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

Optimal. Leaf size=217 \[ \frac{b^{3/2} \left (35 a^2-14 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^4}+\frac{b (4 a-b) (a+3 b) \sinh (c+d x)}{8 a^2 d (a-b)^3 \left (a+b \sinh ^2(c+d x)\right )}+\frac{b (2 a+b) \sinh (c+d x)}{4 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(a-7 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^4}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \]

[Out]

((a - 7*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^4*d) + (b^(3/2)*(35*a^2 - 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c
+ d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^4*d) + (b*(2*a + b)*Sinh[c + d*x])/(4*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^
2)^2) + ((4*a - b)*b*(a + 3*b)*Sinh[c + d*x])/(8*a^2*(a - b)^3*d*(a + b*Sinh[c + d*x]^2)) + (Sech[c + d*x]*Tan
h[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.303491, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{b^{3/2} \left (35 a^2-14 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^4}+\frac{b (4 a-b) (a+3 b) \sinh (c+d x)}{8 a^2 d (a-b)^3 \left (a+b \sinh ^2(c+d x)\right )}+\frac{b (2 a+b) \sinh (c+d x)}{4 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(a-7 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^4}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - 7*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^4*d) + (b^(3/2)*(35*a^2 - 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c
+ d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^4*d) + (b*(2*a + b)*Sinh[c + d*x])/(4*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^
2)^2) + ((4*a - b)*b*(a + 3*b)*Sinh[c + d*x])/(8*a^2*(a - b)^3*d*(a + b*Sinh[c + d*x]^2)) + (Sech[c + d*x]*Tan
h[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2)

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}^3(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2 \left (a+b x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-a+2 b-5 b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{2 (a-b) d}\\ &=\frac{b (2 a+b) \sinh (c+d x)}{4 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-2 \left (2 a^2-8 a b+3 b^2\right )-6 b (2 a+b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{8 a (a-b)^2 d}\\ &=\frac{b (2 a+b) \sinh (c+d x)}{4 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(4 a-b) b (a+3 b) \sinh (c+d x)}{8 a^2 (a-b)^3 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-2 \left (4 a^3-24 a^2 b+11 a b^2-3 b^3\right )-2 (4 a-b) b (a+3 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{16 a^2 (a-b)^3 d}\\ &=\frac{b (2 a+b) \sinh (c+d x)}{4 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(4 a-b) b (a+3 b) \sinh (c+d x)}{8 a^2 (a-b)^3 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(a-7 b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 (a-b)^4 d}+\frac{\left (b^2 \left (35 a^2-14 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a-b)^4 d}\\ &=\frac{(a-7 b) \tan ^{-1}(\sinh (c+d x))}{2 (a-b)^4 d}+\frac{b^{3/2} \left (35 a^2-14 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} (a-b)^4 d}+\frac{b (2 a+b) \sinh (c+d x)}{4 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(4 a-b) b (a+3 b) \sinh (c+d x)}{8 a^2 (a-b)^3 d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}\\ \end{align*}

Mathematica [A]  time = 1.79072, size = 222, normalized size = 1.02 \[ \frac{\frac{2 b^2 (a-b) \sinh (c+d x) \left (26 a^2+b (11 a-3 b) \cosh (2 (c+d x))-21 a b+3 b^2\right )}{a^2 (2 a+b \cosh (2 (c+d x))-b)^2}-\frac{3 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )}{a^{5/2}}+\frac{14 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )}{a^{3/2}}-\frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )}{\sqrt{a}}+4 (a-b) \tanh (c+d x) \text{sech}(c+d x)+8 a \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-56 b \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{8 d (a-b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*b^(3/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + (14*b^(5/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sq
rt[b]])/a^(3/2) - (3*b^(7/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/a^(5/2) + 8*a*ArcTan[Tanh[(c + d*x)/2]]
- 56*b*ArcTan[Tanh[(c + d*x)/2]] + (2*(a - b)*b^2*(26*a^2 - 21*a*b + 3*b^2 + (11*a - 3*b)*b*Cosh[2*(c + d*x)])
*Sinh[c + d*x])/(a^2*(2*a - b + b*Cosh[2*(c + d*x)])^2) + 4*(a - b)*Sech[c + d*x]*Tanh[c + d*x])/(8*(a - b)^4*
d)

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Maple [B]  time = 0.104, size = 2307, normalized size = 10.6 \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)^3/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

-13/4/d*b^2/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*a*tanh(1
/2*d*x+1/2*c)^7+39/4/d*b^2/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*
b+a)^2*a*tanh(1/2*d*x+1/2*c)^5-39/4/d*b^2/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/
2*d*x+1/2*c)^2*b+a)^2*a*tanh(1/2*d*x+1/2*c)^3+13/4/d*b^2/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c
)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*a*tanh(1/2*d*x+1/2*c)+49/8/d*b^3/(a-b)^4/(-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))+49/8/d*b^3/(a-b)^4/(-
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))-5/4/d*b^4/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/
a*tanh(1/2*d*x+1/2*c)^7+71/4/d*b^4/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1
/2*c)^2*b+a)^2/a*tanh(1/2*d*x+1/2*c)^5-3/d*b^5/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*ta
nh(1/2*d*x+1/2*c)^2*b+a)^2/a^2*tanh(1/2*d*x+1/2*c)^5-71/4/d*b^4/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*
x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh(1/2*d*x+1/2*c)^3+3/d*b^5/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2
*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a^2*tanh(1/2*d*x+1/2*c)^3+5/4/d*b^4/(a-b)^4/(tanh(1/2*
d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh(1/2*d*x+1/2*c)+7/4/d*b^3/(a-b)^
4/a/((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))-3
/8/d*b^4/(a-b)^4/a^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))-7/4/d*b^3/(a-b)^4/a/((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))+3/8/d*b^4/(a-b)^4/a^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))-35/8/d*b^2/(a-b)^4*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))-35/8/d*b^2/(a-b)^4*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))
-17/8/d*b^4/(a-b)^4/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))+3/8/d*b^5/(a-b)^4/a^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*a
rctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-17/8/d*b^4/(a-b)^4/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))+3/8/d*b^5/(
a-b)^4/a^2/(-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/(a-b)^4/(tanh(1/2*d*x+1/2*c)^2+1)^2*a*tanh(1/2*d*x+1/2*c)-9/2/d*b^3/(a-b)^4/(tanh(1
/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)+49/2/d*b^3/(a-b
)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^3-1/
d/(a-b)^4/(tanh(1/2*d*x+1/2*c)^2+1)^2*a*tanh(1/2*d*x+1/2*c)^3-1/d/(a-b)^4/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2
*d*x+1/2*c)*b+1/d/(a-b)^4/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3*b-49/2/d*b^3/(a-b)^4/(tanh(1/2*d*x
+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^5+35/8/d*b^2/(a-b)^4/
((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))+9/2/d*
b^3/(a-b)^4/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2
*c)^7-35/8/d*b^2/(a-b)^4/((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))-7/d/(a-b)^4*arctan(tanh(1/2*d*x+1/2*c))*b+1/d/(a-b)^4*arctan(tanh(1/2*d*x+1/2*c))*a

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

(a*e^c - 7*b*e^c)*arctan(e^(d*x + c))*e^(-c)/(a^4*d - 4*a^3*b*d + 6*a^2*b^2*d - 4*a*b^3*d + b^4*d) + 1/4*((4*a
^2*b^2*e^(11*c) + 11*a*b^3*e^(11*c) - 3*b^4*e^(11*c))*e^(11*d*x) + (32*a^3*b*e^(9*c) + 32*a^2*b^2*e^(9*c) - 31
*a*b^3*e^(9*c) + 3*b^4*e^(9*c))*e^(9*d*x) + 2*(32*a^4*e^(7*c) - 48*a^3*b*e^(7*c) + 46*a^2*b^2*e^(7*c) - 21*a*b
^3*e^(7*c) + 3*b^4*e^(7*c))*e^(7*d*x) - 2*(32*a^4*e^(5*c) - 48*a^3*b*e^(5*c) + 46*a^2*b^2*e^(5*c) - 21*a*b^3*e
^(5*c) + 3*b^4*e^(5*c))*e^(5*d*x) - (32*a^3*b*e^(3*c) + 32*a^2*b^2*e^(3*c) - 31*a*b^3*e^(3*c) + 3*b^4*e^(3*c))
*e^(3*d*x) - (4*a^2*b^2*e^c + 11*a*b^3*e^c - 3*b^4*e^c)*e^(d*x))/(a^5*b^2*d - 3*a^4*b^3*d + 3*a^3*b^4*d - a^2*
b^5*d + (a^5*b^2*d*e^(12*c) - 3*a^4*b^3*d*e^(12*c) + 3*a^3*b^4*d*e^(12*c) - a^2*b^5*d*e^(12*c))*e^(12*d*x) + 2
*(4*a^6*b*d*e^(10*c) - 13*a^5*b^2*d*e^(10*c) + 15*a^4*b^3*d*e^(10*c) - 7*a^3*b^4*d*e^(10*c) + a^2*b^5*d*e^(10*
c))*e^(10*d*x) + (16*a^7*d*e^(8*c) - 48*a^6*b*d*e^(8*c) + 47*a^5*b^2*d*e^(8*c) - 13*a^4*b^3*d*e^(8*c) - 3*a^3*
b^4*d*e^(8*c) + a^2*b^5*d*e^(8*c))*e^(8*d*x) + 4*(8*a^7*d*e^(6*c) - 28*a^6*b*d*e^(6*c) + 37*a^5*b^2*d*e^(6*c)
- 23*a^4*b^3*d*e^(6*c) + 7*a^3*b^4*d*e^(6*c) - a^2*b^5*d*e^(6*c))*e^(6*d*x) + (16*a^7*d*e^(4*c) - 48*a^6*b*d*e
^(4*c) + 47*a^5*b^2*d*e^(4*c) - 13*a^4*b^3*d*e^(4*c) - 3*a^3*b^4*d*e^(4*c) + a^2*b^5*d*e^(4*c))*e^(4*d*x) + 2*
(4*a^6*b*d*e^(2*c) - 13*a^5*b^2*d*e^(2*c) + 15*a^4*b^3*d*e^(2*c) - 7*a^3*b^4*d*e^(2*c) + a^2*b^5*d*e^(2*c))*e^
(2*d*x)) + 8*integrate(1/32*((35*a^2*b^2*e^(3*c) - 14*a*b^3*e^(3*c) + 3*b^4*e^(3*c))*e^(3*d*x) + (35*a^2*b^2*e
^c - 14*a*b^3*e^c + 3*b^4*e^c)*e^(d*x))/(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5 + (a^6*b*e^(4*c)
- 4*a^5*b^2*e^(4*c) + 6*a^4*b^3*e^(4*c) - 4*a^3*b^4*e^(4*c) + a^2*b^5*e^(4*c))*e^(4*d*x) + 2*(2*a^7*e^(2*c) -
9*a^6*b*e^(2*c) + 16*a^5*b^2*e^(2*c) - 14*a^4*b^3*e^(2*c) + 6*a^3*b^4*e^(2*c) - a^2*b^5*e^(2*c))*e^(2*d*x)), x
)

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Fricas [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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Exception raised: TypeError