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

Optimal. Leaf size=154 \[ \frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{9/2} d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (a \cosh ^2(c+d x)+b\right )}-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d} \]

[Out]

-(a+3*b)*cosh(d*x+c)/a^4/d+1/3*cosh(d*x+c)^3/a^3/d+1/4*b^2*(a+b)*cosh(d*x+c)/a^4/d/(b+a*cosh(d*x+c)^2)^2-1/8*b
*(9*a+13*b)*cosh(d*x+c)/a^4/d/(b+a*cosh(d*x+c)^2)+5/8*(3*a+7*b)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^
(9/2)/d

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Rubi [A]  time = 0.22, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4133, 455, 1814, 1153, 205} \[ \frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (a \cosh ^2(c+d x)+b\right )}-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{9/2} d}+\frac {\cosh ^3(c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(9/2)*d) - ((a + 3*b)*Cosh[c + d*x])/(a^4
*d) + Cosh[c + d*x]^3/(3*a^3*d) + (b^2*(a + b)*Cosh[c + d*x])/(4*a^4*d*(b + a*Cosh[c + d*x]^2)^2) - (b*(9*a +
13*b)*Cosh[c + d*x])/(8*a^4*d*(b + a*Cosh[c + d*x]^2))

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]

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 4133

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

Rubi steps

\begin {align*} \int \frac {\sinh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^6 \left (1-x^2\right )}{\left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {-b^2 (a+b)+4 a b (a+b) x^2-4 a^2 (a+b) x^4+4 a^3 x^6}{\left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 a^4 d}\\ &=\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-b^2 (7 a+11 b)+8 a b (a+2 b) x^2-8 a^2 b x^4}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^4 b d}\\ &=\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \left (8 b (a+3 b)-8 a b x^2-\frac {5 \left (3 a b^2+7 b^3\right )}{b+a x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{8 a^4 b d}\\ &=-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {(5 b (3 a+7 b)) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^4 d}\\ &=\frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{9/2} d}-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [C]  time = 10.38, size = 1217, normalized size = 7.90 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sinh[c + d*x]^3/(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*((24*(3*a - 4*b)*(ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Co
sh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh
[(d*x)/2]))/Sqrt[b]] + ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + C
osh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]))/(a^(3/2)*b^(5/2)) - (54
*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)
/2])/Sqrt[b]]))/(Sqrt[a]*b^(5/2)) - (36*Cosh[c + d*x]*(3*a + 10*b + 3*a*Cosh[2*(c + d*x)]))/(b^2*(a + 2*b + a*
Cosh[2*(c + d*x)])^2) + (48*Cosh[c + d*x]*(3*a^2 + 6*a*b + 8*b^2 + a*(3*a - 4*b)*Cosh[2*(c + d*x)]))/(a*b^2*(a
+ 2*b + a*Cosh[2*(c + d*x)])^2) + (3*(3*a^4 - 40*a^3*b + 720*a^2*b^2 + 6720*a*b^3 + 8960*b^4)*ArcTan[((Sqrt[a
] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(
Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 3*(3*a^4 - 40*a^3*b + 720*a^2*b^2 + 6720*a*b^3 + 8960*b^4)*Ar
cTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[
a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (2*Sqrt[a]*Sqrt[b]*Cosh[c + d*x]*(9*a^5 - 90*a^4
*b - 10144*a^3*b^2 - 48672*a^2*b^3 - 85120*a*b^4 - 53760*b^5 + a*(9*a^4 - 120*a^3*b - 12432*a^2*b^2 - 47936*a*
b^3 - 44800*b^4)*Cosh[2*(c + d*x)] - 128*a^2*b^2*(15*a + 28*b)*Cosh[4*(c + d*x)] + 128*a^3*b^2*Cosh[6*(c + d*x
)]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2)/(a^(9/2)*b^(5/2)) + (9*((-3*(a^3 - 8*a^2*b + 80*a*b^2 + 320*b^3)*ArcTa
n[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a +
b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(5/2) - (3*(a^3 - 8*a^2*b + 80*a*b^2 + 320*b^3)*Ar
cTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[
a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(5/2) + 512*Sqrt[a]*Cosh[c]*Cosh[d*x] - (8*Sqrt
[a]*(a^3 + 24*a^2*b + 80*a*b^2 + 64*b^3)*Cosh[c + d*x])/(b*(a + 2*b + a*Cosh[2*(c + d*x)])^2) - (2*Sqrt[a]*(3*
a^3 - 24*a^2*b - 400*a*b^2 - 576*b^3)*Cosh[c + d*x])/(b^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + 512*Sqrt[a]*Sinh[
c]*Sinh[d*x]))/a^(7/2)))/(49152*d*(a + b*Sech[c + d*x]^2)^3)

