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

Optimal. Leaf size=114 \[ -\frac{b (a+b) \cosh (c+d x)}{2 a^3 d \left (a \cosh ^2(c+d x)+b\right )}-\frac{(a+2 b) \cosh (c+d x)}{a^3 d}+\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{2 a^{7/2} d}+\frac{\cosh ^3(c+d x)}{3 a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.144245, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4133, 455, 1153, 205} \[ -\frac{b (a+b) \cosh (c+d x)}{2 a^3 d \left (a \cosh ^2(c+d x)+b\right )}-\frac{(a+2 b) \cosh (c+d x)}{a^3 d}+\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{2 a^{7/2} d}+\frac{\cosh ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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
]

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

Mathematica [C]  time = 4.67833, size = 861, normalized size = 7.55 \[ \frac{(\cosh (2 (c+d x)) a+a+2 b)^2 \text{sech}^4(c+d x) \left (\frac{9 \tan ^{-1}\left (\frac{\left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac{d x}{2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right ) a^3}{b^{3/2}}+\frac{9 \tan ^{-1}\left (\frac{\left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac{d x}{2}\right )+\cosh (c) \left (i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )+\sqrt{a}\right )}{\sqrt{b}}\right ) a^3}{b^{3/2}}-\frac{9 \tan ^{-1}\left (\frac{\sqrt{a}-i \sqrt{a+b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b}}\right ) a^3}{b^{3/2}}-\frac{9 \tan ^{-1}\left (\frac{i \sqrt{a+b} \tanh \left (\frac{1}{2} (c+d x)\right )+\sqrt{a}}{\sqrt{b}}\right ) a^3}{b^{3/2}}+32 \cosh (3 c) \cosh (3 d x) a^{3/2}-288 \sinh (c) \sinh (d x) a^{3/2}+32 \sinh (3 c) \sinh (3 d x) a^{3/2}-\frac{384 b \cosh (c+d x) a^{3/2}}{\cosh (2 (c+d x)) a+a+2 b}+576 \sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac{d x}{2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right ) a+576 \sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac{d x}{2}\right )+\cosh (c) \left (i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )+\sqrt{a}\right )}{\sqrt{b}}\right ) a-96 (3 a+8 b) \cosh (c) \cosh (d x) \sqrt{a}-768 b \sinh (c) \sinh (d x) \sqrt{a}-\frac{384 b^2 \cosh (c+d x) \sqrt{a}}{\cosh (2 (c+d x)) a+a+2 b}+960 b^{3/2} \tan ^{-1}\left (\frac{\left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac{d x}{2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )+960 b^{3/2} \tan ^{-1}\left (\frac{\left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac{d x}{2}\right )+\cosh (c) \left (i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )+\sqrt{a}\right )}{\sqrt{b}}\right )\right )}{1536 a^{7/2} d \left (b \text{sech}^2(c+d x)+a\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*((9*a^3*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Si
nh[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^(3/2) + 576*a*Sqrt[b]*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]] + 960*b^(3/2
)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*S
qrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (9*a^3*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]])/b^(3/2) + 576*a*Sqrt[b]*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]
] + 960*b^(3/2)*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]] - (9*a^3*ArcTan[(Sqrt[a] - I*Sqr
t[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]])/b^(3/2) - (9*a^3*ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt
[b]])/b^(3/2) - 96*Sqrt[a]*(3*a + 8*b)*Cosh[c]*Cosh[d*x] + 32*a^(3/2)*Cosh[3*c]*Cosh[3*d*x] - (384*a^(3/2)*b*C
osh[c + d*x])/(a + 2*b + a*Cosh[2*(c + d*x)]) - (384*Sqrt[a]*b^2*Cosh[c + d*x])/(a + 2*b + a*Cosh[2*(c + d*x)]
) - 288*a^(3/2)*Sinh[c]*Sinh[d*x] - 768*Sqrt[a]*b*Sinh[c]*Sinh[d*x] + 32*a^(3/2)*Sinh[3*c]*Sinh[3*d*x]))/(1536
*a^(7/2)*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B]  time = 0.082, size = 561, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/d/a^2/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/d/a^2/(tanh(1/2*d*x+1/2*c)+1)^2-1/2/d/a^2/(tanh(1/2*d*x+1/2*c)+1)-2/d/
a^3/(tanh(1/2*d*x+1/2*c)+1)*b-1/d*b/a^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)*tanh(1/2*d*x+1/2*c)^2+1/d*b^2/a^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)*tanh(1/2*d*x+1/2*c)^2-1/d*b/a^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)-1/d*b^2/a^3/(tan
h(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)+3/2/d*b/
a^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))+5/2/d*b^2/a^3/(a*b)^(1/2)*arct
