\(\int \sinh ^2(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx\) [59]
Optimal result
Integrand size = 23, antiderivative size = 129 \[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=-\frac {1}{2} \left (a^2+7 b^2\right ) x-\frac {4 a b \log (\cosh (c+d x))}{d}+\frac {3 b^2 \tanh (c+d x)}{d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {2 b^2 \tanh ^3(c+d x)}{3 d}+\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {\cosh (c+d x) \sinh (c+d x) \left (a^2+b^2+2 a b \tanh (c+d x)\right )}{2 d}
\]
[Out]
-1/2*(a^2+7*b^2)*x-4*a*b*ln(cosh(d*x+c))/d+3*b^2*tanh(d*x+c)/d+a*b*tanh(d*x+c)^2/d+2/3*b^2*tanh(d*x+c)^3/d+1/5
*b^2*tanh(d*x+c)^5/d+1/2*cosh(d*x+c)*sinh(d*x+c)*(a^2+b^2+2*a*b*tanh(d*x+c))/d
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.23, number of
steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3744, 1818, 1816, 647, 31}
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {\left (a^2+7 b^2\right ) \tanh (c+d x)}{2 d}+\frac {\sinh ^2(c+d x) \left (\left (a^2+b^2\right ) \tanh (c+d x)+2 a b\right )}{2 d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {(a+b) (a+7 b) \log (1-\tanh (c+d x))}{4 d}-\frac {(a-7 b) (a-b) \log (\tanh (c+d x)+1)}{4 d}+\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {2 b^2 \tanh ^3(c+d x)}{3 d}
\]
[In]
Int[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^2,x]
[Out]
((a + b)*(a + 7*b)*Log[1 - Tanh[c + d*x]])/(4*d) - ((a - 7*b)*(a - b)*Log[1 + Tanh[c + d*x]])/(4*d) + ((a^2 +
7*b^2)*Tanh[c + d*x])/(2*d) + (a*b*Tanh[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/
(5*d) + (Sinh[c + d*x]^2*(2*a*b + (a^2 + b^2)*Tanh[c + d*x]))/(2*d)
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 647
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q)),
Int[1/(-q + c*x), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[(-a)*c]
Rule 1816
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 1818
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Rule 3744
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]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 \left (a+b x^3\right )^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\sinh ^2(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {x \left (-4 a b-\left (a^2+3 b^2\right ) x-4 a b x^2-2 b^2 x^3-2 b^2 x^5\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d} \\ & = \frac {\sinh ^2(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \left (a^2+7 b^2+4 a b x+4 b^2 x^2+2 b^2 x^4-\frac {a^2+7 b^2+8 a b x}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d} \\ & = \frac {\left (a^2+7 b^2\right ) \tanh (c+d x)}{2 d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {2 b^2 \tanh ^3(c+d x)}{3 d}+\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {\sinh ^2(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{2 d}-\frac {\text {Subst}\left (\int \frac {a^2+7 b^2+8 a b x}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d} \\ & = \frac {\left (a^2+7 b^2\right ) \tanh (c+d x)}{2 d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {2 b^2 \tanh ^3(c+d x)}{3 d}+\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {\sinh ^2(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{2 d}+\frac {((a-7 b) (a-b)) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\tanh (c+d x)\right )}{4 d}-\frac {((a+b) (a+7 b)) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\tanh (c+d x)\right )}{4 d} \\ & = \frac {(a+b) (a+7 b) \log (1-\tanh (c+d x))}{4 d}-\frac {(a-7 b) (a-b) \log (1+\tanh (c+d x))}{4 d}+\frac {\left (a^2+7 b^2\right ) \tanh (c+d x)}{2 d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {2 b^2 \tanh ^3(c+d x)}{3 d}+\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {\sinh ^2(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{2 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 2.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.06
