3.57 \(\int \sinh ^4(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx\)

Optimal. Leaf size=170 \[ \frac {\sinh (c+d x) \cosh ^3(c+d x) \left (a^2+2 a b \tanh (c+d x)+b^2\right )}{4 d}-\frac {\sinh (c+d x) \cosh (c+d x) \left (5 a^2+20 a b \tanh (c+d x)+17 b^2\right )}{8 d}+\frac {3}{8} x \left (a^2+21 b^2\right )-\frac {a b \tanh ^2(c+d x)}{d}+\frac {6 a b \log (\cosh (c+d x))}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^3(c+d x)}{d}-\frac {6 b^2 \tanh (c+d x)}{d} \]

[Out]

3/8*(a^2+21*b^2)*x+6*a*b*ln(cosh(d*x+c))/d-6*b^2*tanh(d*x+c)/d-a*b*tanh(d*x+c)^2/d-b^2*tanh(d*x+c)^3/d-1/5*b^2
*tanh(d*x+c)^5/d+1/4*cosh(d*x+c)^3*sinh(d*x+c)*(a^2+b^2+2*a*b*tanh(d*x+c))/d-1/8*cosh(d*x+c)*sinh(d*x+c)*(5*a^
2+17*b^2+20*a*b*tanh(d*x+c))/d

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Rubi [A]  time = 0.30, antiderivative size = 206, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3663, 1804, 1802, 633, 31} \[ -\frac {3 \left (a^2+21 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 \left (a^2+16 a b+21 b^2\right ) \log (1-\tanh (c+d x))}{16 d}+\frac {3 \left (a^2-16 a b+21 b^2\right ) \log (\tanh (c+d x)+1)}{16 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a^2+16 a b \tanh (c+d x)+13 b^2\right )}{8 d}+\frac {\sinh ^4(c+d x) \left (\left (a^2+b^2\right ) \tanh (c+d x)+2 a b\right )}{4 d}-\frac {3 a b \tanh ^2(c+d x)}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^3(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(a^2 + 16*a*b + 21*b^2)*Log[1 - Tanh[c + d*x]])/(16*d) + (3*(a^2 - 16*a*b + 21*b^2)*Log[1 + Tanh[c + d*x]]
)/(16*d) - (3*(a^2 + 21*b^2)*Tanh[c + d*x])/(8*d) - (3*a*b*Tanh[c + d*x]^2)/d - (b^2*Tanh[c + d*x]^3)/d - (b^2
*Tanh[c + d*x]^5)/(5*d) - (Sinh[c + d*x]^2*Tanh[c + d*x]*(a^2 + 13*b^2 + 16*a*b*Tanh[c + d*x]))/(8*d) + (Sinh[
c + d*x]^4*(2*a*b + (a^2 + b^2)*Tanh[c + d*x]))/(4*d)

Rule 31

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

Rule 633

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 1802

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 1804

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 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]

Rubi steps

\begin {align*} \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b x^3\right )^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\sinh ^4(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (-8 a b-\left (a^2+5 b^2\right ) x-8 a b x^2-4 b^2 x^3-4 b^2 x^5\right )}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a^2+13 b^2+16 a b \tanh (c+d x)\right )}{8 d}+\frac {\sinh ^4(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 \left (a^2+13 b^2\right )+48 a b x+16 b^2 x^2+8 b^2 x^4\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a^2+13 b^2+16 a b \tanh (c+d x)\right )}{8 d}+\frac {\sinh ^4(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}+\frac {\operatorname {Subst}\left (\int \left (-3 \left (a^2+21 b^2\right )-48 a b x-24 b^2 x^2-8 b^2 x^4+\frac {3 \left (a^2+21 b^2+16 a b x\right )}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {3 \left (a^2+21 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 a b \tanh ^2(c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a^2+13 b^2+16 a b \tanh (c+d x)\right )}{8 d}+\frac {\sinh ^4(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}+\frac {3 \operatorname {Subst}\left (\int \frac {a^2+21 b^2+16 a b x}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {3 \left (a^2+21 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 a b \tanh ^2(c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a^2+13 b^2+16 a b \tanh (c+d x)\right )}{8 d}+\frac {\sinh ^4(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\left (3 \left (a^2-16 a b+21 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\tanh (c+d x)\right )}{16 d}+\frac {\left (3 \left (a^2+16 a b+21 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=-\frac {3 \left (a^2+16 a b+21 b^2\right ) \log (1-\tanh (c+d x))}{16 d}+\frac {3 \left (a^2-16 a b+21 b^2\right ) \log (1+\tanh (c+d x))}{16 d}-\frac {3 \left (a^2+21 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 a b \tanh ^2(c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a^2+13 b^2+16 a b \tanh (c+d x)\right )}{8 d}+\frac {\sinh ^4(c+d x) \left (2 a b+\left (a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}\\ \end {align*}

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Mathematica [A]  time = 2.83, size = 156, normalized size = 0.92 \[ \frac {60 \left (a^2+21 b^2\right ) (c+d x)-40 \left (a^2+4 b^2\right ) \sinh (2 (c+d x))+5 \left (a^2+b^2\right ) \sinh (4 (c+d x))-200 a b \cosh (2 (c+d x))+10 a b \cosh (4 (c+d x))+160 a b \text {sech}^2(c+d x)+960 a b \log (\cosh (c+d x))-1152 b^2 \tanh (c+d x)-32 b^2 \tanh (c+d x) \text {sech}^4(c+d x)+224 b^2 \tanh (c+d x) \text {sech}^2(c+d x)}{160 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*(a^2 + 21*b^2)*(c + d*x) - 200*a*b*Cosh[2*(c + d*x)] + 10*a*b*Cosh[4*(c + d*x)] + 960*a*b*Log[Cosh[c + d*x
]] + 160*a*b*Sech[c + d*x]^2 - 40*(a^2 + 4*b^2)*Sinh[2*(c + d*x)] + 5*(a^2 + b^2)*Sinh[4*(c + d*x)] - 1152*b^2
*Tanh[c + d*x] + 224*b^2*Sech[c + d*x]^2*Tanh[c + d*x] - 32*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(160*d)

