3.222 \(\int \text{csch}^8(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=133 \[ -\frac{a^2 (a+b) \coth ^3(c+d x)}{d}+\frac{a^2 (a+3 b) \coth (c+d x)}{d}-\frac{a^3 \coth ^7(c+d x)}{7 d}+\frac{3 a^3 \coth ^5(c+d x)}{5 d}+\frac{3}{8} b^2 x (8 a+b)+\frac{b^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac{5 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d} \]
[Out]
(3*b^2*(8*a + b)*x)/8 + (a^2*(a + 3*b)*Coth[c + d*x])/d - (a^2*(a + b)*Coth[c + d*x]^3)/d + (3*a^3*Coth[c + d*
x]^5)/(5*d) - (a^3*Coth[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]^3*S
inh[c + d*x])/(4*d)
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Rubi [A] time = 0.288891, antiderivative size = 133, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{3217, 1259, 1805, 1802, 207} \[ -\frac{a^2 (a+b) \coth ^3(c+d x)}{d}+\frac{a^2 (a+3 b) \coth (c+d x)}{d}-\frac{a^3 \coth ^7(c+d x)}{7 d}+\frac{3 a^3 \coth ^5(c+d x)}{5 d}+\frac{3}{8} b^2 x (8 a+b)+\frac{b^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac{5 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^8*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(3*b^2*(8*a + b)*x)/8 + (a^2*(a + 3*b)*Coth[c + d*x])/d - (a^2*(a + b)*Coth[c + d*x]^3)/d + (3*a^3*Coth[c + d*
x]^5)/(5*d) - (a^3*Coth[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]^3*S
inh[c + d*x])/(4*d)
Rule 3217
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rule 1259
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
Rule 1805
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
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 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \text{csch}^8(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-2 a x^2+(a+b) x^4\right )^3}{x^8 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{4 a^3-20 a^3 x^2+4 a^2 (10 a+3 b) x^4-4 a^2 (10 a+9 b) x^6+\left (20 a^3+36 a^2 b+12 a b^2-b^3\right ) x^8-4 (a+b)^3 x^{10}}{x^8 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac{5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-8 a^3+32 a^3 x^2-24 a^2 (2 a+b) x^4+16 a^2 (2 a+3 b) x^6-\left (8 a^3+24 a^2 b+24 a b^2+3 b^3\right ) x^8}{x^8 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac{5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{8 a^3}{x^8}+\frac{24 a^3}{x^6}-\frac{24 a^2 (a+b)}{x^4}+\frac{8 a^2 (a+3 b)}{x^2}+\frac{3 b^2 (8 a+b)}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{a^2 (a+3 b) \coth (c+d x)}{d}-\frac{a^2 (a+b) \coth ^3(c+d x)}{d}+\frac{3 a^3 \coth ^5(c+d x)}{5 d}-\frac{a^3 \coth ^7(c+d x)}{7 d}-\frac{5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac{\left (3 b^2 (8 a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{3}{8} b^2 (8 a+b) x+\frac{a^2 (a+3 b) \coth (c+d x)}{d}-\frac{a^2 (a+b) \coth ^3(c+d x)}{d}+\frac{3 a^3 \coth ^5(c+d x)}{5 d}-\frac{a^3 \coth ^7(c+d x)}{7 d}-\frac{5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.691789, size = 106, normalized size = 0.8 \[ \frac{35 b^2 (12 (8 a+b) (c+d x)-8 b \sinh (2 (c+d x))+b \sinh (4 (c+d x)))-32 a^2 \coth (c+d x) \left ((8 a+35 b) \text{csch}^2(c+d x)-2 (8 a+35 b)+5 a \text{csch}^6(c+d x)-6 a \text{csch}^4(c+d x)\right )}{1120 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^8*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(-32*a^2*Coth[c + d*x]*(-2*(8*a + 35*b) + (8*a + 35*b)*Csch[c + d*x]^2 - 6*a*Csch[c + d*x]^4 + 5*a*Csch[c + d*
x]^6) + 35*b^2*(12*(8*a + b)*(c + d*x) - 8*b*Sinh[2*(c + d*x)] + b*Sinh[4*(c + d*x)]))/(1120*d)
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Maple [A] time = 0.089, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{16}{35}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{7}}+{\frac{6\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{35}}-{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{35}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( 2/3-1/3\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ){\rm coth} \left (dx+c\right )+3\,a{b}^{2} \left ( dx+c \right ) +{b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(a^3*(16/35-1/7*csch(d*x+c)^6+6/35*csch(d*x+c)^4-8/35*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(2/3-1/3*csch(d*x
+c)^2)*coth(d*x+c)+3*a*b^2*(d*x+c)+b^3*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c))
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Maxima [B] time = 1.14781, size = 725, normalized size = 5.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
1/64*b^3*(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) + 3*a*b^
2*x + 32/35*a^3*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^
(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) - 21*e^(-4*d*x - 4*c)
/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10
*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*
d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14
*d*x - 14*c) - 1)) - 1/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c
) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1))) + 4*a^2*b*(3*e^(-2*d*x - 2*c)/(d*
(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4
*c) + e^(-6*d*x - 6*c) - 1)))
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Fricas [B] time = 1.49259, size = 2464, normalized size = 18.53 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
1/2240*(35*b^3*cosh(d*x + c)^11 + 385*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - 525*b^3*cosh(d*x + c)^9 + 525*(11*b
^3*cosh(d*x + c)^3 - 9*b^3*cosh(d*x + c))*sinh(d*x + c)^8 + (1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c)^7
- 8*(128*a^3 + 560*a^2*b - 105*(8*a*b^2 + b^3)*d*x)*sinh(d*x + c)^7 + 7*(2310*b^3*cosh(d*x + c)^5 - 6300*b^3*
cosh(d*x + c)^3 + (1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 7*(1024*a^3 + 4480*a^2*b
+ 975*b^3)*cosh(d*x + c)^5 + 56*(128*a^3 + 560*a^2*b - 105*(8*a*b^2 + b^3)*d*x - 3*(128*a^3 + 560*a^2*b - 105
*(8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*(330*b^3*cosh(d*x + c)^7 - 1890*b^3*cosh(d*x + c)^
5 + (1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c)^3 - (1024*a^3 + 4480*a^2*b + 975*b^3)*cosh(d*x + c))*sinh
(d*x + c)^4 + 42*(512*a^3 + 1600*a^2*b + 215*b^3)*cosh(d*x + c)^3 - 56*(5*(128*a^3 + 560*a^2*b - 105*(8*a*b^2
+ b^3)*d*x)*cosh(d*x + c)^4 + 384*a^3 + 1680*a^2*b - 315*(8*a*b^2 + b^3)*d*x - 10*(128*a^3 + 560*a^2*b - 105*(
8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(275*b^3*cosh(d*x + c)^9 - 2700*b^3*cosh(d*x + c)^7 +
3*(1024*a^3 + 4480*a^2*b + 2695*b^3)*cosh(d*x + c)^5 - 10*(1024*a^3 + 4480*a^2*b + 975*b^3)*cosh(d*x + c)^3 +
18*(512*a^3 + 1600*a^2*b + 215*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 70*(512*a^3 + 576*a^2*b + 63*b^3)*cosh(d
*x + c) - 56*((128*a^3 + 560*a^2*b - 105*(8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^6 - 5*(128*a^3 + 560*a^2*b - 105*(
8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 - 640*a^3 - 2800*a^2*b + 525*(8*a*b^2 + b^3)*d*x + 9*(128*a^3 + 560*a^2*b
- 105*(8*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^7 + 7*(3*d*cosh(d*x + c)^2 - d)*si
nh(d*x + c)^5 + 7*(5*d*cosh(d*x + c)^4 - 10*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^3 + 7*(d*cosh(d*x + c)^6 -
5*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 - 5*d)*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(csch(d*x+c)**8*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
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Giac [B] time = 1.70678, size = 356, normalized size = 2.68 \begin{align*} \frac{3 \,{\left (8 \, a b^{2} + b^{3}\right )}{\left (d x + c\right )}}{8 \, d} - \frac{{\left (144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac{b^{3} d e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{3} d e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d^{2}} - \frac{4 \,{\left (105 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 455 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 280 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 770 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 168 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 245 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a^{3} - 35 \, a^{2} b\right )}}{35 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^8*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
3/8*(8*a*b^2 + b^3)*(d*x + c)/d - 1/64*(144*a*b^2*e^(4*d*x + 4*c) + 18*b^3*e^(4*d*x + 4*c) - 8*b^3*e^(2*d*x +
2*c) + b^3)*e^(-4*d*x - 4*c)/d + 1/64*(b^3*d*e^(4*d*x + 4*c) - 8*b^3*d*e^(2*d*x + 2*c))/d^2 - 4/35*(105*a^2*b*
e^(10*d*x + 10*c) - 455*a^2*b*e^(8*d*x + 8*c) + 280*a^3*e^(6*d*x + 6*c) + 770*a^2*b*e^(6*d*x + 6*c) - 168*a^3*
e^(4*d*x + 4*c) - 630*a^2*b*e^(4*d*x + 4*c) + 56*a^3*e^(2*d*x + 2*c) + 245*a^2*b*e^(2*d*x + 2*c) - 8*a^3 - 35*
a^2*b)/(d*(e^(2*d*x + 2*c) - 1)^7)