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {-30 a^2 c-210 b^2 c-30 a^2 d x-210 b^2 d x+30 a b \cosh (2 (c+d x))-240 a b \log (\cosh (c+d x))+15 a^2 \sinh (2 (c+d x))+15 b^2 \sinh (2 (c+d x))+232 b^2 \tanh (c+d x)+12 b^2 \text {sech}^4(c+d x) \tanh (c+d x)-4 b \text {sech}^2(c+d x) (15 a+16 b \tanh (c+d x))}{60 d}
\]
[In]
Integrate[Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^2,x]
[Out]
(-30*a^2*c - 210*b^2*c - 30*a^2*d*x - 210*b^2*d*x + 30*a*b*Cosh[2*(c + d*x)] - 240*a*b*Log[Cosh[c + d*x]] + 15
*a^2*Sinh[2*(c + d*x)] + 15*b^2*Sinh[2*(c + d*x)] + 232*b^2*Tanh[c + d*x] + 12*b^2*Sech[c + d*x]^4*Tanh[c + d*
x] - 4*b*Sech[c + d*x]^2*(15*a + 16*b*Tanh[c + d*x]))/(60*d)
Maple [A] (verified)
Time = 2.02 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{4}}{2 \cosh \left (d x +c \right )^{2}}-2 \ln \left (\cosh \left (d x +c \right )\right )+\tanh \left (d x +c \right )^{2}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{7}}{2 \cosh \left (d x +c \right )^{5}}-\frac {7 d x}{2}-\frac {7 c}{2}+\frac {7 \tanh \left (d x +c \right )}{2}+\frac {7 \tanh \left (d x +c \right )^{3}}{6}+\frac {7 \tanh \left (d x +c \right )^{5}}{10}\right )}{d}\) |
\(130\) |
| default |
\(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{4}}{2 \cosh \left (d x +c \right )^{2}}-2 \ln \left (\cosh \left (d x +c \right )\right )+\tanh \left (d x +c \right )^{2}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{7}}{2 \cosh \left (d x +c \right )^{5}}-\frac {7 d x}{2}-\frac {7 c}{2}+\frac {7 \tanh \left (d x +c \right )}{2}+\frac {7 \tanh \left (d x +c \right )^{3}}{6}+\frac {7 \tanh \left (d x +c \right )^{5}}{10}\right )}{d}\) |
\(130\) |
| risch |
\(-\frac {a^{2} x}{2}+4 a b x -\frac {7 b^{2} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}+\frac {{\mathrm e}^{2 d x +2 c} a b}{4 d}+\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}+\frac {{\mathrm e}^{-2 d x -2 c} a b}{4 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}+\frac {8 a b c}{d}-\frac {4 b \left (15 a \,{\mathrm e}^{8 d x +8 c}+45 b \,{\mathrm e}^{8 d x +8 c}+45 a \,{\mathrm e}^{6 d x +6 c}+120 b \,{\mathrm e}^{6 d x +6 c}+45 a \,{\mathrm e}^{4 d x +4 c}+170 b \,{\mathrm e}^{4 d x +4 c}+15 \,{\mathrm e}^{2 d x +2 c} a +100 b \,{\mathrm e}^{2 d x +2 c}+29 b \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}-\frac {4 a b \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) |
\(265\) |
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[In]
int(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
[Out]
1/d*(a^2*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+2*a*b*(1/2*sinh(d*x+c)^4/cosh(d*x+c)^2-2*ln(cosh(d*x+c))+
tanh(d*x+c)^2)+b^2*(1/2*sinh(d*x+c)^7/cosh(d*x+c)^5-7/2*d*x-7/2*c+7/2*tanh(d*x+c)+7/6*tanh(d*x+c)^3+7/10*tanh(
d*x+c)^5))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3649 vs. \(2 (121) = 242\).
