3.1.41 \(\int \frac {\sinh ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [41]

Optimal. Leaf size=242 \[ \frac {3 \left (a^2+12 a b+16 b^2\right ) x}{8 a^5}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^5 \sqrt {a+b} d}-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

3/8*(a^2+12*a*b+16*b^2)*x/a^5-3/8*(5*a^2+20*a*b+16*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*b^(1/2)/a^5/d
/(a+b)^(1/2)-1/8*(5*a+8*b)*cosh(d*x+c)*sinh(d*x+c)/a^2/d/(a+b-b*tanh(d*x+c)^2)^2+1/4*cosh(d*x+c)^3*sinh(d*x+c)
/a/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(7*a+12*b)*tanh(d*x+c)/a^3/d/(a+b-b*tanh(d*x+c)^2)^2-3/2*b*(a+2*b)*tanh(d*x
+c)/a^4/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]
time = 0.28, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4217, 481, 541, 536, 212, 214} \begin {gather*} -\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {(5 a+8 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^5 d \sqrt {a+b}}+\frac {3 x \left (a^2+12 a b+16 b^2\right )}{8 a^5}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(3*(a^2 + 12*a*b + 16*b^2)*x)/(8*a^5) - (3*Sqrt[b]*(5*a^2 + 20*a*b + 16*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/S
qrt[a + b]])/(8*a^5*Sqrt[a + b]*d) - ((5*a + 8*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d*(a + b - b*Tanh[c + d*
x]^2)^2) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*a*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(7*a + 12*b)*Tanh[c + d*
x])/(8*a^3*d*(a + b - b*Tanh[c + d*x]^2)^2) - (3*b*(a + 2*b)*Tanh[c + d*x])/(2*a^4*d*(a + b - b*Tanh[c + d*x]^
2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

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 541

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 4217

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1
+ ff^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rubi steps

\begin {align*} \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^3 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a+b+(4 a+7 b) x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-(a+b) (3 a+8 b)-5 b (5 a+8 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{8 a^2 d}\\ &=-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {12 (a+b)^2 (a+4 b)+12 b (a+b) (7 a+12 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{32 a^3 (a+b) d}\\ &=-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-24 (a+b)^2 \left (a^2+8 a b+8 b^2\right )-96 b (a+b)^2 (a+2 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{64 a^4 (a+b)^2 d}\\ &=-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\left (3 \left (a^2+12 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^5 d}-\frac {\left (3 b \left (5 a^2+20 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^5 d}\\ &=\frac {3 \left (a^2+12 a b+16 b^2\right ) x}{8 a^5}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^5 \sqrt {a+b} d}-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3080\) vs. \(2(242)=484\).
time = 20.18, size = 3080, normalized size = 12.73 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(3*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])
/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(
c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(16384*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((a +
2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(
a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*S
inh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(16384*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3)
- (3*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640
*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqr
t[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])
+ (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh
[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(
c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4
*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c
+ 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 684*a^4*b^2*Sinh[2*c] +
2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x
] - 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*x] - 2560*a^2*b^4*Sinh[2*d*x] - 102
4*a*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x
)] - 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Si
nh[4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d
*x] + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(65536*a^3*b^2*(a + b)^2*d*(a + b*Sech
[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a
^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x]
- a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sq
rt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(-1536*b^2*(a + b)^2*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3)*d*x*Cosh
[2*c] - 3072*a*b^2*(a^2 + 3*a*b + 2*b^2)^2*d*x*Cosh[2*d*x] - 768*a^5*b^2*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^4*b^
3*d*x*Cosh[2*(c + 2*d*x)] - 3840*a^3*b^4*d*x*Cosh[2*(c + 2*d*x)] - 1536*a^2*b^5*d*x*Cosh[2*(c + 2*d*x)] - 3072
*a^5*b^2*d*x*Cosh[4*c + 2*d*x] - 18432*a^4*b^3*d*x*Cosh[4*c + 2*d*x] - 39936*a^3*b^4*d*x*Cosh[4*c + 2*d*x] - 3
6864*a^2*b^5*d*x*Cosh[4*c + 2*d*x] - 12288*a*b^6*d*x*Cosh[4*c + 2*d*x] - 768*a^5*b^2*d*x*Cosh[6*c + 4*d*x] - 3
072*a^4*b^3*d*x*Cosh[6*c + 4*d*x] - 3840*a^3*b^4*d*x*Cosh[6*c + 4*d*x] - 1536*a^2*b^5*d*x*Cosh[6*c + 4*d*x] +
9*a^7*Sinh[2*c] - 54*a^6*b*Sinh[2*c] - 2392*a^5*b^2*Sinh[2*c] - 13968*a^4*b^3*Sinh[2*c] - 36480*a^3*b^4*Sinh[2
*c] - 50432*a^2*b^5*Sinh[2*c] - 35840*a*b^6*Sinh[2*c] - 10240*b^7*Sinh[2*c] - 9*a^7*Sinh[2*d*x] + 56*a^6*b*Sin
h[2*d*x] + 2552*a^5*b^2*Sinh[2*d*x] + 13184*a^4*b^3*Sinh[2*d*x] + 27072*a^3*b^4*Sinh[2*d*x] + 24576*a^2*b^5*Si
nh[2*d*x] + 8192*a*b^6*Sinh[2*d*x] - 3*a^7*Sinh[2*(c + 2*d*x)] + 26*a^6*b*Sinh[2*(c + 2*d*x)] + 992*a^5*b^2*Si
nh[2*(c + 2*d*x)] + 3648*a^4*b^3*Sinh[2*(c + 2*d*x)] + 4480*a^3*b^4*Sinh[2*(c + 2*d*x)] + 1792*a^2*b^5*Sinh[2*
(c + 2*d*x)] + 3*a^7*Sinh[4*c + 2*d*x] - 24*a^6*b*Sinh[4*c + 2*d*x] - 600*a^5*b^2*Sinh[4*c + 2*d*x] - 3200*a^4
*b^3*Sinh[4*c + 2*d*x] - 6720*a^3*b^4*Sinh[4*c + 2*d*x] - 6144*a^2*b^5*Sinh[4*c + 2*d*x] - 2048*a*b^6*Sinh[4*c
+ 2*d*x] + 256*a^5*b^2*Sinh[6*c + 4*d*x] + 1024*a^4*b^3*Sinh[6*c + 4*d*x] + 1280*a^3*b^4*Sinh[6*c + 4*d*x] +
512*a^2*b^5*Sinh[6*c + 4*d*x] + 64*a^5*b^2*Sinh[4*c + 6*d*x] + 128*a^4*b^3*Sinh[4*c + 6*d*x] + 64*a^3*b^4*Sinh
[4*c + 6*d*x] + 64*a^5*b^2*Sinh[8*c + 6*d*x] + 128*a^4*b^3*Sinh[8*c + 6*d*x] + 64*a^3*b^4*Sinh[8*c + 6*d*x]))/
(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(32768*a^4*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*b + a*Cosh
[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*a^2*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*S
inh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(
Cosh[c] - Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(
-3*a^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c
+ 2*d*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*
Cosh[2*(c + d*x)])^2)))/(8192*b^2*(a + b)^2*d*(...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(537\) vs. \(2(222)=444\).
time = 3.27, size = 538, normalized size = 2.22 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4/a^3/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/a^3/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(a+12*b)/a^4/(tanh(1/2*d*x+1/2*c)
-1)^2-1/8*(3*a+12*b)/a^4/(tanh(1/2*d*x+1/2*c)-1)+1/8/a^5*(-3*a^2-36*a*b-48*b^2)*ln(tanh(1/2*d*x+1/2*c)-1)+2*b/
a^5*(((-9/8*a^3-21/8*a^2*b-3/2*a*b^2)*tanh(1/2*d*x+1/2*c)^7-1/8*(27*a^2+35*a*b-12*b^2)*a*tanh(1/2*d*x+1/2*c)^5
-1/8*(27*a^2+35*a*b-12*b^2)*a*tanh(1/2*d*x+1/2*c)^3+(-9/8*a^3-21/8*a^2*b-3/2*a*b^2)*tanh(1/2*d*x+1/2*c))/(a*ta
nh(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+1/8*(15
*a^2+60*a*b+48*b^2)*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/
2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+
b)^(1/2))))-1/4/a^3/(tanh(1/2*d*x+1/2*c)+1)^4+1/2/a^3/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-a-12*b)/a^4/(tanh(1/2*d*
x+1/2*c)+1)^2-1/8*(3*a+12*b)/a^4/(tanh(1/2*d*x+1/2*c)+1)+1/8/a^5*(3*a^2+36*a*b+48*b^2)*ln(tanh(1/2*d*x+1/2*c)+
1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2468 vs. \(2 (234) = 468\).
time = 0.62, size = 2468, normalized size = 10.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-3/256*(5*a^4*b + 100*a^3*b^2 + 320*a^2*b^3 + 352*a*b^4 + 128*b^5)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((
a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^7 + 2*a^6*b + a^5*b^2)*sqrt((a + b)*b)*d) -
3/64*(5*a^3*b + 30*a^2*b^2 + 40*a*b^3 + 16*b^4)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*
d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) + 3/256*(5*a^4*b + 10
0*a^3*b^2 + 320*a^2*b^3 + 352*a*b^4 + 128*b^5)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2
*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^7 + 2*a^6*b + a^5*b^2)*sqrt((a + b)*b)*d) + 3/64*(5*a^3*b + 30
*a^2*b^2 + 40*a*b^3 + 16*b^4)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a +
2*b + 2*sqrt((a + b)*b)))/((a^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) + 3/128*(15*a^2*b + 20*a*b^2 + 8*b^3)
*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((
a^5 + 2*a^4*b + a^3*b^2)*sqrt((a + b)*b)*d) + 1/64*(9*a^5*b + 110*a^4*b^2 + 216*a^3*b^3 + 112*a^2*b^4 + (9*a^5
*b + 228*a^4*b^2 + 920*a^3*b^3 + 1216*a^2*b^4 + 512*a*b^5)*e^(6*d*x + 6*c) + (27*a^5*b + 594*a^4*b^2 + 2816*a^
3*b^3 + 5696*a^2*b^4 + 5248*a*b^5 + 1792*b^6)*e^(4*d*x + 4*c) + (27*a^5*b + 476*a^4*b^2 + 1720*a^3*b^3 + 2176*
a^2*b^4 + 896*a*b^5)*e^(2*d*x + 2*c))/((a^9 + 2*a^8*b + a^7*b^2 + (a^9 + 2*a^8*b + a^7*b^2)*e^(8*d*x + 8*c) +
4*(a^9 + 4*a^8*b + 5*a^7*b^2 + 2*a^6*b^3)*e^(6*d*x + 6*c) + 2*(3*a^9 + 14*a^8*b + 27*a^7*b^2 + 24*a^6*b^3 + 8*
a^5*b^4)*e^(4*d*x + 4*c) + 4*(a^9 + 4*a^8*b + 5*a^7*b^2 + 2*a^6*b^3)*e^(2*d*x + 2*c))*d) - 1/64*(9*a^5*b + 110
*a^4*b^2 + 216*a^3*b^3 + 112*a^2*b^4 + (27*a^5*b + 476*a^4*b^2 + 1720*a^3*b^3 + 2176*a^2*b^4 + 896*a*b^5)*e^(-
2*d*x - 2*c) + (27*a^5*b + 594*a^4*b^2 + 2816*a^3*b^3 + 5696*a^2*b^4 + 5248*a*b^5 + 1792*b^6)*e^(-4*d*x - 4*c)
+ (9*a^5*b + 228*a^4*b^2 + 920*a^3*b^3 + 1216*a^2*b^4 + 512*a*b^5)*e^(-6*d*x - 6*c))/((a^9 + 2*a^8*b + a^7*b^
2 + 4*(a^9 + 4*a^8*b + 5*a^7*b^2 + 2*a^6*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^9 + 14*a^8*b + 27*a^7*b^2 + 24*a^6*b^3
+ 8*a^5*b^4)*e^(-4*d*x - 4*c) + 4*(a^9 + 4*a^8*b + 5*a^7*b^2 + 2*a^6*b^3)*e^(-6*d*x - 6*c) + (a^9 + 2*a^8*b +
a^7*b^2)*e^(-8*d*x - 8*c))*d) + 1/16*(9*a^4*b + 32*a^3*b^2 + 20*a^2*b^3 + 3*(3*a^4*b + 34*a^3*b^2 + 64*a^2*b^
3 + 32*a*b^4)*e^(6*d*x + 6*c) + (27*a^4*b + 264*a^3*b^2 + 740*a^2*b^3 + 832*a*b^4 + 320*b^5)*e^(4*d*x + 4*c) +
(27*a^4*b + 194*a^3*b^2 + 336*a^2*b^3 + 160*a*b^4)*e^(2*d*x + 2*c))/((a^8 + 2*a^7*b + a^6*b^2 + (a^8 + 2*a^7*
b + a^6*b^2)*e^(8*d*x + 8*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(6*d*x + 6*c) + 2*(3*a^8 + 14*a^7*b
+ 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*e^(4*d*x + 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(2*d*x +
2*c))*d) - 1/16*(9*a^4*b + 32*a^3*b^2 + 20*a^2*b^3 + (27*a^4*b + 194*a^3*b^2 + 336*a^2*b^3 + 160*a*b^4)*e^(-2
*d*x - 2*c) + (27*a^4*b + 264*a^3*b^2 + 740*a^2*b^3 + 832*a*b^4 + 320*b^5)*e^(-4*d*x - 4*c) + 3*(3*a^4*b + 34*
a^3*b^2 + 64*a^2*b^3 + 32*a*b^4)*e^(-6*d*x - 6*c))/((a^8 + 2*a^7*b + a^6*b^2 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 +