an(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))-1/3/d/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/d/a^2/(tan
h(1/2*d*x+1/2*c)-1)^2+1/2/d/a^2/(tanh(1/2*d*x+1/2*c)-1)+2/d/a^3/(tanh(1/2*d*x+1/2*c)-1)*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} e^{\left (10 \, d x + 10 \, c\right )} + a^{2} -{\left (7 \, a^{2} e^{\left (8 \, c\right )} + 20 \, a b e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 2 \,{\left (13 \, a^{2} e^{\left (6 \, c\right )} + 66 \, a b e^{\left (6 \, c\right )} + 60 \, b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \,{\left (13 \, a^{2} e^{\left (4 \, c\right )} + 66 \, a b e^{\left (4 \, c\right )} + 60 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} -{\left (7 \, a^{2} e^{\left (2 \, c\right )} + 20 \, a b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{24 \,{\left (a^{4} d e^{\left (7 \, d x + 7 \, c\right )} + a^{4} d e^{\left (3 \, d x + 3 \, c\right )} + 2 \,{\left (a^{4} d e^{\left (5 \, c\right )} + 2 \, a^{3} b d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )}\right )}} + \frac{1}{8} \, \int \frac{8 \,{\left ({\left (3 \, a b e^{\left (3 \, c\right )} + 5 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (3 \, a b e^{c} + 5 \, b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{4} e^{\left (4 \, d x + 4 \, c\right )} + a^{4} + 2 \,{\left (a^{4} e^{\left (2 \, c\right )} + 2 \, a^{3} b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a^2*e^(10*d*x + 10*c) + a^2 - (7*a^2*e^(8*c) + 20*a*b*e^(8*c))*e^(8*d*x) - 2*(13*a^2*e^(6*c) + 66*a*b*e^
(6*c) + 60*b^2*e^(6*c))*e^(6*d*x) - 2*(13*a^2*e^(4*c) + 66*a*b*e^(4*c) + 60*b^2*e^(4*c))*e^(4*d*x) - (7*a^2*e^
(2*c) + 20*a*b*e^(2*c))*e^(2*d*x))/(a^4*d*e^(7*d*x + 7*c) + a^4*d*e^(3*d*x + 3*c) + 2*(a^4*d*e^(5*c) + 2*a^3*b
*d*e^(5*c))*e^(5*d*x)) + 1/8*integrate(8*((3*a*b*e^(3*c) + 5*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c + 5*b^2*e^c)*
e^(d*x))/(a^4*e^(4*d*x + 4*c) + a^4 + 2*(a^4*e^(2*c) + 2*a^3*b*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 3.98992, size = 9646, normalized size = 84.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(a^2*cosh(d*x + c)^10 + 10*a^2*cosh(d*x + c)*sinh(d*x + c)^9 + a^2*sinh(d*x + c)^10 - (7*a^2 + 20*a*b)*c
osh(d*x + c)^8 + (45*a^2*cosh(d*x + c)^2 - 7*a^2 - 20*a*b)*sinh(d*x + c)^8 + 8*(15*a^2*cosh(d*x + c)^3 - (7*a^
2 + 20*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^6 + 2*(105*a^2*cosh(d*
x + c)^4 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^6 + 4*(63*a^2*cosh(d*
x + c)^5 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)^3 - 3*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 -
2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^4 + 2*(105*a^2*cosh(d*x + c)^6 - 35*(7*a^2 + 20*a*b)*cosh(d*x + c)
^4 - 15*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^4 + 8*(15*a^2*cos
h(d*x + c)^7 - 7*(7*a^2 + 20*a*b)*cosh(d*x + c)^5 - 5*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (13*a^2 + 6
6*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (7*a^2 + 20*a*b)*cosh(d*x + c)^2 + (45*a^2*cosh(d*x + c)^8 -
28*(7*a^2 + 20*a*b)*cosh(d*x + c)^6 - 30*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^4 - 12*(13*a^2 + 66*a*b + 60
*b^2)*cosh(d*x + c)^2 - 7*a^2 - 20*a*b)*sinh(d*x + c)^2 + 6*((3*a^2 + 5*a*b)*cosh(d*x + c)^7 + 7*(3*a^2 + 5*a*
b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 + 5*a*b)*sinh(d*x + c)^7 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)
^5 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^2 + 6*a^2 + 22*a*b + 20*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + 5*a*b)*cos
h(d*x + c)^3 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^2 + 5*a*b)*cosh(d*x + c)^3 +
(35*(3*a^2 + 5*a*b)*cosh(d*x + c)^4 + 20*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^2 + 3*a^2 + 5*a*b)*sinh(d*x +
c)^3 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^5 + 20*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 5*a*b)
*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2 + 5*a*b)*cosh(d*x + c)^6 + 10*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x +
c)^4 + 3*(3*a^2 + 5*a*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))*sin
h(d*x + c) + a)) + a^2 + 2*(5*a^2*cosh(d*x + c)^9 - 4*(7*a^2 + 20*a*b)*cosh(d*x + c)^7 - 6*(13*a^2 + 66*a*b +
60*b^2)*cosh(d*x + c)^5 - 4*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (7*a^2 + 20*a*b)*cosh(d*x + c))*sinh(
d*x + c))/(a^4*d*cosh(d*x + c)^7 + 7*a^4*d*cosh(d*x + c)*sinh(d*x + c)^6 + a^4*d*sinh(d*x + c)^7 + a^4*d*cosh(
