorname{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 \operatorname{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 \operatorname{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*}
Mathematica [A] time = 0.281514, size = 125, normalized size = 1.18 \[ \frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left (\frac{5 (a+b) \sinh (c+d x)+3 a \sinh ^3(c+d x)}{(a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{5/2}}\right )}{64 d \left (a+b \text{sech}^2(c+d x)\right )^3} \]
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] time = 0.077, size = 592, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}
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)
[Out]
-5/4/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^7+3/4/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)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^5*a+15/4/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)^2*b/(a+b)^2*tanh(1/2*d*x+1/2*c)^5-3/4/
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)^2/
(a+b)^2*tanh(1/2*d*x+1/2*c)^3*a-15/4/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)^2*b/(a+b)^2*tanh(1/2*d*x+1/2*c)^3+5/4/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)+3/8/d/(a^2+
2*a*b+b^2)/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+3/8/d/(a^2+2*
a*b+b^2)/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)-2*b^(1/2))/a^(1/2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (11 \, a e^{\left (5 \, c\right )} + 20 \, b e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (11 \, a e^{\left (3 \, c\right )} + 20 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 3 \, a e^{\left (7 \, d x + 7 \, c\right )} - 3 \, a e^{\left (d x + c\right )}}{4 \,{\left (a^{4} d + 2 \, a^{3} b d + a^{2} b^{2} d +{\left (a^{4} d e^{\left (8 \, c\right )} + 2 \, a^{3} b d e^{\left (8 \, c\right )} + a^{2} b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (a^{4} d e^{\left (6 \, c\right )} + 4 \, a^{3} b d e^{\left (6 \, c\right )} + 5 \, a^{2} b^{2} d e^{\left (6 \, c\right )} + 2 \, a b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (3 \, a^{4} d e^{\left (4 \, c\right )} + 14 \, a^{3} b d e^{\left (4 \, c\right )} + 27 \, a^{2} b^{2} d e^{\left (4 \, c\right )} + 24 \, a b^{3} d e^{\left (4 \, c\right )} + 8 \, b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (a^{4} d e^{\left (2 \, c\right )} + 4 \, a^{3} b d e^{\left (2 \, c\right )} + 5 \, a^{2} b^{2} d e^{\left (2 \, c\right )} + 2 \, a b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + 32 \, \int \frac{3 \,{\left (e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}\right )}}{128 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2} +{\left (a^{3} e^{\left (4 \, c\right )} + 2 \, a^{2} b e^{\left (4 \, c\right )} + a b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{3} e^{\left (2 \, c\right )} + 4 \, a^{2} b e^{\left (2 \, c\right )} + 5 \, a b^{2} e^{\left (2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}
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] time = 2.7666, size = 12078, normalized size = 113.94 \begin{align*} \text{result too large to display} \end{align*}
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^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))*si
nh(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))*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cos
h(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)) + 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))*sinh(d*x + c))*sqr
t(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a + b)) - 6*(a^3 + a^2*b)*cosh(d*x +
c) + 2*(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*co
sh(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*cosh(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))]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
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