3.327 \(\int \frac{\text{sech}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=138 \[ \frac{\left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d (a-b)^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^3}+\frac{\tanh (c+d x) \text{sech}^3(c+d x)}{4 d (a-b)}+\frac{(3 a-7 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d (a-b)^2} \]

[Out]

((3*a^2 - 10*a*b + 15*b^2)*ArcTan[Sinh[c + d*x]])/(8*(a - b)^3*d) - (b^(5/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sq
rt[a]])/(Sqrt[a]*(a - b)^3*d) + ((3*a - 7*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*(a - b)^2*d) + (Sech[c + d*x]^3*T
anh[c + d*x])/(4*(a - b)*d)

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Rubi [A]  time = 0.161852, antiderivative size = 138, 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 = {3190, 414, 527, 522, 203, 205} \[ \frac{\left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d (a-b)^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^3}+\frac{\tanh (c+d x) \text{sech}^3(c+d x)}{4 d (a-b)}+\frac{(3 a-7 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2 - 10*a*b + 15*b^2)*ArcTan[Sinh[c + d*x]])/(8*(a - b)^3*d) - (b^(5/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sq
rt[a]])/(Sqrt[a]*(a - b)^3*d) + ((3*a - 7*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*(a - b)^2*d) + (Sech[c + d*x]^3*T
anh[c + d*x])/(4*(a - b)*d)

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\text{sech}^3(c+d x) \tanh (c+d x)}{4 (a-b) d}-\frac{\operatorname{Subst}\left (\int \frac{-3 a+4 b-3 b x^2}{\left (1+x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{4 (a-b) d}\\ &=\frac{(3 a-7 b) \text{sech}(c+d x) \tanh (c+d x)}{8 (a-b)^2 d}+\frac{\text{sech}^3(c+d x) \tanh (c+d x)}{4 (a-b) d}+\frac{\operatorname{Subst}\left (\int \frac{3 a^2-7 a b+8 b^2+(3 a-7 b) b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 (a-b)^2 d}\\ &=\frac{(3 a-7 b) \text{sech}(c+d x) \tanh (c+d x)}{8 (a-b)^2 d}+\frac{\text{sech}^3(c+d x) \tanh (c+d x)}{4 (a-b) d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{(a-b)^3 d}+\frac{\left (3 a^2-10 a b+15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 (a-b)^3 d}\\ &=\frac{\left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 (a-b)^3 d}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a-b)^3 d}+\frac{(3 a-7 b) \text{sech}(c+d x) \tanh (c+d x)}{8 (a-b)^2 d}+\frac{\text{sech}^3(c+d x) \tanh (c+d x)}{4 (a-b) d}\\ \end{align*}

Mathematica [A]  time = 0.50705, size = 139, normalized size = 1.01 \[ \frac{2 \sqrt{a} \left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+\sqrt{a} \left (3 a^2-10 a b+7 b^2\right ) \tanh (c+d x) \text{sech}(c+d x)+8 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )+2 \sqrt{a} (a-b)^2 \tanh (c+d x) \text{sech}^3(c+d x)}{8 \sqrt{a} d (a-b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b^(5/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[Tanh[(c + d*x)
/2]] + Sqrt[a]*(3*a^2 - 10*a*b + 7*b^2)*Sech[c + d*x]*Tanh[c + d*x] + 2*Sqrt[a]*(a - b)^2*Sech[c + d*x]^3*Tanh
[c + d*x])/(8*Sqrt[a]*(a - b)^3*d)