d*x + c)^3 + 2*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^5 + (21*a^4*d*cosh(d*x + c)^2 + 2*(a^4 + 2*a^3*b)*d)*sinh(d*x +
c)^5 + 5*(7*a^4*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (35*a^4*d*cosh(d*x +
c)^4 + a^4*d + 20*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (21*a^4*d*cosh(d*x + c)^5 + 3*a^4*d*co
sh(d*x + c) + 20*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (7*a^4*d*cosh(d*x + c)^6 + 3*a^4*d*cosh(
d*x + c)^2 + 10*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^4)*sinh(d*x + c)), 1/24*(a^2*cosh(d*x + c)^10 + 10*a^2*cosh(d*
x + c)*sinh(d*x + c)^9 + a^2*sinh(d*x + c)^10 - (7*a^2 + 20*a*b)*cosh(d*x + c)^8 + (45*a^2*cosh(d*x + c)^2 - 7
*a^2 - 20*a*b)*sinh(d*x + c)^8 + 8*(15*a^2*cosh(d*x + c)^3 - (7*a^2 + 20*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 -
2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^6 + 2*(105*a^2*cosh(d*x + c)^4 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)
^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^6 + 4*(63*a^2*cosh(d*x + c)^5 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)
^3 - 3*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^
4 + 2*(105*a^2*cosh(d*x + c)^6 - 35*(7*a^2 + 20*a*b)*cosh(d*x + c)^4 - 15*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x
+ c)^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^4 + 8*(15*a^2*cosh(d*x + c)^7 - 7*(7*a^2 + 20*a*b)*cosh(d*x +
c)^5 - 5*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)
^3 - (7*a^2 + 20*a*b)*cosh(d*x + c)^2 + (45*a^2*cosh(d*x + c)^8 - 28*(7*a^2 + 20*a*b)*cosh(d*x + c)^6 - 30*(13
*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^4 - 12*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^2 - 7*a^2 - 20*a*b)*sinh
(d*x + c)^2 - 12*((3*a^2 + 5*a*b)*cosh(d*x + c)^7 + 7*(3*a^2 + 5*a*b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 +
5*a*b)*sinh(d*x + c)^7 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^2 +
6*a^2 + 22*a*b + 20*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + 5*a*b)*cosh(d*x + c)^3 + 2*(3*a^2 + 11*a*b + 10*b^2)*
cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^2 + 5*a*b)*cosh(d*x + c)^3 + (35*(3*a^2 + 5*a*b)*cosh(d*x + c)^4 + 20*(3
*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^2 + 3*a^2 + 5*a*b)*sinh(d*x + c)^3 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^5
+ 20*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 5*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2
+ 5*a*b)*cosh(d*x + c)^6 + 10*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^4 + 3*(3*a^2 + 5*a*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) + 12*((3*a^2 + 5*a*b)
*cosh(d*x + c)^7 + 7*(3*a^2 + 5*a*b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 + 5*a*b)*sinh(d*x + c)^7 + 2*(3*a^
2 + 11*a*b + 10*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^2 + 6*a^2 + 22*a*b + 20*b^2)*sinh(d*x
+ c)^5 + 5*(7*(3*a^2 + 5*a*b)*cosh(d*x + c)^3 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 +
(3*a^2 + 5*a*b)*cosh(d*x + c)^3 + (35*(3*a^2 + 5*a*b)*cosh(d*x + c)^4 + 20*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x
+ c)^2 + 3*a^2 + 5*a*b)*sinh(d*x + c)^3 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^5 + 20*(3*a^2 + 11*a*b + 10*b^2)*c
osh(d*x + c)^3 + 3*(3*a^2 + 5*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2 + 5*a*b)*cosh(d*x + c)^6 + 10*(3
*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^4 + 3*(3*a^2 + 5*a*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) + a^2 + 2*(5*a^2*cosh(d*x + c)^9 - 4*(7*a^2 + 20*a*b)*cosh
(d*x + c)^7 - 6*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^5 - 4*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (7
*a^2 + 20*a*b)*cosh(d*x + c))*sinh(d*x + c))/(a^4*d*cosh(d*x + c)^7 + 7*a^4*d*cosh(d*x + c)*sinh(d*x + c)^6 +
a^4*d*sinh(d*x + c)^7 + a^4*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^5 + (21*a^4*d*cosh(d*x + c)^
2 + 2*(a^4 + 2*a^3*b)*d)*sinh(d*x + c)^5 + 5*(7*a^4*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b)*d*cosh(d*x + c))*sin
h(d*x + c)^4 + (35*a^4*d*cosh(d*x + c)^4 + a^4*d + 20*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (21
*a^4*d*cosh(d*x + c)^5 + 3*a^4*d*cosh(d*x + c) + 20*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (7*a^
4*d*cosh(d*x + c)^6 + 3*a^4*d*cosh(d*x + c)^2 + 10*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^4)*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(sinh(d*x+c)**3/(a+b*sech(d*x+c)**2)**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(sinh(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError