3.101 \(\int \frac{\text{sech}^7(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=153 \[ -\frac{\sqrt{a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 b^3 d (a+b)^{5/2}}-\frac{a (4 a+7 b) \sinh (c+d x)}{8 b^2 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{a \sinh (c+d x)}{4 b d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^3 d} \]

[Out]

ArcTan[Sinh[c + d*x]]/(b^3*d) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]
)/(8*b^3*(a + b)^(5/2)*d) - (a*Sinh[c + d*x])/(4*b*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) - (a*(4*a + 7*b)*S
inh[c + d*x])/(8*b^2*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))

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Rubi [A]  time = 0.204683, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4147, 414, 527, 522, 203, 205} \[ -\frac{\sqrt{a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 b^3 d (a+b)^{5/2}}-\frac{a (4 a+7 b) \sinh (c+d x)}{8 b^2 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{a \sinh (c+d x)}{4 b d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

ArcTan[Sinh[c + d*x]]/(b^3*d) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]
)/(8*b^3*(a + b)^(5/2)*d) - (a*Sinh[c + d*x])/(4*b*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) - (a*(4*a + 7*b)*S
inh[c + d*x])/(8*b^2*(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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

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

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]

Rubi steps

\begin{align*} \int \frac{\text{sech}^7(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b+a x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{a+4 b-3 a x^2}{\left (1+x^2\right ) \left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 b (a+b) d}\\ &=-\frac{a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac{a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{4 a^2+9 a b+8 b^2-a (4 a+7 b) x^2}{\left (1+x^2\right ) \left (a+b+a x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 b^2 (a+b)^2 d}\\ &=-\frac{a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac{a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b^3 d}-\frac{\left (a \left (8 a^2+20 a b+15 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{8 b^3 (a+b)^2 d}\\ &=\frac{\tan ^{-1}(\sinh (c+d x))}{b^3 d}-\frac{\sqrt{a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 b^3 (a+b)^{5/2} d}-\frac{a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac{a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 4.60144, size = 247, normalized size = 1.61 \[ \frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{\sqrt{a} \left (8 a^2+20 a b+15 b^2\right ) (\cosh (c)-\sinh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )}{(a+b)^{5/2} \sqrt{(\cosh (c)-\sinh (c))^2}}-\frac{8 a b^2 \tanh (c+d x)}{a+b}-\frac{2 a b (4 a+7 b) \tanh (c+d x) (a \cosh (2 (c+d x))+a+2 b)}{(a+b)^2}+16 \text{sech}(c+d x) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2\right )}{64 b^3 d \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*(16*ArcTan[Tanh[(c + d*x)/2]]*(a + 2*b + a*Cosh[2*(c + d*x)])
^2*Sech[c + d*x] + (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c
])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*(Cosh[c] - Sinh[c]))/((a +
b)^(5/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) - (8*a*b^2*Tanh[c + d*x])/(a + b) - (2*a*b*(4*a + 7*b)*(a + 2*b + a*Cos
h[2*(c + d*x)])*Tanh[c + d*x])/(a + b)^2))/(64*b^3*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B]  time = 0.084, size = 1202, normalized size = 7.9 \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)^7/(a+b*sech(d*x+c)^2)^3,x)

[Out]

1/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^2/(a+b)/b^2*tanh(1/2*d*x+1/2*c)^7+9/4/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)*tanh(1/2*d*x+1/2*c)^7*a+1/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^3/(a+b)^2/b^2*tanh(1/2*d*x+1/2*c
)^5-11/4/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/b*tanh(1/2*d*x+1/2*c)^5*a^2-27/4/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-1/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^3/(a+b)^2/b^2*tanh(1/2*d
*x+1/2*c)^3+11/4/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/b*tanh(1/2*d*x+1/2*c)^3*a^2+27/4/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-1/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^2/(a+b)/b^2*tanh
(1/2*d*x+1/2*c)-9/4/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)*tanh(1/2*d*x+1/2*c)*a-1/d*a^(5/2)/b^3/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*ta
nh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-5/2/d*a^(3/2)/b^2/(a^2+2*a*b+b^2)/(a+b)^(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))-15/8/d*a^(1/2)/b/(a^2+2*a*b+b^2)/(a+b)^(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))-1/d*a^(5/2)/b^3/(a^2+2*a*b+b^2)/(a+b)^(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))-5/2/d*a^(3/2)/b^2/(a^2+2*a*b+b^2)/(a+b)^(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))-15/8/d*a^(1/2)/b/(a^2+2*a*b+b^2)/(a+b)^(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))+2/d/b^3*arctan(tanh(1/2*d*x+1/2*c))

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Maxima [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)^7/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 3.80044, size = 18826, normalized size = 123.05 \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)^7/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^7 + 28*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(
4*a^3*b + 7*a^2*b^2)*sinh(d*x + c)^7 + 4*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^5 + 4*(4*a^3*b + 31*a
^2*b^2 + 36*a*b^3 + 21*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(4*a^3*b + 7*a^2*b^2)*co
sh(d*x + c)^3 + (4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(4*a^3*b + 31*a^2*b^2 + 3
6*a*b^3)*cosh(d*x + c)^3 + 4*(35*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^4 - 4*a^3*b - 31*a^2*b^2 - 36*a*b^3 + 10*
(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)
^5 + 10*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c))
*sinh(d*x + c)^2 - ((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^8 + 8*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d
*x + c)*sinh(d*x + c)^7 + (8*a^4 + 20*a^3*b + 15*a^2*b^2)*sinh(d*x + c)^8 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 +
30*a*b^3)*cosh(d*x + c)^6 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3 + 7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^3 + 3*(8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120
*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^4 + 24*a^4 + 124*a^3*b + 269*a^2*b
^2 + 280*a*b^3 + 120*b^4 + 30*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*
a^4 + 20*a^3*b + 15*a^2*b^2 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^5 + 10*(8*a^4 + 36*a^3*b + 55
*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c))
*sinh(d*x + c)^3 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 20*a^3*b + 15*
a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^4 + 8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3 + 3*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^7 + 3*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh
(d*x + c)^5 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^3 + (8*a^4 + 36*a^3*b + 5
5*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a/(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)*si
nh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((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 - (a + b)*cosh(d*x + c) + (3*(a + b)*cosh(d
*x + c)^2 - a - b)*sinh(d*x + c))*sqrt(-a/(a + b)) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(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)) - 32*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^8
+ 8*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^8 + 4*(
a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^6 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 7*(a^4 + 2*a^3
*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + 3*(a^4 + 4*a
^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b
^4)*cosh(d*x + c)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 + 3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b
^3 + 8*b^4 + 30*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 2*a^3*b + a^2*b
^2 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3
+ (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 4*a^3*b + 5*a^
2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*
b^2 + 2*a*b^3)*cosh(d*x + c)^4 + a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a
*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^7 + 3*(a^4 + 4*a^3
*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^5 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c)^3
+ (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) -
4*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c) + 4*(7*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^6 + 5*(4*a^3*b + 31*a^2*b^2
+ 36*a*b^3)*cosh(d*x + c)^4 - 4*a^3*b - 7*a^2*b^2 - 3*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c))/((a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^8 + 8*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*sinh(d*x + c)^8 + 4*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 +
2*a*b^6)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 5
*a^2*b^5 + 2*a*b^6)*d)*sinh(d*x + c)^6 + 2*(3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*d*cosh(d*x
+ c)^4 + 8*(7*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^3 + 3*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^
6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^4 + 30*(a^4*b^3 +
4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^2 + (3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*
d)*sinh(d*x + c)^4 + 4*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b^3 + 2*a^3*b
^4 + a^2*b^5)*d*cosh(d*x + c)^5 + 10*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^3 + (3*a^4*b^
3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b^3 + 2*a^3*b^4 +
a^2*b^5)*d*cosh(d*x + c)^6 + 15*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^4 + 3*(3*a^4*b^3
+ 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*d*cosh(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*
d)*sinh(d*x + c)^2 + (a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d + 8*((a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^7
+ 3*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^5 + (3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*
a*b^6 + 8*b^7)*d*cosh(d*x + c)^3 + (a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c))*sinh(d*x + c))
, -1/8*(2*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^7 + 14*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(
4*a^3*b + 7*a^2*b^2)*sinh(d*x + c)^7 + 2*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^5 + 2*(4*a^3*b + 31*a
^2*b^2 + 36*a*b^3 + 21*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(4*a^3*b + 7*a^2*b^2)*co
sh(d*x + c)^3 + (4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(4*a^3*b + 31*a^2*b^2 + 3
6*a*b^3)*cosh(d*x + c)^3 + 2*(35*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^4 - 4*a^3*b - 31*a^2*b^2 - 36*a*b^3 + 10*
(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(21*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)
^5 + 10*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c))
*sinh(d*x + c)^2 + ((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^8 + 8*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d
*x + c)*sinh(d*x + c)^7 + (8*a^4 + 20*a^3*b + 15*a^2*b^2)*sinh(d*x + c)^8 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 +
30*a*b^3)*cosh(d*x + c)^6 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3 + 7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^3 + 3*(8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120
*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^4 + 24*a^4 + 124*a^3*b + 269*a^2*b
^2 + 280*a*b^3 + 120*b^4 + 30*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*
a^4 + 20*a^3*b + 15*a^2*b^2 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^5 + 10*(8*a^4 + 36*a^3*b + 55
*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c))
*sinh(d*x + c)^3 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 20*a^3*b + 15*
a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^4 + 8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3 + 3*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^7 + 3*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh
(d*x + c)^5 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^3 + (8*a^4 + 36*a^3*b + 5
5*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a/(a + b))*arctan(1/2*sqrt(a/(a + b))*(cosh(d*x + c)
+ sinh(d*x + c))) + ((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^8 + 8*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(
d*x + c)*sinh(d*x + c)^7 + (8*a^4 + 20*a^3*b + 15*a^2*b^2)*sinh(d*x + c)^8 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2
+ 30*a*b^3)*cosh(d*x + c)^6 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3 + 7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*
cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^3 + 3*(8*a^4 + 36*a^3*b
+ 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 12
0*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^4 + 24*a^4 + 124*a^3*b + 269*a^2*
b^2 + 280*a*b^3 + 120*b^4 + 30*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8
*a^4 + 20*a^3*b + 15*a^2*b^2 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^5 + 10*(8*a^4 + 36*a^3*b + 5
5*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)
)*sinh(d*x + c)^3 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 20*a^3*b + 15
*a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^4 + 8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3 + 3*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^7 + 3*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cos
h(d*x + c)^5 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^3 + (8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a/(a + b))*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(
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)*s
inh(d*x + c))*sqrt(a/(a + b))/a) - 16*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 + 2*a^3*b + a^2*b^2)
*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^8 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*
a*b^3)*cosh(d*x + c)^6 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + 3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*co
sh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 2*(35*(a
^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 + 3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4 + 30*(a^4 + 4*a^3*b
+ 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 2*a^3*b + a^2*b^2 + 8*(7*(a^4 + 2*a^3*b + a^2
*b^2)*cosh(d*x + c)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 + (3*a^4 + 14*a^3*b + 27*a^2*
b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)
^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4
+ a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^7 + 3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh
(d*x + c)^5 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2
+ 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(4*a^3*b + 7*a^2*b^2)*cosh(
d*x + c) + 2*(7*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^6 + 5*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^4 -
4*a^3*b - 7*a^2*b^2 - 3*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^4*b^3 + 2*a^3*b^
4 + a^2*b^5)*d*cosh(d*x + c)^8 + 8*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b^3
+ 2*a^3*b^4 + a^2*b^5)*d*sinh(d*x + c)^8 + 4*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^6 + 4
*(7*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d)*sinh(d*
x + c)^6 + 2*(3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*d*cosh(d*x + c)^4 + 8*(7*(a^4*b^3 + 2*a^
3*b^4 + a^2*b^5)*d*cosh(d*x + c)^3 + 3*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c))*sinh(d*x +
c)^5 + 2*(35*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^4 + 30*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^
6)*d*cosh(d*x + c)^2 + (3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*d)*sinh(d*x + c)^4 + 4*(a^4*b^
3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^
5 + 10*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^3 + (3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 +
24*a*b^6 + 8*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^6 +
15*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^4 + 3*(3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24
*a*b^6 + 8*b^7)*d*cosh(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d)*sinh(d*x + c)^2 + (a^4*b^3
+ 2*a^3*b^4 + a^2*b^5)*d + 8*((a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^7 + 3*(a^4*b^3 + 4*a^3*b^4 + 5*a
^2*b^5 + 2*a*b^6)*d*cosh(d*x + c)^5 + (3*a^4*b^3 + 14*a^3*b^4 + 27*a^2*b^5 + 24*a*b^6 + 8*b^7)*d*cosh(d*x + c)
^3 + (a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 + 2*a*b^6)*d*cosh(d*x + c))*sinh(d*x + c))]

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

[Out]

Timed out

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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)^7/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError