### 3.75 $$\int \frac{\sinh ^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$$

Optimal. Leaf size=384 $-\frac{a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 d \left (a^2-b^2\right )^2}+\frac{b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac{a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac{a^{2/3} \sqrt [3]{b} \left (-3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\tanh (c+d x))}-\frac{1}{4 d (a-b) (\tanh (c+d x)+1)}+\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 d (a+b)^2}-\frac{(a+2 b) \log (\tanh (c+d x)+1)}{4 d (a-b)^2}$

[Out]

(a^(2/3)*b^(1/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3)
)])/(Sqrt[3]*(a^2 - b^2)^2*d) + ((a - 2*b)*Log[1 - Tanh[c + d*x]])/(4*(a + b)^2*d) - ((a + 2*b)*Log[1 + Tanh[c
+ d*x]])/(4*(a - b)^2*d) + (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Tanh[c +
d*x]])/(3*(a^2 - b^2)^2*d) - (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*(a^2 - b^2)^2*d) + (b*(2*a^2 + b^2)*Log[a + b*Tanh[c + d*x]^3])/(
3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Tanh[c + d*x])) - 1/(4*(a - b)*d*(1 + Tanh[c + d*x]))

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Rubi [A]  time = 0.629447, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.435, Rules used = {3663, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} $-\frac{a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 d \left (a^2-b^2\right )^2}+\frac{b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac{a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac{a^{2/3} \sqrt [3]{b} \left (-3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\tanh (c+d x))}-\frac{1}{4 d (a-b) (\tanh (c+d x)+1)}+\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 d (a+b)^2}-\frac{(a+2 b) \log (\tanh (c+d x)+1)}{4 d (a-b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a^(2/3)*b^(1/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3)
)])/(Sqrt[3]*(a^2 - b^2)^2*d) + ((a - 2*b)*Log[1 - Tanh[c + d*x]])/(4*(a + b)^2*d) - ((a + 2*b)*Log[1 + Tanh[c
+ d*x]])/(4*(a - b)^2*d) + (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Tanh[c +
d*x]])/(3*(a^2 - b^2)^2*d) - (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*(a^2 - b^2)^2*d) + (b*(2*a^2 + b^2)*Log[a + b*Tanh[c + d*x]^3])/(
3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Tanh[c + d*x])) - 1/(4*(a - b)*d*(1 + Tanh[c + d*x]))

Rule 3663

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\sinh ^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (1-x^2\right )^2 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4 (a+b) (-1+x)^2}+\frac{a-2 b}{4 (a+b)^2 (-1+x)}+\frac{1}{4 (a-b) (1+x)^2}+\frac{-a-2 b}{4 (a-b)^2 (1+x)}+\frac{b \left (3 a^2 b-a \left (a^2+2 b^2\right ) x+b \left (2 a^2+b^2\right ) x^2\right )}{\left (a^2-b^2\right )^2 \left (a+b x^3\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac{(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac{1}{4 (a+b) d (1-\tanh (c+d x))}-\frac{1}{4 (a-b) d (1+\tanh (c+d x))}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^2 b-a \left (a^2+2 b^2\right ) x+b \left (2 a^2+b^2\right ) x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac{(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac{1}{4 (a+b) d (1-\tanh (c+d x))}-\frac{1}{4 (a-b) d (1+\tanh (c+d x))}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^2 b-a \left (a^2+2 b^2\right ) x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac{\left (b^2 \left (2 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac{(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac{b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac{1}{4 (a+b) d (1-\tanh (c+d x))}-\frac{1}{4 (a-b) d (1+\tanh (c+d x))}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (6 a^2 b^{4/3}-a^{4/3} \left (a^2+2 b^2\right )\right )+\sqrt [3]{b} \left (-3 a^2 b^{4/3}-a^{4/3} \left (a^2+2 b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac{\left (a^{2/3} b^{2/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}\\ &=\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac{(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac{a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac{b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac{1}{4 (a+b) d (1-\tanh (c+d x))}-\frac{1}{4 (a-b) d (1+\tanh (c+d x))}-\frac{\left (a b^{2/3} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d}-\frac{\left (a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}\\ &=\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac{(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac{a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac{a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}+\frac{b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac{1}{4 (a+b) d (1-\tanh (c+d x))}-\frac{1}{4 (a-b) d (1+\tanh (c+d x))}-\frac{\left (a^{2/3} \sqrt [3]{b} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac{a^{2/3} \sqrt [3]{b} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \left (a^2-b^2\right )^2 d}+\frac{(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac{(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac{a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac{a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}+\frac{b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac{1}{4 (a+b) d (1-\tanh (c+d x))}-\frac{1}{4 (a-b) d (1+\tanh (c+d x))}\\ \end{align*}

Mathematica [C]  time = 3.39126, size = 423, normalized size = 1.1 $-\frac{4 b \text{RootSum}\left [\text{\#1}^3 a+3 \text{\#1}^2 a+\text{\#1}^3 b-3 \text{\#1}^2 b+3 \text{\#1} a+3 \text{\#1} b+a-b\& ,\frac{-4 \text{\#1}^2 a^2 \log \left (e^{2 (c+d x)}-\text{\#1}\right )+8 \text{\#1}^2 a^2 c+8 \text{\#1}^2 a^2 d x+4 \text{\#1}^2 a b \log \left (e^{2 (c+d x)}-\text{\#1}\right )-8 \text{\#1}^2 a b c-8 \text{\#1}^2 a b d x-\text{\#1}^2 b^2 \log \left (e^{2 (c+d x)}-\text{\#1}\right )+2 \text{\#1}^2 b^2 c+2 \text{\#1}^2 b^2 d x-2 a^2 \log \left (e^{2 (c+d x)}-\text{\#1}\right )-2 \text{\#1} a^2 \log \left (e^{2 (c+d x)}-\text{\#1}\right )+4 \text{\#1} a^2 c+4 \text{\#1} a^2 d x-b^2 \log \left (e^{2 (c+d x)}-\text{\#1}\right )+2 \text{\#1} b^2 \log \left (e^{2 (c+d x)}-\text{\#1}\right )-4 \text{\#1} b^2 c-4 \text{\#1} b^2 d x+4 a^2 c+4 a^2 d x+2 b^2 c+2 b^2 d x}{\text{\#1}^2 a-\text{\#1}^2 b+2 \text{\#1} a+2 \text{\#1} b+a-b}\& \right ]+6 \left (a^2-3 a b+2 b^2\right ) (c+d x)-3 a (a+b) \sinh (2 (c+d x))+3 b (a+b) \cosh (2 (c+d x))}{12 d (a-b) (a+b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(6*(a^2 - 3*a*b + 2*b^2)*(c + d*x) + 3*b*(a + b)*Cosh[2*(c + d*x)] + 4*b*RootSum[a - b + 3*a*#1 + 3*b*#1 + 3*
a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (4*a^2*c + 2*b^2*c + 4*a^2*d*x + 2*b^2*d*x - 2*a^2*Log[E^(2*(c + d*x))
- #1] - b^2*Log[E^(2*(c + d*x)) - #1] + 4*a^2*c*#1 - 4*b^2*c*#1 + 4*a^2*d*x*#1 - 4*b^2*d*x*#1 - 2*a^2*Log[E^(
2*(c + d*x)) - #1]*#1 + 2*b^2*Log[E^(2*(c + d*x)) - #1]*#1 + 8*a^2*c*#1^2 - 8*a*b*c*#1^2 + 2*b^2*c*#1^2 + 8*a^
2*d*x*#1^2 - 8*a*b*d*x*#1^2 + 2*b^2*d*x*#1^2 - 4*a^2*Log[E^(2*(c + d*x)) - #1]*#1^2 + 4*a*b*Log[E^(2*(c + d*x)
) - #1]*#1^2 - b^2*Log[E^(2*(c + d*x)) - #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ] - 3*a*(a +
b)*Sinh[2*(c + d*x)])/(12*(a - b)*(a + b)^2*d)

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Maple [C]  time = 0.109, size = 356, normalized size = 0.9 \begin{align*} -4\,{\frac{1}{d \left ( 8\,a-8\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+8\,{\frac{1}{d \left ( 16\,a-16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}-{\frac{a}{2\,d \left ( a-b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{b}{d \left ( a-b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+4\,{\frac{1}{d \left ( 8\,a+8\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+8\,{\frac{1}{d \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+{\frac{a}{2\,d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{b}{d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{b}{3\,d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{a \left ( 2\,{a}^{2}+{b}^{2} \right ){{\it \_R}}^{5}-3\,{{\it \_R}}^{4}{a}^{2}b+6\,a \left ({a}^{2}+{b}^{2} \right ){{\it \_R}}^{3}+4\,b \left ( 2\,{a}^{2}+{b}^{2} \right ){{\it \_R}}^{2}-3\,a{b}^{2}{\it \_R}+3\,{a}^{2}b}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/d/(8*a-8*b)/(tanh(1/2*d*x+1/2*c)+1)^2+8/d/(16*a-16*b)/(tanh(1/2*d*x+1/2*c)+1)-1/2/d/(a-b)^2*ln(tanh(1/2*d*x
+1/2*c)+1)*a-1/d/(a-b)^2*ln(tanh(1/2*d*x+1/2*c)+1)*b+4/d/(8*a+8*b)/(tanh(1/2*d*x+1/2*c)-1)^2+8/d/(16*a+16*b)/(
tanh(1/2*d*x+1/2*c)-1)+1/2/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)-1)*a-1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)-1)*b+1/3/d
*b/(a-b)^2/(a+b)^2*sum((a*(2*a^2+b^2)*_R^5-3*_R^4*a^2*b+6*a*(a^2+b^2)*_R^3+4*b*(2*a^2+b^2)*_R^2-3*a*b^2*_R+3*a
^2*b)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a
))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a^2*b*(integrate(((a + b)*e^(4*d*x + 4*c) + 3*(a - b)*e^(2*d*x + 2*c) + 3*a + 3*b)*e^(2*d*x + 2*c)/((a + b)*
e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^4 - 2*a^2*b^2 + b^4) -
(d*x + c)/((a^4 - 2*a^2*b^2 + b^4)*d)) + 2*b^3*(integrate(((a + b)*e^(4*d*x + 4*c) + 3*(a - b)*e^(2*d*x + 2*c
) + 3*a + 3*b)*e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c
) + a - b), x)/(a^4 - 2*a^2*b^2 + b^4) - (d*x + c)/((a^4 - 2*a^2*b^2 + b^4)*d)) - 8*a^2*b*integrate(e^(4*d*x +
4*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2
*b - a*b^2 - b^3) + 8*a*b^2*integrate(e^(4*d*x + 4*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3
*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3) - 2*b^3*integrate(e^(4*d*x + 4*c)/((a + b)*e
^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3
) - 4*a^2*b*integrate(e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*
x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3) + 4*b^3*integrate(e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) +
3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3) - 1/8*(4*(a^2*
d*e^(2*c) - 3*a*b*d*e^(2*c) + 2*b^2*d*e^(2*c))*x*e^(2*d*x) + a^2 + 2*a*b + b^2 - (a^2*e^(4*c) - b^2*e^(4*c))*e
^(4*d*x))*e^(-2*d*x)/(a^3*d*e^(2*c) + a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c) - b^3*d*e^(2*c))

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Fricas [C]  time = 16.0621, size = 22579, normalized size = 58.8 \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)^2/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/72*(36*(a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3)*d*x*cosh(d*x + c)^2 - 9*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^
4 - 36*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^3 - 9*(a^3 - a^2*b - a*b^2 + b^3)*sinh(d*x + c)
^4 + 9*a^3 + 9*a^2*b - 9*a*b^2 - 9*b^3 - 4*((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b
^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^2)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2
+ b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((
a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*
d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3
) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2
*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b
^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*log(1/
18*(a^5 + 2*a^4*b - 2*a^3*b^2 - 4*a^2*b^3 + a*b^4 + 2*b^5)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*
b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*
d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*
b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*
b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4
*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^
2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + a^3 + 2*a^2*b + 2*
a*b^2 + 4*b^3 - 1/3*(a^4 + 3*a^3*b + 13*a^2*b^2 + 6*a*b^3 + 4*b^4)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) -
(2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*
a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3
+ 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18
*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3
)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((
a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + (a^3 + 8*a*b^2
)*cosh(d*x + c)^2 + 2*(a^3 + 8*a*b^2)*cosh(d*x + c)*sinh(d*x + c) + (a^3 + 8*a*b^2)*sinh(d*x + c)^2) + 18*(2*(
a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3)*d*x - 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*(18*
(2*a^2*b + b^3)*cosh(d*x + c)^2 + 36*(2*a^2*b + b^3)*cosh(d*x + c)*sinh(d*x + c) + 18*(2*a^2*b + b^3)*sinh(d*x
+ c)^2 - ((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)
+ (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^2)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a
^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2
)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 -
2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/
((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^
2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1
/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d)) - 3*sqrt(1/3)*((a^4 - 2*a^2*b^2 + b^4)*
d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d
*x + c)^2)*sqrt((288*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a
^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18
*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3
)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((
a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^
2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4
*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2
*b^2*d + b^4*d))^2*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b +
b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2
+ b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(
a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b +
b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d
- 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^
4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*
b^4 - 4*a^2*b^6 + b^8)*d^2)))*log(-1/36*(a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d
^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*
a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/
(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2
- b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d
+ b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3
) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2
*d + b^4*d))^2*d^2 + a^4 - 3*a^3*b + 10*a^2*b^2 - 15*a*b^3 - 2*b^4 + 1/6*(a^5 + 4*a^4*b + 16*a^3*b^2 + 19*a^2*
b^3 + 10*a*b^4 + 4*b^5)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b
^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d
+ b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^
3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*
d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*
b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) +
6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^2 + 2*(
a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)*sinh(d*x + c) + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*sinh(d*x
+ c)^2 + 1/12*sqrt(1/3)*((a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d^2 - 2*a^2*b^2*
d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^
2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*
b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^
(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/2
7*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 +
8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^
2 + 6*(a^5 - 2*a^4*b - 2*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*b^5)*d)*sqrt((288*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a
^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a
^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2
)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 -
2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/
((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^
2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1
/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*
((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1
)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2
*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*
a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d -
2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d
^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d
- 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))) - 2*(18*(2*a^2*b + b^3)*co
sh(d*x + c)^2 + 36*(2*a^2*b + b^3)*cosh(d*x + c)*sinh(d*x + c) + 18*(2*a^2*b + b^3)*sinh(d*x + c)^2 - ((a^4 -
2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^
2 + b^4)*d*sinh(d*x + c)^2)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d
+ b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b
^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^
4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*
b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1
/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1
) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d)) + 3*sqrt(1/3)*((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2
+ 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^2)*sqrt((2
88*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a^4*d^2 - 2*a^2*b^2
*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b
^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2
*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))
^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/
27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2
+ 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2
*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2
*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d
- 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^
2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^
2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^
4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*s
qrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 +
b^8)*d^2)))*log(-1/36*(a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2
+ b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/(
(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2
*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/
3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(
2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*
b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2
+ a^4 - 3*a^3*b + 10*a^2*b^2 - 15*a*b^3 - 2*b^4 + 1/6*(a^5 + 4*a^4*b + 16*a^3*b^2 + 19*a^2*b^3 + 10*a*b^4 + 4
*b^5)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(
3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*
(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8
*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^
4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2
*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/
(a^4*d - 2*a^2*b^2*d + b^4*d))*d + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^2 + 2*(a^4 + a^3*b + 8*a^
2*b^2 + 8*a*b^3)*cosh(d*x + c)*sinh(d*x + c) + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*sinh(d*x + c)^2 - 1/12*sqrt
(1/3)*((a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (
2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^
2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 +
1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(
2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^
3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^
2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^2 + 6*(a^5 - 2*a^4
*b - 2*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*b^5)*d)*sqrt((288*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a^8 - 4*a^6*b^2 + 6
*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d
+ b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b
^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^
4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*
b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1
/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1
) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*((b^2/(a^4*d^2 - 2
*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b
+ b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d
- 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)
^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*
d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/
54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b
^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))) + 36*(2*(a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3)*d*
x*cosh(d*x + c) - (a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^3)*sinh(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cosh(
d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)
^2)

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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)**2/(a+b*tanh(d*x+c)**3),x)

[Out]

Timed out

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Giac [A]  time = 1.78404, size = 301, normalized size = 0.78 \begin{align*} -\frac{\frac{12 \,{\left (a + 2 \, b\right )} d x}{a^{2} - 2 \, a b + b^{2}} - \frac{3 \,{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-2 \, d x\right )}}{a^{2} e^{\left (2 \, c\right )} - 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}} - \frac{8 \,{\left (2 \, a^{2} b + b^{3}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{3 \, e^{\left (2 \, d x + 10 \, c\right )}}{a e^{\left (8 \, c\right )} + b e^{\left (8 \, c\right )}}}{24 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*(a + 2*b)*d*x/(a^2 - 2*a*b + b^2) - 3*(2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) - a + b)*e^(-2*d*x)
/(a^2*e^(2*c) - 2*a*b*e^(2*c) + b^2*e^(2*c)) - 8*(2*a^2*b + b^3)*log(abs(a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c)
+ 3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) + a - b))/(a^4 - 2*a^
2*b^2 + b^4) - 3*e^(2*d*x + 10*c)/(a*e^(8*c) + b*e^(8*c)))/d