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fricas [B]  time = 0.56, size = 8667, normalized size = 56.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(2*a^3*cosh(d*x + c)^14 + 28*a^3*cosh(d*x + c)*sinh(d*x + c)^13 + 2*a^3*sinh(d*x + c)^14 - 2*(5*a^3 + 28
*a^2*b)*cosh(d*x + c)^12 + 2*(91*a^3*cosh(d*x + c)^2 - 5*a^3 - 28*a^2*b)*sinh(d*x + c)^12 + 8*(91*a^3*cosh(d*x
+ c)^3 - 3*(5*a^3 + 28*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^11 - 2*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x +
c)^10 + 2*(1001*a^3*cosh(d*x + c)^4 - 39*a^3 - 290*a^2*b - 350*a*b^2 - 66*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^2)
*sinh(d*x + c)^10 + 4*(1001*a^3*cosh(d*x + c)^5 - 110*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^3 - 5*(39*a^3 + 290*a^2
*b + 350*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 10*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^8
+ 2*(3003*a^3*cosh(d*x + c)^6 - 495*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^4 - 85*a^3 - 730*a^2*b - 1410*a*b^2 - 84
0*b^3 - 45*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(429*a^3*cosh(d*x + c)^7 - 9
9*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^5 - 15*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^3 - 5*(17*a^3 + 146*a
^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 10*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cos
h(d*x + c)^6 + 2*(3003*a^3*cosh(d*x + c)^8 - 924*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^6 - 210*(39*a^3 + 290*a^2*b
+ 350*a*b^2)*cosh(d*x + c)^4 - 85*a^3 - 730*a^2*b - 1410*a*b^2 - 840*b^3 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2
+ 168*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(1001*a^3*cosh(d*x + c)^9 - 396*(5*a^3 + 28*a^2*b)*cosh(d*x +
c)^7 - 126*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^5 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*
cosh(d*x + c)^3 - 15*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(39*a^3 + 2
90*a^2*b + 350*a*b^2)*cosh(d*x + c)^4 + 2*(1001*a^3*cosh(d*x + c)^10 - 495*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^8
- 210*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^6 - 350*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d
*x + c)^4 - 39*a^3 - 290*a^2*b - 350*a*b^2 - 75*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^2)*si
nh(d*x + c)^4 + 8*(91*a^3*cosh(d*x + c)^11 - 55*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^9 - 30*(39*a^3 + 290*a^2*b +
350*a*b^2)*cosh(d*x + c)^7 - 70*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^5 - 25*(17*a^3 + 146*
a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^3 - (39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3
+ 2*a^3 - 2*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^2 + 2*(91*a^3*cosh(d*x + c)^12 - 66*(5*a^3 + 28*a^2*b)*cosh(d*x
+ c)^10 - 45*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^8 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)
*cosh(d*x + c)^6 - 75*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^4 - 5*a^3 - 28*a^2*b - 6*(39*a^
3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 15*((3*a^3 + 7*a^2*b)*cosh(d*x + c)^11 + 11*(3*a
^3 + 7*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^10 + (3*a^3 + 7*a^2*b)*sinh(d*x + c)^11 + 4*(3*a^3 + 13*a^2*b + 14*a
*b^2)*cosh(d*x + c)^9 + (12*a^3 + 52*a^2*b + 56*a*b^2 + 55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^9
+ 3*(55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^3 + 12*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 +
2*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^7 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^4 + 9*a^3 +
45*a^2*b + 80*a*b^2 + 56*b^3 + 72*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 14*(33*(3*a
^3 + 7*a^2*b)*cosh(d*x + c)^5 + 24*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^3 + (9*a^3 + 45*a^2*b + 80*a*b^
2 + 56*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 4*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 2*(231*(3*a^3 +
7*a^2*b)*cosh(d*x + c)^6 + 252*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^4 + 6*a^3 + 26*a^2*b + 28*a*b^2 +
21*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x
+ c)^7 + 252*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 35*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d
*x + c)^3 + 10*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^3 + 7*a^2*b)*cosh(d*x + c)^
3 + (165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^8 + 336*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^6 + 70*(9*a^3 + 4
5*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^4 + 3*a^3 + 7*a^2*b + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c
)^2)*sinh(d*x + c)^3 + (55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^9 + 144*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)
^7 + 42*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^5 + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c
)^3 + 3*(3*a^3 + 7*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (11*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^10 + 36*(3*a^3
+ 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^8 + 14*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^6 + 20*(3*a^3
+ 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-b/a)*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*(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)
+ 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) + 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)) + 4*(7*a^3*cosh(d*x + c)^13 - 6*(5*a^3 + 28*a^2*b)*cosh(d
*x + c)^11 - 5*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^9 - 20*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3
)*cosh(d*x + c)^7 - 15*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^5 - 2*(39*a^3 + 290*a^2*b + 35
0*a*b^2)*cosh(d*x + c)^3 - (5*a^3 + 28*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(a^6*d*cosh(d*x + c)^11 + 11*a^6*d
*cosh(d*x + c)*sinh(d*x + c)^10 + a^6*d*sinh(d*x + c)^11 + 4*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^9 + a^6*d*cosh(d*
x + c)^3 + (55*a^6*d*cosh(d*x + c)^2 + 4*(a^6 + 2*a^5*b)*d)*sinh(d*x + c)^9 + 2*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*
d*cosh(d*x + c)^7 + 3*(55*a^6*d*cosh(d*x + c)^3 + 12*(a^6 + 2*a^5*b)*d*cosh(d*x + c))*sinh(d*x + c)^8 + 2*(165
*a^6*d*cosh(d*x + c)^4 + 72*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^2 + (3*a^6 + 8*a^5*b + 8*a^4*b^2)*d)*sinh(d*x + c)
^7 + 4*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^5 + 14*(33*a^6*d*cosh(d*x + c)^5 + 24*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^3
+ (3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(231*a^6*d*cosh(d*x + c)^6 + 252*(a^6 +
2*a^5*b)*d*cosh(d*x + c)^4 + 21*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^2 + 2*(a^6 + 2*a^5*b)*d)*sinh(d*
x + c)^5 + 2*(165*a^6*d*cosh(d*x + c)^7 + 252*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^5 + 35*(3*a^6 + 8*a^5*b + 8*a^4*
b^2)*d*cosh(d*x + c)^3 + 10*(a^6 + 2*a^5*b)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (165*a^6*d*cosh(d*x + c)^8 + 33
6*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^6 + a^6*d + 70*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^4 + 40*(a^6 + 2
*a^5*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (55*a^6*d*cosh(d*x + c)^9 + 144*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^7
+ 3*a^6*d*cosh(d*x + c) + 42*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^5 + 40*(a^6 + 2*a^5*b)*d*cosh(d*x
+ c)^3)*sinh(d*x + c)^2 + (11*a^6*d*cosh(d*x + c)^10 + 36*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^8 + 3*a^6*d*cosh(d*x
+ c)^2 + 14*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^6 + 20*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^4)*sinh(d*x
+ c)), 1/24*(a^3*cosh(d*x + c)^14 + 14*a^3*cosh(d*x + c)*sinh(d*x + c)^13 + a^3*sinh(d*x + c)^14 - (5*a^3 + 28
*a^2*b)*cosh(d*x + c)^12 + (91*a^3*cosh(d*x + c)^2 - 5*a^3 - 28*a^2*b)*sinh(d*x + c)^12 + 4*(91*a^3*cosh(d*x +
c)^3 - 3*(5*a^3 + 28*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^11 - (39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^
10 + (1001*a^3*cosh(d*x + c)^4 - 39*a^3 - 290*a^2*b - 350*a*b^2 - 66*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^2)*sinh(
d*x + c)^10 + 2*(1001*a^3*cosh(d*x + c)^5 - 110*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^3 - 5*(39*a^3 + 290*a^2*b + 3
50*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 5*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^8 + (300
3*a^3*cosh(d*x + c)^6 - 495*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^4 - 85*a^3 - 730*a^2*b - 1410*a*b^2 - 840*b^3 - 4
5*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(429*a^3*cosh(d*x + c)^7 - 99*(5*a^3 +
28*a^2*b)*cosh(d*x + c)^5 - 15*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^3 - 5*(17*a^3 + 146*a^2*b + 282
*a*b^2 + 168*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 5*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^
6 + (3003*a^3*cosh(d*x + c)^8 - 924*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^6 - 210*(39*a^3 + 290*a^2*b + 350*a*b^2)*
cosh(d*x + c)^4 - 85*a^3 - 730*a^2*b - 1410*a*b^2 - 840*b^3 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(1001*a^3*cosh(d*x + c)^9 - 396*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^7 - 126*(
39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^5 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)
^3 - 15*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - (39*a^3 + 290*a^2*b + 350*
a*b^2)*cosh(d*x + c)^4 + (1001*a^3*cosh(d*x + c)^10 - 495*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^8 - 210*(39*a^3 + 2
90*a^2*b + 350*a*b^2)*cosh(d*x + c)^6 - 350*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^4 - 39*a^
3 - 290*a^2*b - 350*a*b^2 - 75*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4
*(91*a^3*cosh(d*x + c)^11 - 55*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^9 - 30*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d
*x + c)^7 - 70*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^5 - 25*(17*a^3 + 146*a^2*b + 282*a*b^2
+ 168*b^3)*cosh(d*x + c)^3 - (39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - (5*a^3 +
28*a^2*b)*cosh(d*x + c)^2 + (91*a^3*cosh(d*x + c)^12 - 66*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^10 - 45*(39*a^3 +
290*a^2*b + 350*a*b^2)*cosh(d*x + c)^8 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^6 - 75*(
17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^4 - 5*a^3 - 28*a^2*b - 6*(39*a^3 + 290*a^2*b + 350*a*b
^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 15*((3*a^3 + 7*a^2*b)*cosh(d*x + c)^11 + 11*(3*a^3 + 7*a^2*b)*cosh(d*x
+ c)*sinh(d*x + c)^10 + (3*a^3 + 7*a^2*b)*sinh(d*x + c)^11 + 4*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^9 +
(12*a^3 + 52*a^2*b + 56*a*b^2 + 55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 3*(55*(3*a^3 + 7*a^2*
b)*cosh(d*x + c)^3 + 12*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 2*(9*a^3 + 45*a^2*b + 8
0*a*b^2 + 56*b^3)*cosh(d*x + c)^7 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^4 + 9*a^3 + 45*a^2*b + 80*a*b^2 + 5
6*b^3 + 72*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 14*(33*(3*a^3 + 7*a^2*b)*cosh(d*x
+ c)^5 + 24*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^3 + (9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x +
c))*sinh(d*x + c)^6 + 4*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 2*(231*(3*a^3 + 7*a^2*b)*cosh(d*x + c)
^6 + 252*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^4 + 6*a^3 + 26*a^2*b + 28*a*b^2 + 21*(9*a^3 + 45*a^2*b +
80*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^7 + 252*(3*a^3 +
13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 35*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^3 + 10*(3*a^3 +
13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^3 + 7*a^2*b)*cosh(d*x + c)^3 + (165*(3*a^3 + 7*a^2
*b)*cosh(d*x + c)^8 + 336*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^6 + 70*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56
*b^3)*cosh(d*x + c)^4 + 3*a^3 + 7*a^2*b + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 +
(55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^9 + 144*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^7 + 42*(9*a^3 + 45*a^2
*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^5 + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^3 + 3*(3*a^3 + 7*a^2*
b)*cosh(d*x + c))*sinh(d*x + c)^2 + (11*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^10 + 36*(3*a^3 + 13*a^2*b + 14*a*b^2)*
cosh(d*x + c)^8 + 14*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^6 + 20*(3*a^3 + 13*a^2*b + 14*a*b^2)
*cosh(d*x + c)^4 + 3*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(b/a)*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 + (a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a
+ 4*b)*sinh(d*x + c))*sqrt(b/a)/b) + 15*((3*a^3 + 7*a^2*b)*cosh(d*x + c)^11 + 11*(3*a^3 + 7*a^2*b)*cosh(d*x +
c)*sinh(d*x + c)^10 + (3*a^3 + 7*a^2*b)*sinh(d*x + c)^11 + 4*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^9 + (
12*a^3 + 52*a^2*b + 56*a*b^2 + 55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 3*(55*(3*a^3 + 7*a^2*b)
*cosh(d*x + c)^3 + 12*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 2*(9*a^3 + 45*a^2*b + 80*
a*b^2 + 56*b^3)*cosh(d*x + c)^7 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^4 + 9*a^3 + 45*a^2*b + 80*a*b^2 + 56*
b^3 + 72*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 14*(33*(3*a^3 + 7*a^2*b)*cosh(d*x +
c)^5 + 24*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^3 + (9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)
)*sinh(d*x + c)^6 + 4*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 2*(231*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^6
+ 252*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^4 + 6*a^3 + 26*a^2*b + 28*a*b^2 + 21*(9*a^3 + 45*a^2*b + 80
*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^7 + 252*(3*a^3 + 13
*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 35*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^3 + 10*(3*a^3 + 1
3*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^3 + 7*a^2*b)*cosh(d*x + c)^3 + (165*(3*a^3 + 7*a^2*b