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fricas [B]  time = 0.75, size = 5034, normalized size = 29.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^18 + 90*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^17 + 5*(a^2
+ 2*a*b + b^2)*sinh(d*x + c)^18 - 15*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^16 + 15*(51*(a^2 + 2*a*b + b^2)*cos
h(d*x + c)^2 - a^2 - 10*a*b - 9*b^2)*sinh(d*x + c)^16 + 240*(17*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2 + 1
0*a*b + 9*b^2)*cosh(d*x + c))*sinh(d*x + c)^15 + 30*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*
cosh(d*x + c)^14 + 30*(510*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 - 16*a*b + 21*b^2)*d*x - 60*(a^2 + 10*
a*b + 9*b^2)*cosh(d*x + c)^2 - 5*a^2 - 30*a*b - 25*b^2)*sinh(d*x + c)^14 + 420*(102*(a^2 + 2*a*b + b^2)*cosh(d
*x + c)^5 - 20*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^3 + (4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b
^2)*cosh(d*x + c))*sinh(d*x + c)^13 + 10*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x
+ c)^12 + 10*(9282*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 2730*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^4 + 60*(a^
2 - 16*a*b + 21*b^2)*d*x + 273*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^2 - 31*
a^2 - 82*a*b + 501*b^2)*sinh(d*x + c)^12 + 120*(1326*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 546*(a^2 + 10*a*b +
9*b^2)*cosh(d*x + c)^5 + 91*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^3 + (60*(
a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c))*sinh(d*x + c)^11 + 60*(20*(a^2 - 16*a*b
+ 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cosh(d*x + c)^10 + 30*(7293*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 4
004*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^6 + 1001*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*co
sh(d*x + c)^4 + 40*(a^2 - 16*a*b + 21*b^2)*d*x + 22*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^
2)*cosh(d*x + c)^2 - 6*a^2 + 30*a*b + 614*b^2)*sinh(d*x + c)^10 + 20*(12155*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^
9 - 8580*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^7 + 3003*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^
2)*cosh(d*x + c)^5 + 110*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^3 + 30*(20
*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 60*(20*(a^2 - 16*a*b
+ 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^2)*cosh(d*x + c)^8 + 30*(7293*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 - 6
435*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^8 + 3003*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*co
sh(d*x + c)^6 + 165*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^4 + 40*(a^2 - 1
6*a*b + 21*b^2)*d*x + 90*(20*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cosh(d*x + c)^2 + 6*a^2 +
30*a*b + 922*b^2)*sinh(d*x + c)^8 + 240*(663*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^11 - 715*(a^2 + 10*a*b + 9*b^2
)*cosh(d*x + c)^9 + 429*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^7 + 33*(60*(a^
2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^5 + 30*(20*(a^2 - 16*a*b + 21*b^2)*d*x - 3
*a^2 + 15*a*b + 307*b^2)*cosh(d*x + c)^3 + 2*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^2)*cosh(
d*x + c))*sinh(d*x + c)^7 + 10*(60*(a^2 - 16*a*b + 21*b^2)*d*x + 31*a^2 - 82*a*b + 1803*b^2)*cosh(d*x + c)^6 +
10*(9282*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^12 - 12012*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^10 + 9009*(4*(a^2
- 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^8 + 924*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a
^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^6 + 1260*(20*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cosh
(d*x + c)^4 + 60*(a^2 - 16*a*b + 21*b^2)*d*x + 168*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^2)
*cosh(d*x + c)^2 + 31*a^2 - 82*a*b + 1803*b^2)*sinh(d*x + c)^6 + 60*(714*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^13
- 1092*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^11 + 1001*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2
)*cosh(d*x + c)^9 + 132*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^7 + 252*(20
*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cosh(d*x + c)^5 + 56*(20*(a^2 - 16*a*b + 21*b^2)*d*x
+ 3*a^2 + 15*a*b + 461*b^2)*cosh(d*x + c)^3 + (60*(a^2 - 16*a*b + 21*b^2)*d*x + 31*a^2 - 82*a*b + 1803*b^2)*co
sh(d*x + c))*sinh(d*x + c)^5 + 6*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 25*a^2 - 150*a*b + 893*b^2)*cosh(d*x + c)^4
+ 6*(2550*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^14 - 4550*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^12 + 5005*(4*(a^2
- 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^10 + 825*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*
a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^8 + 2100*(20*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cos
h(d*x + c)^6 + 700*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^2)*cosh(d*x + c)^4 + 20*(a^2 - 16*
a*b + 21*b^2)*d*x + 25*(60*(a^2 - 16*a*b + 21*b^2)*d*x + 31*a^2 - 82*a*b + 1803*b^2)*cosh(d*x + c)^2 + 25*a^2
- 150*a*b + 893*b^2)*sinh(d*x + c)^4 + 8*(510*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^15 - 1050*(a^2 + 10*a*b + 9*b^
2)*cosh(d*x + c)^13 + 1365*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^11 + 275*(6
0*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^9 + 900*(20*(a^2 - 16*a*b + 21*b^2)*d
*x - 3*a^2 + 15*a*b + 307*b^2)*cosh(d*x + c)^7 + 420*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^
2)*cosh(d*x + c)^5 + 25*(60*(a^2 - 16*a*b + 21*b^2)*d*x + 31*a^2 - 82*a*b + 1803*b^2)*cosh(d*x + c)^3 + 3*(20*
(a^2 - 16*a*b + 21*b^2)*d*x + 25*a^2 - 150*a*b + 893*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 15*(a^2 - 10*a*b +
9*b^2)*cosh(d*x + c)^2 + 3*(255*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^16 - 600*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c
)^14 + 910*(4*(a^2 - 16*a*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^12 + 220*(60*(a^2 - 16*a*b
+ 21*b^2)*d*x - 31*a^2 - 82*a*b + 501*b^2)*cosh(d*x + c)^10 + 900*(20*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15
*a*b + 307*b^2)*cosh(d*x + c)^8 + 560*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^2)*cosh(d*x + c
)^6 + 50*(60*(a^2 - 16*a*b + 21*b^2)*d*x + 31*a^2 - 82*a*b + 1803*b^2)*cosh(d*x + c)^4 + 12*(20*(a^2 - 16*a*b
+ 21*b^2)*d*x + 25*a^2 - 150*a*b + 893*b^2)*cosh(d*x + c)^2 + 5*a^2 - 50*a*b + 45*b^2)*sinh(d*x + c)^2 - 5*a^2
+ 10*a*b - 5*b^2 + 1920*(a*b*cosh(d*x + c)^14 + 14*a*b*cosh(d*x + c)*sinh(d*x + c)^13 + a*b*sinh(d*x + c)^14
+ 5*a*b*cosh(d*x + c)^12 + (91*a*b*cosh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^12 + 10*a*b*cosh(d*x + c)^10 + 4*(91
*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)^11 + (1001*a*b*cosh(d*x + c)^4 + 330*a*b*cosh(d*x +
c)^2 + 10*a*b)*sinh(d*x + c)^10 + 10*a*b*cosh(d*x + c)^8 + 2*(1001*a*b*cosh(d*x + c)^5 + 550*a*b*cosh(d*x + c
)^3 + 50*a*b*cosh(d*x + c))*sinh(d*x + c)^9 + (3003*a*b*cosh(d*x + c)^6 + 2475*a*b*cosh(d*x + c)^4 + 450*a*b*c
osh(d*x + c)^2 + 10*a*b)*sinh(d*x + c)^8 + 5*a*b*cosh(d*x + c)^6 + 8*(429*a*b*cosh(d*x + c)^7 + 495*a*b*cosh(d
*x + c)^5 + 150*a*b*cosh(d*x + c)^3 + 10*a*b*cosh(d*x + c))*sinh(d*x + c)^7 + (3003*a*b*cosh(d*x + c)^8 + 4620
*a*b*cosh(d*x + c)^6 + 2100*a*b*cosh(d*x + c)^4 + 280*a*b*cosh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^6 + a*b*cosh(
d*x + c)^4 + 2*(1001*a*b*cosh(d*x + c)^9 + 1980*a*b*cosh(d*x + c)^7 + 1260*a*b*cosh(d*x + c)^5 + 280*a*b*cosh(
d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + (1001*a*b*cosh(d*x + c)^10 + 2475*a*b*cosh(d*x + c)^8 + 2
100*a*b*cosh(d*x + c)^6 + 700*a*b*cosh(d*x + c)^4 + 75*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 + 4*(91*a*b*
cosh(d*x + c)^11 + 275*a*b*cosh(d*x + c)^9 + 300*a*b*cosh(d*x + c)^7 + 140*a*b*cosh(d*x + c)^5 + 25*a*b*cosh(d
*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^3 + (91*a*b*cosh(d*x + c)^12 + 330*a*b*cosh(d*x + c)^10 + 450*a*b
*cosh(d*x + c)^8 + 280*a*b*cosh(d*x + c)^6 + 75*a*b*cosh(d*x + c)^4 + 6*a*b*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
2*(7*a*b*cosh(d*x + c)^13 + 30*a*b*cosh(d*x + c)^11 + 50*a*b*cosh(d*x + c)^9 + 40*a*b*cosh(d*x + c)^7 + 15*a*
b*cosh(d*x + c)^5 + 2*a*b*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c)))
+ 6*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^17 - 40*(a^2 + 10*a*b + 9*b^2)*cosh(d*x + c)^15 + 70*(4*(a^2 - 16*a
*b + 21*b^2)*d*x - 5*a^2 - 30*a*b - 25*b^2)*cosh(d*x + c)^13 + 20*(60*(a^2 - 16*a*b + 21*b^2)*d*x - 31*a^2 - 8
2*a*b + 501*b^2)*cosh(d*x + c)^11 + 100*(20*(a^2 - 16*a*b + 21*b^2)*d*x - 3*a^2 + 15*a*b + 307*b^2)*cosh(d*x +
c)^9 + 80*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 3*a^2 + 15*a*b + 461*b^2)*cosh(d*x + c)^7 + 10*(60*(a^2 - 16*a*b
+ 21*b^2)*d*x + 31*a^2 - 82*a*b + 1803*b^2)*cosh(d*x + c)^5 + 4*(20*(a^2 - 16*a*b + 21*b^2)*d*x + 25*a^2 - 150
*a*b + 893*b^2)*cosh(d*x + c)^3 + 5*(a^2 - 10*a*b + 9*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^14 +
14*d*cosh(d*x + c)*sinh(d*x + c)^13 + d*sinh(d*x + c)^14 + 5*d*cosh(d*x + c)^12 + (91*d*cosh(d*x + c)^2 + 5*d
)*sinh(d*x + c)^12 + 4*(91*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^11 + 10*d*cosh(d*x + c)^10 +
(1001*d*cosh(d*x + c)^4 + 330*d*cosh(d*x + c)^2 + 10*d)*sinh(d*x + c)^10 + 2*(1001*d*cosh(d*x + c)^5 + 550*d*c
osh(d*x + c)^3 + 50*d*cosh(d*x + c))*sinh(d*x + c)^9 + 10*d*cosh(d*x + c)^8 + (3003*d*cosh(d*x + c)^6 + 2475*d
*cosh(d*x + c)^4 + 450*d*cosh(d*x + c)^2 + 10*d)*sinh(d*x + c)^8 + 8*(429*d*cosh(d*x + c)^7 + 495*d*cosh(d*x +
c)^5 + 150*d*cosh(d*x + c)^3 + 10*d*cosh(d*x + c))*sinh(d*x + c)^7 + 5*d*cosh(d*x + c)^6 + (3003*d*cosh(d*x +
c)^8 + 4620*d*cosh(d*x + c)^6 + 2100*d*cosh(d*x + c)^4 + 280*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 2*(10
01*d*cosh(d*x + c)^9 + 1980*d*cosh(d*x + c)^7 + 1260*d*cosh(d*x + c)^5 + 280*d*cosh(d*x + c)^3 + 15*d*cosh(d*x
+ c))*sinh(d*x + c)^5 + d*cosh(d*x + c)^4 + (1001*d*cosh(d*x + c)^10 + 2475*d*cosh(d*x + c)^8 + 2100*d*cosh(d
*x + c)^6 + 700*d*cosh(d*x + c)^4 + 75*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 4*(91*d*cosh(d*x + c)^11 + 275
*d*cosh(d*x + c)^9 + 300*d*cosh(d*x + c)^7 + 140*d*cosh(d*x + c)^5 + 25*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*s
inh(d*x + c)^3 + (91*d*cosh(d*x + c)^12 + 330*d*cosh(d*x + c)^10 + 450*d*cosh(d*x + c)^8 + 280*d*cosh(d*x + c)
^6 + 75*d*cosh(d*x + c)^4 + 6*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(7*d*cosh(d*x + c)^13 + 30*d*cosh(d*x + c
)^11 + 50*d*cosh(d*x + c)^9 + 40*d*cosh(d*x + c)^7 + 15*d*cosh(d*x + c)^5 + 2*d*cosh(d*x + c)^3)*sinh(d*x + c)
)