Time = 0.28 (sec) , antiderivative size = 3649, normalized size of antiderivative = 28.29
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x, algorithm="fricas")
[Out]
1/120*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^14 + 210*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^13 + 15*(
a^2 + 2*a*b + b^2)*sinh(d*x + c)^14 - 15*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^
12 - 15*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 91*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - 5*a^2 - 10*a*b - 5*b^2)*sinh(d
*x + c)^12 + 60*(91*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - 3*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*
b^2)*cosh(d*x + c))*sinh(d*x + c)^11 - 15*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x +
c)^10 + 15*(1001*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 - 20*(a^2 - 8*a*b + 7*b^2)*d*x - 66*(4*(a^2 - 8*a*b + 7*b
^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^2 + 9*a^2 - 10*a*b - 87*b^2)*sinh(d*x + c)^10 + 30*(1001*(a^2
+ 2*a*b + b^2)*cosh(d*x + c)^5 - 110*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^3 -
5*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 15*(40*(a^2 - 8*a*
b + 7*b^2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^8 + 15*(3003*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 49
5*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^4 - 40*(a^2 - 8*a*b + 7*b^2)*d*x - 45*(
20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^2 + 5*a^2 - 66*a*b - 251*b^2)*sinh(d*x +
c)^8 + 120*(429*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 99*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^
2)*cosh(d*x + c)^5 - 15*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^3 - (40*(a^2 -
8*a*b + 7*b^2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 5*(120*(a^2 - 8*a*b + 7*b^2)*d
*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)^6 + 5*(9009*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 2772*(4*(a^2
- 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^6 - 630*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 +
10*a*b + 87*b^2)*cosh(d*x + c)^4 - 120*(a^2 - 8*a*b + 7*b^2)*d*x - 84*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 +
66*a*b + 251*b^2)*cosh(d*x + c)^2 - 15*a^2 - 198*a*b - 1103*b^2)*sinh(d*x + c)^6 + 30*(1001*(a^2 + 2*a*b + b^2
)*cosh(d*x + c)^9 - 396*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^7 - 126*(20*(a^2
- 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^5 - 28*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 + 6
6*a*b + 251*b^2)*cosh(d*x + c)^3 - (120*(a^2 - 8*a*b + 7*b^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^5 - 5*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b + 667*b^2)*cosh(d*x + c)^4 + 5*(3003*(a^
2 + 2*a*b + b^2)*cosh(d*x + c)^10 - 1485*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^
8 - 630*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^6 - 210*(40*(a^2 - 8*a*b + 7*b^
2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^4 - 60*(a^2 - 8*a*b + 7*b^2)*d*x - 15*(120*(a^2 - 8*a*b + 7*b
^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)^2 - 27*a^2 - 30*a*b - 667*b^2)*sinh(d*x + c)^4 + 20*(273*
(a^2 + 2*a*b + b^2)*cosh(d*x + c)^11 - 165*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c
)^9 - 90*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^7 - 42*(40*(a^2 - 8*a*b + 7*b^
2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^5 - 5*(120*(a^2 - 8*a*b + 7*b^2)*d*x + 15*a^2 + 198*a*b + 110
3*b^2)*cosh(d*x + c)^3 - (60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b + 667*b^2)*cosh(d*x + c))*sinh(d*x +
c)^3 - (60*(a^2 - 8*a*b + 7*b^2)*d*x + 75*a^2 - 150*a*b + 1003*b^2)*cosh(d*x + c)^2 + (1365*(a^2 + 2*a*b + b^2
)*cosh(d*x + c)^12 - 990*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^10 - 675*(20*(a^
2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^8 - 420*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2
+ 66*a*b + 251*b^2)*cosh(d*x + c)^6 - 75*(120*(a^2 - 8*a*b + 7*b^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*
x + c)^4 - 60*(a^2 - 8*a*b + 7*b^2)*d*x - 30*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b + 667*b^2)*cosh(d
*x + c)^2 - 75*a^2 + 150*a*b - 1003*b^2)*sinh(d*x + c)^2 - 15*a^2 + 30*a*b - 15*b^2 - 480*(a*b*cosh(d*x + c)^1
2 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^11 + a*b*sinh(d*x + c)^12 + 5*a*b*cosh(d*x + c)^10 + (66*a*b*cosh(d*x +
c)^2 + 5*a*b)*sinh(d*x + c)^10 + 10*a*b*cosh(d*x + c)^8 + 10*(22*a*b*cosh(d*x + c)^3 + 5*a*b*cosh(d*x + c))*s
inh(d*x + c)^9 + 5*(99*a*b*cosh(d*x + c)^4 + 45*a*b*cosh(d*x + c)^2 + 2*a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x
+ c)^6 + 8*(99*a*b*cosh(d*x + c)^5 + 75*a*b*cosh(d*x + c)^3 + 10*a*b*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(462*
a*b*cosh(d*x + c)^6 + 525*a*b*cosh(d*x + c)^4 + 140*a*b*cosh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^6 + 5*a*b*cosh(
d*x + c)^4 + 4*(198*a*b*cosh(d*x + c)^7 + 315*a*b*cosh(d*x + c)^5 + 140*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x
+ c))*sinh(d*x + c)^5 + 5*(99*a*b*cosh(d*x + c)^8 + 210*a*b*cosh(d*x + c)^6 + 140*a*b*cosh(d*x + c)^4 + 30*a*b
*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 + a*b*cosh(d*x + c)^2 + 20*(11*a*b*cosh(d*x + c)^9 + 30*a*b*cosh(d*x +
c)^7 + 28*a*b*cosh(d*x + c)^5 + 10*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^3 + (66*a*b*cosh(d*
x + c)^10 + 225*a*b*cosh(d*x + c)^8 + 280*a*b*cosh(d*x + c)^6 + 150*a*b*cosh(d*x + c)^4 + 30*a*b*cosh(d*x + c)
^2 + a*b)*sinh(d*x + c)^2 + 2*(6*a*b*cosh(d*x + c)^11 + 25*a*b*cosh(d*x + c)^9 + 40*a*b*cosh(d*x + c)^7 + 30*a
*b*cosh(d*x + c)^5 + 10*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x
+ c) - sinh(d*x + c))) + 2*(105*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^13 - 90*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2
- 10*a*b - 5*b^2)*cosh(d*x + c)^11 - 75*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c
)^9 - 60*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^7 - 15*(120*(a^2 - 8*a*b + 7*
b^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)^5 - 10*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b +
667*b^2)*cosh(d*x + c)^3 - (60*(a^2 - 8*a*b + 7*b^2)*d*x + 75*a^2 - 150*a*b + 1003*b^2)*cosh(d*x + c))*sinh(d
*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x + c)^12 + 5*d*cosh(d*x + c)^10
+ (66*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^10 + 10*(22*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)
^9 + 10*d*cosh(d*x + c)^8 + 5*(99*d*cosh(d*x + c)^4 + 45*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8*(99*d*co
sh(d*x + c)^5 + 75*d*cosh(d*x + c)^3 + 10*d*cosh(d*x + c))*sinh(d*x + c)^7 + 10*d*cosh(d*x + c)^6 + 2*(462*d*c
osh(d*x + c)^6 + 525*d*cosh(d*x + c)^4 + 140*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 4*(198*d*cosh(d*x + c)
^7 + 315*d*cosh(d*x + c)^5 + 140*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^5 + 5*d*cosh(d*x + c)^4
+ 5*(99*d*cosh(d*x + c)^8 + 210*d*cosh(d*x + c)^6 + 140*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + d)*sinh(d*
x + c)^4 + 20*(11*d*cosh(d*x + c)^9 + 30*d*cosh(d*x + c)^7 + 28*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + d*c
osh(d*x + c))*sinh(d*x + c)^3 + d*cosh(d*x + c)^2 + (66*d*cosh(d*x + c)^10 + 225*d*cosh(d*x + c)^8 + 280*d*cos
h(d*x + c)^6 + 150*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 2*(6*d*cosh(d*x + c)^11 + 2
5*d*cosh(d*x + c)^9 + 40*d*cosh(d*x + c)^7 + 30*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*si
nh(d*x + c))
Sympy [F]
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \sinh ^{2}{\left (c + d x \right )}\, dx
\]
[In]
integrate(sinh(d*x+c)**2*(a+b*tanh(d*x+c)**3)**2,x)
[Out]
Integral((a + b*tanh(c + d*x)**3)**2*sinh(c + d*x)**2, x)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (121) = 242\).
Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.33
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=-\frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{120} \, b^{2} {\left (\frac {420 \, {\left (d x + c\right )}}{d} + \frac {15 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {1003 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3350 \, e^{\left (-4 \, d x - 4 \, c\right )} + 5590 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3915 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1455 \, e^{\left (-10 \, d x - 10 \, c\right )} + 15}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 10 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )}\right )}}\right )} - \frac {1}{4} \, a b {\left (\frac {16 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} + \frac {16 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}\right )}
\]
[In]
integrate(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x, algorithm="maxima")
[Out]
-1/8*a^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/120*b^2*(420*(d*x + c)/d + 15*e^(-2*d*x - 2*c)/d -
(1003*e^(-2*d*x - 2*c) + 3350*e^(-4*d*x - 4*c) + 5590*e^(-6*d*x - 6*c) + 3915*e^(-8*d*x - 8*c) + 1455*e^(-10*
d*x - 10*c) + 15)/(d*(e^(-2*d*x - 2*c) + 5*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 10*e^(-8*d*x - 8*c) + 5*e^
(-10*d*x - 10*c) + e^(-12*d*x - 12*c)))) - 1/4*a*b*(16*(d*x + c)/d - e^(-2*d*x - 2*c)/d + 16*log(e^(-2*d*x - 2
*c) + 1)/d - (2*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 1)/(d*(e^(-2*d*x - 2*c) + 2*e^(-4*d*x - 4*c) + e^(-6*
d*x - 6*c))))
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (121) = 242\).
Time = 0.42 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.29
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {15 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 30 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 15 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 480 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 60 \, {\left (a^{2} - 8 \, a b + 7 \, b^{2}\right )} {\left (d x + c\right )} + 15 \, {\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 14 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} + 2 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + \frac {8 \, {\left (137 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 625 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 180 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1190 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 480 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1190 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 680 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 625 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 400 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 137 \, a b - 116 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d}
\]
[In]
integrate(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x, algorithm="giac")
[Out]
1/120*(15*a^2*e^(2*d*x + 2*c) + 30*a*b*e^(2*d*x + 2*c) + 15*b^2*e^(2*d*x + 2*c) - 480*a*b*log(e^(2*d*x + 2*c)
+ 1) - 60*(a^2 - 8*a*b + 7*b^2)*(d*x + c) + 15*(2*a^2*e^(2*d*x + 2*c) - 16*a*b*e^(2*d*x + 2*c) + 14*b^2*e^(2*d
*x + 2*c) - a^2 + 2*a*b - b^2)*e^(-2*d*x - 2*c) + 8*(137*a*b*e^(10*d*x + 10*c) + 625*a*b*e^(8*d*x + 8*c) - 180
*b^2*e^(8*d*x + 8*c) + 1190*a*b*e^(6*d*x + 6*c) - 480*b^2*e^(6*d*x + 6*c) + 1190*a*b*e^(4*d*x + 4*c) - 680*b^2
*e^(4*d*x + 4*c) + 625*a*b*e^(2*d*x + 2*c) - 400*b^2*e^(2*d*x + 2*c) + 137*a*b - 116*b^2)/(e^(2*d*x + 2*c) + 1
)^5)/d
Mupad [B] (verification not implemented)
Time = 2.05 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.37
\[
\int \sinh ^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-\frac {4\,\left (3\,b^2+a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-x\,\left (\frac {a^2}{2}-4\,a\,b+\frac {7\,b^2}{2}\right )+\frac {4\,\left (4\,b^2+a\,b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {64\,b^2}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a-b\right )}^2}{8\,d}+\frac {16\,b^2}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32\,b^2}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {4\,a\,b\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}
\]
[In]
int(sinh(c + d*x)^2*(a + b*tanh(c + d*x)^3)^2,x)
[Out]
(exp(2*c + 2*d*x)*(a + b)^2)/(8*d) - (4*(a*b + 3*b^2))/(d*(exp(2*c + 2*d*x) + 1)) - x*(a^2/2 - 4*a*b + (7*b^2)
/2) + (4*(a*b + 4*b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (64*b^2)/(3*d*(3*exp(2*c + 2*d*x) +
3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (exp(- 2*c - 2*d*x)*(a - b)^2)/(8*d) + (16*b^2)/(d*(4*exp(2*c +
2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (32*b^2)/(5*d*(5*exp(2*c + 2*d*x)
+ 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (4*a*b*log(exp(2
*c)*exp(2*d*x) + 1))/d