)*cosh(d*x + c)^8 + 336*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^6 + 70*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b
^3)*cosh(d*x + c)^4 + 3*a^3 + 7*a^2*b + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (5
5*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^9 + 144*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^7 + 42*(9*a^3 + 45*a^2*b
+ 80*a*b^2 + 56*b^3)*cosh(d*x + c)^5 + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^3 + 3*(3*a^3 + 7*a^2*b)
*cosh(d*x + c))*sinh(d*x + c)^2 + (11*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^10 + 36*(3*a^3 + 13*a^2*b + 14*a*b^2)*co
sh(d*x + c)^8 + 14*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^6 + 20*(3*a^3 + 13*a^2*b + 14*a*b^2)*c
osh(d*x + c)^4 + 3*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c) + a
*sinh(d*x + c))*sqrt(b/a)/b) + 2*(7*a^3*cosh(d*x + c)^13 - 6*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^11 - 5*(39*a^3 +
290*a^2*b + 350*a*b^2)*cosh(d*x + c)^9 - 20*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^7 - 15*(
17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^5 - 2*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^3
- (5*a^3 + 28*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(a^6*d*cosh(d*x + c)^11 + 11*a^6*d*cosh(d*x + c)*sinh(d*x
+ c)^10 + a^6*d*sinh(d*x + c)^11 + 4*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^9 + a^6*d*cosh(d*x + c)^3 + (55*a^6*d*cos
h(d*x + c)^2 + 4*(a^6 + 2*a^5*b)*d)*sinh(d*x + c)^9 + 2*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^7 + 3*(5
5*a^6*d*cosh(d*x + c)^3 + 12*(a^6 + 2*a^5*b)*d*cosh(d*x + c))*sinh(d*x + c)^8 + 2*(165*a^6*d*cosh(d*x + c)^4 +
72*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^2 + (3*a^6 + 8*a^5*b + 8*a^4*b^2)*d)*sinh(d*x + c)^7 + 4*(a^6 + 2*a^5*b)*d
*cosh(d*x + c)^5 + 14*(33*a^6*d*cosh(d*x + c)^5 + 24*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^3 + (3*a^6 + 8*a^5*b + 8*
a^4*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(231*a^6*d*cosh(d*x + c)^6 + 252*(a^6 + 2*a^5*b)*d*cosh(d*x + c)
^4 + 21*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^2 + 2*(a^6 + 2*a^5*b)*d)*sinh(d*x + c)^5 + 2*(165*a^6*d*
cosh(d*x + c)^7 + 252*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^5 + 35*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^3 +
10*(a^6 + 2*a^5*b)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (165*a^6*d*cosh(d*x + c)^8 + 336*(a^6 + 2*a^5*b)*d*cosh
(d*x + c)^6 + a^6*d + 70*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^4 + 40*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^
2)*sinh(d*x + c)^3 + (55*a^6*d*cosh(d*x + c)^9 + 144*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^7 + 3*a^6*d*cosh(d*x + c)
+ 42*(3*a^6 + 8*a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^5 + 40*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^2
+ (11*a^6*d*cosh(d*x + c)^10 + 36*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^8 + 3*a^6*d*cosh(d*x + c)^2 + 14*(3*a^6 + 8*
a^5*b + 8*a^4*b^2)*d*cosh(d*x + c)^6 + 20*(a^6 + 2*a^5*b)*d*cosh(d*x + c)^4)*sinh(d*x + c))]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[84,-86]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-42,-12]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-43,-99]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumin
g [a,b]=[-28,94]Warning, need to choose a branch for the root of a polynomial with parameters. This might be w
rong.The choice was done assuming [a,b]=[-7,46]Warning, need to choose a branch for the root of a polynomial w
ith parameters. This might be wrong.The choice was done assuming [a,b]=[-35,-99]Warning, need to choose a bran
ch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[7,50]
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice
was done assuming [a,b]=[-63,-70]Warning, need to choose a branch for the root of a polynomial with parameters
. This might be wrong.The choice was done assuming [a,b]=[-5,48]Warning, need to choose a branch for the root
of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-33,-84]Warning, need
to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assu
ming [a,b]=[90,-39]Warning, need to choose a branch for the root of a polynomial with parameters. This might b
e wrong.The choice was done assuming [a,b]=[70,15]Warning, need to choose a branch for the root of a polynomia
l with parameters. This might be wrong.The choice was done assuming [a,b]=[-11,29]Warning, need to choose a br
anch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[84,
29]Undef/Unsigned Inf encountered in limitEvaluation time: 3.48Limit: Max order reached or unable to make seri
es expansion Error: Bad Argument Value

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maple [B]  time = 0.38, size = 1296, normalized size = 8.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/d/a^3/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)+3/d
/a^4/(tanh(1/2*d*x+1/2*c)-1)*b+1/3/d/a^3/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/d/a^3/(tanh(1/2*d*x+1/2*c)+1)^2-1/2/d/a
^3/(tanh(1/2*d*x+1/2*c)+1)-3/d/a^4/(tanh(1/2*d*x+1/2*c)+1)*b-9/4/d/a^2*b/(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*tanh(1/2*d*x+1/2*c)^6+1/2/d/a^3*b^2/(ta
nh(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*tanh(
1/2*d*x+1/2*c)^6+11/4/d/a^4*b^3/(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*t
anh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^6-27/4/d/a*b/(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)^4-15/4/d/a^2*b^2/(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)*ta
nh(1/2*d*x+1/2*c)^4-5/4/d/a^3*b^3/(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)^4-33/4/d/a^4*b^4/(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)^4-27/4/d/a^
2*b/(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*tanh(1/2*d*x+1/2*c)^2-5/2/d/a^3*b^2/(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*tanh(1/2*d*x+1/2*c)^2+33/4/d/a^4*b^3/(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*tanh(1/2*d*x+1/2*c)^2-9/4/d/a^2*b/(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)^2-5/d/a^3
*b^2/(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)
^2-11/4/d/a^4*b^3/(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+15/8/d/a^3*b/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))+35/8/
d/a^4*b^2/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________