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Maple [B]  time = 0.073, size = 1023, normalized size = 7.4 \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*sinh(d*x+c)^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, a^{2} e^{c} - 10 \, a b e^{c} + 15 \, b^{2} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{4 \,{\left (a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d\right )}} + \frac{{\left (3 \, a e^{\left (7 \, c\right )} - 7 \, b e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} +{\left (11 \, a e^{\left (5 \, c\right )} - 15 \, b e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (11 \, a e^{\left (3 \, c\right )} - 15 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (3 \, a e^{c} - 7 \, b e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{2} d - 2 \, a b d + b^{2} d +{\left (a^{2} d e^{\left (8 \, c\right )} - 2 \, a b d e^{\left (8 \, c\right )} + b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (a^{2} d e^{\left (6 \, c\right )} - 2 \, a b d e^{\left (6 \, c\right )} + b^{2} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 6 \,{\left (a^{2} d e^{\left (4 \, c\right )} - 2 \, a b d e^{\left (4 \, c\right )} + b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (a^{2} d e^{\left (2 \, c\right )} - 2 \, a b d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} - 32 \, \int \frac{b^{3} e^{\left (3 \, d x + 3 \, c\right )} + b^{3} e^{\left (d x + c\right )}}{16 \,{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4} +{\left (a^{3} b e^{\left (4 \, c\right )} - 3 \, a^{2} b^{2} e^{\left (4 \, c\right )} + 3 \, a b^{3} e^{\left (4 \, c\right )} - b^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{4} e^{\left (2 \, c\right )} - 7 \, a^{3} b e^{\left (2 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (2 \, c\right )} - 5 \, a b^{3} e^{\left (2 \, c\right )} + b^{4} 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*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/4*(3*a^2*e^c - 10*a*b*e^c + 15*b^2*e^c)*arctan(e^(d*x + c))*e^(-c)/(a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d) +
1/4*((3*a*e^(7*c) - 7*b*e^(7*c))*e^(7*d*x) + (11*a*e^(5*c) - 15*b*e^(5*c))*e^(5*d*x) - (11*a*e^(3*c) - 15*b*e
^(3*c))*e^(3*d*x) - (3*a*e^c - 7*b*e^c)*e^(d*x))/(a^2*d - 2*a*b*d + b^2*d + (a^2*d*e^(8*c) - 2*a*b*d*e^(8*c) +
b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^2*d*e^(6*c) - 2*a*b*d*e^(6*c) + b^2*d*e^(6*c))*e^(6*d*x) + 6*(a^2*d*e^(4*c) -
2*a*b*d*e^(4*c) + b^2*d*e^(4*c))*e^(4*d*x) + 4*(a^2*d*e^(2*c) - 2*a*b*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) -
32*integrate(1/16*(b^3*e^(3*d*x + 3*c) + b^3*e^(d*x + c))/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4 + (a^3*b*e^(4*c)
- 3*a^2*b^2*e^(4*c) + 3*a*b^3*e^(4*c) - b^4*e^(4*c))*e^(4*d*x) + 2*(2*a^4*e^(2*c) - 7*a^3*b*e^(2*c) + 9*a^2*b
^2*e^(2*c) - 5*a*b^3*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.41039, size = 13526, normalized size = 98.01 \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*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/4*((3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + (3
*a^2 - 10*a*b + 7*b^2)*sinh(d*x + c)^7 + (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 - 10*a*b + 7*
b^2)*cosh(d*x + c)^2 + 11*a^2 - 26*a*b + 15*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)
^3 + (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^3 +
(35*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^4 + 10*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^2 - 11*a^2 + 26*a*b
- 15*b^2)*sinh(d*x + c)^3 + (21*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^5 + 10*(11*a^2 - 26*a*b + 15*b^2)*cosh
(d*x + c)^3 - 3*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b^2*cosh(d*x + c)^8 + 8*b^2*cos
h(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*x + c)^8 + 4*b^2*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + b^2)*sin
h(d*x + c)^6 + 6*b^2*cosh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35
*b^2*cosh(d*x + c)^4 + 30*b^2*cosh(d*x + c)^2 + 3*b^2)*sinh(d*x + c)^4 + 4*b^2*cosh(d*x + c)^2 + 8*(7*b^2*cosh
(d*x + c)^5 + 10*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^2*cosh(d*x + c)^6 + 15*b^
2*cosh(d*x + c)^4 + 9*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*b^2*cosh(d
*x + c)^5 + 3*b^2*cosh(d*x + c)^3 + b^2*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log((b*cosh(d*x + c)^4 + 4*b*
cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a
- b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^3 +
3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sinh(d*x
+ c))*sqrt(-b/a) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)
*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d
*x + c))*sinh(d*x + c) + b)) + ((3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 - 10*a*b + 15*b^2)*cosh(d
*x + c)*sinh(d*x + c)^7 + (3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^8 + 4*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c
)^6 + 4*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2
- 10*a*b + 15*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2 - 1
0*a*b + 15*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^4 + 30*(3*a^2 - 10*a*b + 15*b^
2)*cosh(d*x + c)^2 + 9*a^2 - 30*a*b + 45*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^5
+ 10*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^6 + 15*(3*a^2 - 10
*a*b + 15*b^2)*cosh(d*x + c)^4 + 9*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2)*sinh(d
*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2 + 8*((3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^7 + 3*(3*a^2 - 10*a*b + 15*b^
2)*cosh(d*x + c)^5 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^3 + (3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*si
nh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c) + (7*(3*a^2 - 10*a
*b + 7*b^2)*cosh(d*x + c)^6 + 5*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^4 - 3*(11*a^2 - 26*a*b + 15*b^2)*cosh
(d*x + c)^2 - 3*a^2 + 10*a*b - 7*b^2)*sinh(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^8 + 8*(a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sinh(d*x + c
)^8 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x +