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giac [B]  time = 0.71, size = 376, normalized size = 2.21 \[ \frac {120 \, {\left (a^{2} - 16 \, a b + 21 \, b^{2}\right )} d x + 1920 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 5 \, {\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 288 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 378 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 32 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 5 \, {\left (a^{2} e^{\left (4 \, d x + 36 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 36 \, c\right )} + b^{2} e^{\left (4 \, d x + 36 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 34 \, c\right )} - 40 \, a b e^{\left (2 \, d x + 34 \, c\right )} - 32 \, b^{2} e^{\left (2 \, d x + 34 \, c\right )}\right )} e^{\left (-32 \, c\right )} - \frac {32 \, {\left (137 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 645 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 200 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1250 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 600 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1250 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 840 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 645 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 520 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 137 \, a b - 144 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*(120*(a^2 - 16*a*b + 21*b^2)*d*x + 1920*a*b*log(e^(2*d*x + 2*c) + 1) - 5*(18*a^2*e^(4*d*x + 4*c) - 288*a
*b*e^(4*d*x + 4*c) + 378*b^2*e^(4*d*x + 4*c) - 8*a^2*e^(2*d*x + 2*c) + 40*a*b*e^(2*d*x + 2*c) - 32*b^2*e^(2*d*
x + 2*c) + a^2 - 2*a*b + b^2)*e^(-4*d*x - 4*c) + 5*(a^2*e^(4*d*x + 36*c) + 2*a*b*e^(4*d*x + 36*c) + b^2*e^(4*d
*x + 36*c) - 8*a^2*e^(2*d*x + 34*c) - 40*a*b*e^(2*d*x + 34*c) - 32*b^2*e^(2*d*x + 34*c))*e^(-32*c) - 32*(137*a
*b*e^(10*d*x + 10*c) + 645*a*b*e^(8*d*x + 8*c) - 200*b^2*e^(8*d*x + 8*c) + 1250*a*b*e^(6*d*x + 6*c) - 600*b^2*
e^(6*d*x + 6*c) + 1250*a*b*e^(4*d*x + 4*c) - 840*b^2*e^(4*d*x + 4*c) + 645*a*b*e^(2*d*x + 2*c) - 520*b^2*e^(2*
d*x + 2*c) + 137*a*b - 144*b^2)/(e^(2*d*x + 2*c) + 1)^5)/d

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maple [A]  time = 0.32, size = 243, normalized size = 1.43 \[ \frac {a^{2} \cosh \left (d x +c \right ) \left (\sinh ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 a^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8 d}+\frac {3 a^{2} x}{8}+\frac {3 a^{2} c}{8 d}+\frac {a b \left (\sinh ^{6}\left (d x +c \right )\right )}{2 d \cosh \left (d x +c \right )^{2}}-\frac {3 a b \left (\sinh ^{4}\left (d x +c \right )\right )}{2 d \cosh \left (d x +c \right )^{2}}+\frac {6 a b \ln \left (\cosh \left (d x +c \right )\right )}{d}-\frac {3 a b \left (\tanh ^{2}\left (d x +c \right )\right )}{d}+\frac {b^{2} \left (\sinh ^{9}\left (d x +c \right )\right )}{4 d \cosh \left (d x +c \right )^{5}}-\frac {9 b^{2} \left (\sinh ^{7}\left (d x +c \right )\right )}{8 d \cosh \left (d x +c \right )^{5}}+\frac {63 b^{2} x}{8}+\frac {63 c \,b^{2}}{8 d}-\frac {63 b^{2} \tanh \left (d x +c \right )}{8 d}-\frac {21 b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )}{8 d}-\frac {63 b^{2} \left (\tanh ^{5}\left (d x +c \right )\right )}{40 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d*a^2*cosh(d*x+c)*sinh(d*x+c)^3-3/8*a^2*cosh(d*x+c)*sinh(d*x+c)/d+3/8*a^2*x+3/8/d*a^2*c+1/2/d*a*b*sinh(d*x
+c)^6/cosh(d*x+c)^2-3/2/d*a*b*sinh(d*x+c)^4/cosh(d*x+c)^2+6*a*b*ln(cosh(d*x+c))/d-3*a*b*tanh(d*x+c)^2/d+1/4/d*
b^2*sinh(d*x+c)^9/cosh(d*x+c)^5-9/8/d*b^2*sinh(d*x+c)^7/cosh(d*x+c)^5+63/8*b^2*x+63/8/d*c*b^2-63/8*b^2*tanh(d*
x+c)/d-21/8*b^2*tanh(d*x+c)^3/d-63/40*b^2*tanh(d*x+c)^5/d

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maxima [B]  time = 0.41, size = 379, normalized size = 2.23 \[ \frac {1}{64} \, a^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{320} \, b^{2} {\left (\frac {2520 \, {\left (d x + c\right )}}{d} + \frac {5 \, {\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{d} - \frac {135 \, e^{\left (-2 \, d x - 2 \, c\right )} + 5358 \, e^{\left (-4 \, d x - 4 \, c\right )} + 18190 \, e^{\left (-6 \, d x - 6 \, c\right )} + 28455 \, e^{\left (-8 \, d x - 8 \, c\right )} + 19995 \, e^{\left (-10 \, d x - 10 \, c\right )} + 6560 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 5 \, e^{\left (-6 \, d x - 6 \, c\right )} + 10 \, e^{\left (-8 \, d x - 8 \, c\right )} + 10 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )}\right )}}\right )} + \frac {1}{32} \, a b {\left (\frac {192 \, {\left (d x + c\right )}}{d} - \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} + \frac {192 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {18 \, e^{\left (-2 \, d x - 2 \, c\right )} + 39 \, e^{\left (-4 \, d x - 4 \, c\right )} - 108 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*a^2*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) + 1/320*
b^2*(2520*(d*x + c)/d + 5*(32*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c))/d - (135*e^(-2*d*x - 2*c) + 5358*e^(-4*d*x
- 4*c) + 18190*e^(-6*d*x - 6*c) + 28455*e^(-8*d*x - 8*c) + 19995*e^(-10*d*x - 10*c) + 6560*e^(-12*d*x - 12*c)
- 5)/(d*(e^(-4*d*x - 4*c) + 5*e^(-6*d*x - 6*c) + 10*e^(-8*d*x - 8*c) + 10*e^(-10*d*x - 10*c) + 5*e^(-12*d*x -
12*c) + e^(-14*d*x - 14*c)))) + 1/32*a*b*(192*(d*x + c)/d - (20*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c))/d + 192*l
og(e^(-2*d*x - 2*c) + 1)/d - (18*e^(-2*d*x - 2*c) + 39*e^(-4*d*x - 4*c) - 108*e^(-6*d*x - 6*c) - 1)/(d*(e^(-4*
d*x - 4*c) + 2*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c))))

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mupad [B]  time = 0.44, size = 359, normalized size = 2.11 \[ x\,\left (\frac {3\,a^2}{8}-6\,a\,b+\frac {63\,b^2}{8}\right )+\frac {4\,\left (5\,b^2+a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a^2-5\,a\,b+4\,b^2\right )}{8\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2+5\,a\,b+4\,b^2\right )}{8\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2}{64\,d}-\frac {4\,\left (5\,b^2+a\,b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {24\,b^2}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,{\left (a-b\right )}^2}{64\,d}-\frac {16\,b^2}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {32\,b^2}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {6\,a\,b\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((3*a^2)/8 - 6*a*b + (63*b^2)/8) + (4*(a*b + 5*b^2))/(d*(exp(2*c + 2*d*x) + 1)) + (exp(- 2*c - 2*d*x)*(a^2 -
5*a*b + 4*b^2))/(8*d) - (exp(2*c + 2*d*x)*(5*a*b + a^2 + 4*b^2))/(8*d) + (exp(4*c + 4*d*x)*(a + b)^2)/(64*d)
- (4*(a*b + 5*b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (24*b^2)/(d*(3*exp(2*c + 2*d*x) + 3*exp(
4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (exp(- 4*c - 4*d*x)*(a - b)^2)/(64*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)) + (6*a*b*log(exp(2*c)*e
xp(2*d*x) + 1))/d

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

[Out]

Timed out

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