2*a^5*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*e^(-4*d*x - 4*c) + 4*
(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(-6*d*x - 6*c) + (a^8 + 2*a^7*b + a^6*b^2)*e^(-8*d*x - 8*c))*d) - 3/
32*(9*a^3*b + 6*a^2*b^2 + (27*a^3*b + 68*a^2*b^2 + 32*a*b^3)*e^(-2*d*x - 2*c) + 3*(9*a^3*b + 30*a^2*b^2 + 40*a
*b^3 + 16*b^4)*e^(-4*d*x - 4*c) + (9*a^3*b + 28*a^2*b^2 + 16*a*b^3)*e^(-6*d*x - 6*c))/((a^7 + 2*a^6*b + a^5*b^
2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3
+ 8*a^3*b^4)*e^(-4*d*x - 4*c) + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-6*d*x - 6*c) + (a^7 + 2*a^6*b +
a^5*b^2)*e^(-8*d*x - 8*c))*d) + 3/8*(d*x + c)/(a^3*d) - 1/8*e^(2*d*x + 2*c)/(a^3*d) + 1/8*e^(-2*d*x - 2*c)/(a
^3*d) + 3/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^4*d) - 3/4*b*log(2*(a + 2*b)*e^(-2*d
*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^4*d) + 1/64*(a*e^(4*d*x + 4*c) - 24*b*e^(2*d*x + 2*c))/(a^4*d) + 1/64*(
24*b*e^(-2*d*x - 2*c) - a*e^(-4*d*x - 4*c))/(a^4*d) + 3/8*(a*b + 4*b^2)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^
(2*d*x + 2*c) + a)/(a^5*d) - 3/8*(a*b + 4*b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^5
*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6038 vs. \(2 (234) = 468\).
time = 0.58, size = 12353, normalized size = 51.05 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/64*(a^4*cosh(d*x + c)^16 + 16*a^4*cosh(d*x + c)*sinh(d*x + c)^15 + a^4*sinh(d*x + c)^16 - 4*(a^4 + 4*a^3*b)
*cosh(d*x + c)^14 + 4*(30*a^4*cosh(d*x + c)^2 - a^4 - 4*a^3*b)*sinh(d*x + c)^14 + 56*(10*a^4*cosh(d*x + c)^3 -
(a^4 + 4*a^3*b)*cosh(d*x + c))*sinh(d*x + c)^13 - 2*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16
*a^2*b^2)*d*x)*cosh(d*x + c)^12 + 2*(910*a^4*cosh(d*x + c)^4 - 13*a^4 - 72*a^3*b - 88*a^2*b^2 + 12*(a^4 + 12*a
^3*b + 16*a^2*b^2)*d*x - 182*(a^4 + 4*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 8*(546*a^4*cosh(d*x + c)^5 -
182*(a^4 + 4*a^3*b)*cosh(d*x + c)^3 - 3*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x
)*cosh(d*x + c))*sinh(d*x + c)^11 - 4*(9*a^4 + 24*a^3*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*
b^2 + 32*a*b^3)*d*x)*cosh(d*x + c)^10 + 4*(2002*a^4*cosh(d*x + c)^6 - 1001*(a^4 + 4*a^3*b)*cosh(d*x + c)^4 - 9
*a^4 - 24*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x - 33*(13*a^4 + 72*a^
3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 8*(1430*a^4*cosh(
d*x + c)^7 - 1001*(a^4 + 4*a^3*b)*cosh(d*x + c)^5 - 55*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b +
16*a^2*b^2)*d*x)*cosh(d*x + c)^3 - 5*(9*a^4 + 24*a^3*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*b
^2 + 32*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 16*(27*a^3*b + 114*a^2*b^2 + 184*a*b^3 + 112*b^4 + 3*(3*a
^4 + 44*a^3*b + 152*a^2*b^2 + 224*a*b^3 + 128*b^4)*d*x)*cosh(d*x + c)^8 + 2*(6435*a^4*cosh(d*x + c)^8 - 6006*(
a^4 + 4*a^3*b)*cosh(d*x + c)^6 - 495*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*c
osh(d*x + c)^4 + 216*a^3*b + 912*a^2*b^2 + 1472*a*b^3 + 896*b^4 + 24*(3*a^4 + 44*a^3*b + 152*a^2*b^2 + 224*a*b
^3 + 128*b^4)*d*x - 90*(9*a^4 + 24*a^3*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)
*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(715*a^4*cosh(d*x + c)^9 - 858*(a^4 + 4*a^3*b)*cosh(d*x + c)^7 - 9
9*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x + c)^5 - 30*(9*a^4 + 24*a^3
*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x)*cosh(d*x + c)^3 + 8*(27*a^3*b +
114*a^2*b^2 + 184*a*b^3 + 112*b^4 + 3*(3*a^4 + 44*a^3*b + 152*a^2*b^2 + 224*a*b^3 + 128*b^4)*d*x)*cosh(d*x + c
))*sinh(d*x + c)^7 + 4*(9*a^4 + 168*a^3*b + 496*a^2*b^2 + 416*a*b^3 + 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b
^3)*d*x)*cosh(d*x + c)^6 + 4*(2002*a^4*cosh(d*x + c)^10 - 3003*(a^4 + 4*a^3*b)*cosh(d*x + c)^8 - 462*(13*a^4 +
72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x + c)^6 - 210*(9*a^4 + 24*a^3*b - 16*a^
2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x)*cosh(d*x + c)^4 + 9*a^4 + 168*a^3*b + 496*
a^2*b^2 + 416*a*b^3 + 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x + 112*(27*a^3*b + 114*a^2*b^2 + 184*a*b^
3 + 112*b^4 + 3*(3*a^4 + 44*a^3*b + 152*a^2*b^2 + 224*a*b^3 + 128*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 +
8*(546*a^4*cosh(d*x + c)^11 - 1001*(a^4 + 4*a^3*b)*cosh(d*x + c)^9 - 198*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12
*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x + c)^7 - 126*(9*a^4 + 24*a^3*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4
+ 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x)*cosh(d*x + c)^5 + 112*(27*a^3*b + 114*a^2*b^2 + 184*a*b^3 + 112*b^4 +
3*(3*a^4 + 44*a^3*b + 152*a^2*b^2 + 224*a*b^3 + 128*b^4)*d*x)*cosh(d*x + c)^3 + 3*(9*a^4 + 168*a^3*b + 496*a^
2*b^2 + 416*a*b^3 + 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(13*a^
4 + 144*a^3*b + 200*a^2*b^2 + 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x + c)^4 + 2*(910*a^4*cosh(d*x + c)
^12 - 2002*(a^4 + 4*a^3*b)*cosh(d*x + c)^10 - 495*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^
2*b^2)*d*x)*cosh(d*x + c)^8 - 420*(9*a^4 + 24*a^3*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*b^2
+ 32*a*b^3)*d*x)*cosh(d*x + c)^6 + 560*(27*a^3*b + 114*a^2*b^2 + 184*a*b^3 + 112*b^4 + 3*(3*a^4 + 44*a^3*b + 1
52*a^2*b^2 + 224*a*b^3 + 128*b^4)*d*x)*cosh(d*x + c)^4 + 13*a^4 + 144*a^3*b + 200*a^2*b^2 + 12*(a^4 + 12*a^3*b
+ 16*a^2*b^2)*d*x + 30*(9*a^4 + 168*a^3*b + 496*a^2*b^2 + 416*a*b^3 + 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*
b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - a^4 + 8*(70*a^4*cosh(d*x + c)^13 - 182*(a^4 + 4*a^3*b)*cosh(d*x +
c)^11 - 55*(13*a^4 + 72*a^3*b + 88*a^2*b^2 - 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x + c)^9 - 60*(9*a^
4 + 24*a^3*b - 16*a^2*b^2 - 32*a*b^3 - 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3)*d*x)*cosh(d*x + c)^7 + 112*
(27*a^3*b + 114*a^2*b^2 + 184*a*b^3 + 112*b^4 + 3*(3*a^4 + 44*a^3*b + 152*a^2*b^2 + 224*a*b^3 + 128*b^4)*d*x)*
cosh(d*x + c)^5 + 10*(9*a^4 + 168*a^3*b + 496*a^2*b^2 + 416*a*b^3 + 24*(a^4 + 14*a^3*b + 40*a^2*b^2 + 32*a*b^3
)*d*x)*cosh(d*x + c)^3 + (13*a^4 + 144*a^3*b + 200*a^2*b^2 + 12*(a^4 + 12*a^3*b + 16*a^2*b^2)*d*x)*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(a^4 + 4*a^3*b)*cosh(d*x + c)^2 + 4*(30*a^4*cosh(d*x + c)^14 - 91*(a^4 + 4*a^3*b)*cosh
(d*x + c)^12 - 33*(13*a^4 + 72*a^3*b + 88*a^2*b...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (234) = 468\).
time = 1.75, size = 518, normalized size = 2.14 \begin {gather*} \frac {\frac {24 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {24 \, {\left (5 \, a^{2} b + 20 \, a b^{2} + 16 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{5}} + \frac {a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )}}{a^{6}} - \frac {6 \, a^{4} e^{\left (12 \, d x + 12 \, c\right )} + 72 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 96 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 16 \, a^{4} e^{\left (10 \, d x + 10 \, c\right )} + 168 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 384 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 256 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 5 \, a^{4} e^{\left (8 \, d x + 8 \, c\right )} - 64 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} - 192 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 256 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 256 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 20 \, a^{4} e^{\left (6 \, d x + 6 \, c\right )} - 360 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 1024 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 896 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} - 216 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 304 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a^{4} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + a^{4}}{{\left (a e^{\left (6 \, d x + 6 \, c\right )} + 2 \, a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} + a e^{\left (2 \, d x + 2 \, c\right )}\right )}^{2} a^{5}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/64*(24*(a^2 + 12*a*b + 16*b^2)*(d*x + c)/a^5 - 24*(5*a^2*b + 20*a*b^2 + 16*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c
) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^5) + (a^3*e^(4*d*x + 4*c) - 8*a^3*e^(2*d*x + 2*c) - 24*a^2*
b*e^(2*d*x + 2*c))/a^6 - (6*a^4*e^(12*d*x + 12*c) + 72*a^3*b*e^(12*d*x + 12*c) + 96*a^2*b^2*e^(12*d*x + 12*c)
+ 16*a^4*e^(10*d*x + 10*c) + 168*a^3*b*e^(10*d*x + 10*c) + 384*a^2*b^2*e^(10*d*x + 10*c) + 256*a*b^3*e^(10*d*x
+ 10*c) + 5*a^4*e^(8*d*x + 8*c) - 64*a^3*b*e^(8*d*x + 8*c) - 192*a^2*b^2*e^(8*d*x + 8*c) - 256*a*b^3*e^(8*d*x
+ 8*c) - 256*b^4*e^(8*d*x + 8*c) - 20*a^4*e^(6*d*x + 6*c) - 360*a^3*b*e^(6*d*x + 6*c) - 1024*a^2*b^2*e^(6*d*x
+ 6*c) - 896*a*b^3*e^(6*d*x + 6*c) - 20*a^4*e^(4*d*x + 4*c) - 216*a^3*b*e^(4*d*x + 4*c) - 304*a^2*b^2*e^(4*d*
x + 4*c) - 4*a^4*e^(2*d*x + 2*c) - 16*a^3*b*e^(2*d*x + 2*c) + a^4)/((a*e^(6*d*x + 6*c) + 2*a*e^(4*d*x + 4*c) +
4*b*e^(4*d*x + 4*c) + a*e^(2*d*x + 2*c))^2*a^5))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^4/(a + b/cosh(c + d*x)^2)^3,x)

[Out]

int((cosh(c + d*x)^6*sinh(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^3, x)

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