c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^6 + 6*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^
4 + 8*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)
)*sinh(d*x + c)^5 + 2*(35*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 30*(a^3 - 3*a^2*b + 3*a*b^2 - b^
3)*d*cosh(d*x + c)^2 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^4 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3
)*d*cosh(d*x + c)^2 + 8*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^5 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b
^3)*d*cosh(d*x + c)^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3 - 3*a^2
*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^6 + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 9*(a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^2 + (a^3 - 3*a^2*b + 3
*a*b^2 - b^3)*d + 8*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^7 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*c
osh(d*x + c)^5 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^3 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(
d*x + c))*sinh(d*x + c)), 1/4*((3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x
+ c)*sinh(d*x + c)^6 + (3*a^2 - 10*a*b + 7*b^2)*sinh(d*x + c)^7 + (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^5 +
(21*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^2 + 11*a^2 - 26*a*b + 15*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 - 10*a
*b + 7*b^2)*cosh(d*x + c)^3 + (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a^2 - 26*a*b + 1
5*b^2)*cosh(d*x + c)^3 + (35*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^4 + 10*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x
+ c)^2 - 11*a^2 + 26*a*b - 15*b^2)*sinh(d*x + c)^3 + (21*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^5 + 10*(11*a^
2 - 26*a*b + 15*b^2)*cosh(d*x + c)^3 - 3*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(b^2*co
sh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*x + c)^8 + 4*b^2*cosh(d*x + c)^6 + 4*(7*b^2*c
osh(d*x + c)^2 + b^2)*sinh(d*x + c)^6 + 6*b^2*cosh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)
)*sinh(d*x + c)^5 + 2*(35*b^2*cosh(d*x + c)^4 + 30*b^2*cosh(d*x + c)^2 + 3*b^2)*sinh(d*x + c)^4 + 4*b^2*cosh(d
*x + c)^2 + 8*(7*b^2*cosh(d*x + c)^5 + 10*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^
2*cosh(d*x + c)^6 + 15*b^2*cosh(d*x + c)^4 + 9*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(
d*x + c)^7 + 3*b^2*cosh(d*x + c)^5 + 3*b^2*cosh(d*x + c)^3 + b^2*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arcta
n(1/2*sqrt(b/a)*(cosh(d*x + c) + sinh(d*x + c))) - 4*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^
7 + b^2*sinh(d*x + c)^8 + 4*b^2*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^6 + 6*b^2*cosh
(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*b^2*cosh(d*x + c)^4 + 30
*b^2*cosh(d*x + c)^2 + 3*b^2)*sinh(d*x + c)^4 + 4*b^2*cosh(d*x + c)^2 + 8*(7*b^2*cosh(d*x + c)^5 + 10*b^2*cosh
(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^2*cosh(d*x + c)^6 + 15*b^2*cosh(d*x + c)^4 + 9*b^2
*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*b^2*cosh(d*x + c)^5 + 3*b^2*cosh(d*
x + c)^3 + b^2*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(
d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - b)*sinh(d*x + c))*sqrt
(b/a)/b) + ((3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)*sinh(d*x + c
)^7 + (3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^8 + 4*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 -
10*a*b + 15*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 - 10*a*b + 15*b^2)*c
osh(d*x + c)^3 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2 - 10*a*b + 15*b^2)*cosh
(d*x + c)^4 + 2*(35*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^4 + 30*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 +
9*a^2 - 30*a*b + 45*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^5 + 10*(3*a^2 - 10*a*
b + 15*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^2 - 10*a*b +
15*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^6 + 15*(3*a^2 - 10*a*b + 15*b^2)*cosh(
d*x + c)^4 + 9*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^2 + 3*a^2 -
10*a*b + 15*b^2 + 8*((3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^7 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^5 +
3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^3 + (3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c))*arctan(
cosh(d*x + c) + sinh(d*x + c)) - (3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c) + (7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x
+ c)^6 + 5*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^4 - 3*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^2 - 3*a^2
+ 10*a*b - 7*b^2)*sinh(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^8 + 8*(a^3 - 3*a^2*b + 3*a*b
^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sinh(d*x + c)^8 + 4*(a^3 - 3*a^2
*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^6 + 6*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 8*(7*(a^3 - 3*a^
2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 +
2*(35*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 30*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^2
+ 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^4 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^2
+ 8*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^5 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^
3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*
d*cosh(d*x + c)^6 + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 9*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d
*cosh(d*x + c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d + 8*
((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^7 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^5 + 3*(
a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^3 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*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)**5/(a+b*sinh(d*x+c)**2),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)^5/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError