3.211 \(\int \text {csch}^3(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=148 \[ \frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d} \]
[Out]
1/2*a^3*arctanh(cosh(d*x+c))/d+b*(3*a^2+3*a*b+b^2)*cosh(d*x+c)/d-2/3*b^2*(3*a+2*b)*cosh(d*x+c)^3/d+3/5*b^2*(a+
2*b)*cosh(d*x+c)^5/d-4/7*b^3*cosh(d*x+c)^7/d+1/9*b^3*cosh(d*x+c)^9/d-1/2*a^3*coth(d*x+c)*csch(d*x+c)/d
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Rubi [A] time = 0.21, antiderivative size = 148, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used =
{3215, 1157, 1810, 206} \[ \frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(a^3*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(3*a^2 + 3*a*b + b^2)*Cosh[c + d*x])/d - (2*b^2*(3*a + 2*b)*Cosh[c + d
*x]^3)/(3*d) + (3*b^2*(a + 2*b)*Cosh[c + d*x]^5)/(5*d) - (4*b^3*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)
/(9*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d)
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 1157
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Rule 1810
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 3215
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {-a^3-6 a^2 b-6 a b^2-2 b^3+2 b \left (3 a^2+9 a b+5 b^2\right ) x^2-2 b^2 (9 a+10 b) x^4+2 b^2 (3 a+10 b) x^6-10 b^3 x^8+2 b^3 x^{10}}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\operatorname {Subst}\left (\int \left (-2 b \left (3 a^2+3 a b+b^2\right )+4 b^2 (3 a+2 b) x^2-6 b^2 (a+2 b) x^4+8 b^3 x^6-2 b^3 x^8-\frac {a^3}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 155, normalized size = 1.05 \[ \frac {-10080 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-10080 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-40320 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+1890 b \left (128 a^2+80 a b+21 b^2\right ) \cosh (c+d x)+3024 a b^2 \cosh (5 (c+d x))-1260 b^2 (20 a+7 b) \cosh (3 (c+d x))+2268 b^3 \cosh (5 (c+d x))-405 b^3 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(1890*b*(128*a^2 + 80*a*b + 21*b^2)*Cosh[c + d*x] - 1260*b^2*(20*a + 7*b)*Cosh[3*(c + d*x)] + 3024*a*b^2*Cosh[
5*(c + d*x)] + 2268*b^3*Cosh[5*(c + d*x)] - 405*b^3*Cosh[7*(c + d*x)] + 35*b^3*Cosh[9*(c + d*x)] - 10080*a^3*C
sch[(c + d*x)/2]^2 - 40320*a^3*Log[Tanh[(c + d*x)/2]] - 10080*a^3*Sech[(c + d*x)/2]^2)/(80640*d)
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fricas [B] time = 1.94, size = 4895, normalized size = 33.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
1/161280*(35*b^3*cosh(d*x + c)^22 + 770*b^3*cosh(d*x + c)*sinh(d*x + c)^21 + 35*b^3*sinh(d*x + c)^22 - 475*b^3
*cosh(d*x + c)^20 + 5*(1617*b^3*cosh(d*x + c)^2 - 95*b^3)*sinh(d*x + c)^20 + 100*(539*b^3*cosh(d*x + c)^3 - 95
*b^3*cosh(d*x + c))*sinh(d*x + c)^19 + (3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^18 + (256025*b^3*cosh(d*x + c)^4
- 90250*b^3*cosh(d*x + c)^2 + 3024*a*b^2 + 3113*b^3)*sinh(d*x + c)^18 + 6*(153615*b^3*cosh(d*x + c)^5 - 90250*
b^3*cosh(d*x + c)^3 + 3*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - 9*(3472*a*b^2 + 1529*b^3)*co
sh(d*x + c)^16 + 3*(870485*b^3*cosh(d*x + c)^6 - 767125*b^3*cosh(d*x + c)^4 - 10416*a*b^2 - 4587*b^3 + 51*(302
4*a*b^2 + 3113*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 48*(124355*b^3*cosh(d*x + c)^7 - 153425*b^3*cosh(d*x +
c)^5 + 17*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^3 - 3*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c))*sinh(d*x + c)^15
+ 126*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^14 + 6*(1865325*b^3*cosh(d*x + c)^8 - 3068500*b^3*cos
h(d*x + c)^6 + 510*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^4 + 40320*a^2*b + 34104*a*b^2 + 9933*b^3 - 180*(3472*
a*b^2 + 1529*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 4*(4352425*b^3*cosh(d*x + c)^9 - 9205500*b^3*cosh(d*x +
c)^7 + 2142*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^5 - 1260*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^3 + 441*(1920
*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 - 630*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3
)*cosh(d*x + c)^12 + 2*(11316305*b^3*cosh(d*x + c)^10 - 29917875*b^3*cosh(d*x + c)^8 + 9282*(3024*a*b^2 + 3113
*b^3)*cosh(d*x + c)^6 - 8190*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^4 - 80640*a^3 - 120960*a^2*b - 88200*a*b^2
- 24255*b^3 + 5733*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 8*(3086265*b^3*cosh
(d*x + c)^11 - 9972625*b^3*cosh(d*x + c)^9 + 3978*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^7 - 4914*(3472*a*b^2 +
1529*b^3)*cosh(d*x + c)^5 + 5733*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^3 - 945*(256*a^3 + 384*a^2
*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 630*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh
(d*x + c)^10 + 2*(11316305*b^3*cosh(d*x + c)^12 - 43879550*b^3*cosh(d*x + c)^10 + 21879*(3024*a*b^2 + 3113*b^3
)*cosh(d*x + c)^8 - 36036*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^6 + 63063*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*
cosh(d*x + c)^4 - 80640*a^3 - 120960*a^2*b - 88200*a*b^2 - 24255*b^3 - 20790*(256*a^3 + 384*a^2*b + 280*a*b^2
+ 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 4*(4352425*b^3*cosh(d*x + c)^13 - 19945250*b^3*cosh(d*x + c)^11
+ 12155*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^9 - 25740*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^7 + 63063*(1920*
a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^5 - 34650*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)
^3 - 1575*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 126*(1920*a^2*b + 1624*a
*b^2 + 473*b^3)*cosh(d*x + c)^8 + 6*(1865325*b^3*cosh(d*x + c)^14 - 9972625*b^3*cosh(d*x + c)^12 + 7293*(3024*
a*b^2 + 3113*b^3)*cosh(d*x + c)^10 - 19305*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^8 + 63063*(1920*a^2*b + 1624*
a*b^2 + 473*b^3)*cosh(d*x + c)^6 - 51975*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^4 + 40320*a^
2*b + 34104*a*b^2 + 9933*b^3 - 4725*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^
8 + 48*(124355*b^3*cosh(d*x + c)^15 - 767125*b^3*cosh(d*x + c)^13 + 663*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^
11 - 2145*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^9 + 9009*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^7 -
10395*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^5 - 1575*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77
*b^3)*cosh(d*x + c)^3 + 21*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 9*(3472*a*b^2
+ 1529*b^3)*cosh(d*x + c)^6 + 3*(870485*b^3*cosh(d*x + c)^16 - 6137000*b^3*cosh(d*x + c)^14 + 6188*(3024*a*b^2
+ 3113*b^3)*cosh(d*x + c)^12 - 24024*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^10 + 126126*(1920*a^2*b + 1624*a*b
^2 + 473*b^3)*cosh(d*x + c)^8 - 194040*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^6 - 44100*(256
*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^4 - 10416*a*b^2 - 4587*b^3 + 1176*(1920*a^2*b + 1624*a*b^
2 + 473*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 6*(153615*b^3*cosh(d*x + c)^17 - 1227400*b^3*cosh(d*x + c)^15
+ 1428*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^13 - 6552*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^11 + 42042*(1920*
a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^9 - 83160*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)
^7 - 26460*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^5 + 1176*(1920*a^2*b + 1624*a*b^2 + 473*b^
3)*cosh(d*x + c)^3 - 9*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 475*b^3*cosh(d*x + c)^2 + (302
4*a*b^2 + 3113*b^3)*cosh(d*x + c)^4 + (256025*b^3*cosh(d*x + c)^18 - 2301375*b^3*cosh(d*x + c)^16 + 3060*(3024
*a*b^2 + 3113*b^3)*cosh(d*x + c)^14 - 16380*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^12 + 126126*(1920*a^2*b + 16
24*a*b^2 + 473*b^3)*cosh(d*x + c)^10 - 311850*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^8 - 132
300*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^6 + 8820*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh
(d*x + c)^4 + 3024*a*b^2 + 3113*b^3 - 135*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(13475*
b^3*cosh(d*x + c)^19 - 135375*b^3*cosh(d*x + c)^17 + 204*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^15 - 1260*(3472
*a*b^2 + 1529*b^3)*cosh(d*x + c)^13 + 11466*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^11 - 34650*(256*
a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^9 - 18900*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(
d*x + c)^7 + 1764*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^5 - 45*(3472*a*b^2 + 1529*b^3)*cosh(d*x +
c)^3 + (3024*a*b^2 + 3113*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 35*b^3 + (8085*b^3*cosh(d*x + c)^20 - 90250*b^
3*cosh(d*x + c)^18 + 153*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^16 - 1080*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)
^14 + 11466*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^12 - 41580*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77
*b^3)*cosh(d*x + c)^10 - 28350*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^8 + 3528*(1920*a^2*b +
1624*a*b^2 + 473*b^3)*cosh(d*x + c)^6 - 135*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^4 - 475*b^3 + 6*(3024*a*b^2
+ 3113*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 80640*(a^3*cosh(d*x + c)^13 + 13*a^3*cosh(d*x + c)*sinh(d*x +
c)^12 + a^3*sinh(d*x + c)^13 - 2*a^3*cosh(d*x + c)^11 + a^3*cosh(d*x + c)^9 + 2*(39*a^3*cosh(d*x + c)^2 - a^3)
*sinh(d*x + c)^11 + 22*(13*a^3*cosh(d*x + c)^3 - a^3*cosh(d*x + c))*sinh(d*x + c)^10 + (715*a^3*cosh(d*x + c)^
4 - 110*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^9 + 3*(429*a^3*cosh(d*x + c)^5 - 110*a^3*cosh(d*x + c)^3 + 3*
a^3*cosh(d*x + c))*sinh(d*x + c)^8 + 12*(143*a^3*cosh(d*x + c)^6 - 55*a^3*cosh(d*x + c)^4 + 3*a^3*cosh(d*x + c
)^2)*sinh(d*x + c)^7 + 12*(143*a^3*cosh(d*x + c)^7 - 77*a^3*cosh(d*x + c)^5 + 7*a^3*cosh(d*x + c)^3)*sinh(d*x
+ c)^6 + 3*(429*a^3*cosh(d*x + c)^8 - 308*a^3*cosh(d*x + c)^6 + 42*a^3*cosh(d*x + c)^4)*sinh(d*x + c)^5 + (715
*a^3*cosh(d*x + c)^9 - 660*a^3*cosh(d*x + c)^7 + 126*a^3*cosh(d*x + c)^5)*sinh(d*x + c)^4 + 2*(143*a^3*cosh(d*
x + c)^10 - 165*a^3*cosh(d*x + c)^8 + 42*a^3*cosh(d*x + c)^6)*sinh(d*x + c)^3 + 2*(39*a^3*cosh(d*x + c)^11 - 5
5*a^3*cosh(d*x + c)^9 + 18*a^3*cosh(d*x + c)^7)*sinh(d*x + c)^2 + (13*a^3*cosh(d*x + c)^12 - 22*a^3*cosh(d*x +
c)^10 + 9*a^3*cosh(d*x + c)^8)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 80640*(a^3*cosh(d*x +
c)^13 + 13*a^3*cosh(d*x + c)*sinh(d*x + c)^12 + a^3*sinh(d*x + c)^13 - 2*a^3*cosh(d*x + c)^11 + a^3*cosh(d*x +
c)^9 + 2*(39*a^3*cosh(d*x + c)^2 - a^3)*sinh(d*x + c)^11 + 22*(13*a^3*cosh(d*x + c)^3 - a^3*cosh(d*x + c))*si
nh(d*x + c)^10 + (715*a^3*cosh(d*x + c)^4 - 110*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^9 + 3*(429*a^3*cosh(d
*x + c)^5 - 110*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^8 + 12*(143*a^3*cosh(d*x + c)^6 - 55*
a^3*cosh(d*x + c)^4 + 3*a^3*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 12*(143*a^3*cosh(d*x + c)^7 - 77*a^3*cosh(d*x +
c)^5 + 7*a^3*cosh(d*x + c)^3)*sinh(d*x + c)^6 + 3*(429*a^3*cosh(d*x + c)^8 - 308*a^3*cosh(d*x + c)^6 + 42*a^3
*cosh(d*x + c)^4)*sinh(d*x + c)^5 + (715*a^3*cosh(d*x + c)^9 - 660*a^3*cosh(d*x + c)^7 + 126*a^3*cosh(d*x + c)
^5)*sinh(d*x + c)^4 + 2*(143*a^3*cosh(d*x + c)^10 - 165*a^3*cosh(d*x + c)^8 + 42*a^3*cosh(d*x + c)^6)*sinh(d*x
+ c)^3 + 2*(39*a^3*cosh(d*x + c)^11 - 55*a^3*cosh(d*x + c)^9 + 18*a^3*cosh(d*x + c)^7)*sinh(d*x + c)^2 + (13*
a^3*cosh(d*x + c)^12 - 22*a^3*cosh(d*x + c)^10 + 9*a^3*cosh(d*x + c)^8)*sinh(d*x + c))*log(cosh(d*x + c) + sin
h(d*x + c) - 1) + 2*(385*b^3*cosh(d*x + c)^21 - 4750*b^3*cosh(d*x + c)^19 + 9*(3024*a*b^2 + 3113*b^3)*cosh(d*x
+ c)^17 - 72*(3472*a*b^2 + 1529*b^3)*cosh(d*x + c)^15 + 882*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)
^13 - 3780*(256*a^3 + 384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^11 - 3150*(256*a^3 + 384*a^2*b + 280*a*b^2
+ 77*b^3)*cosh(d*x + c)^9 + 504*(1920*a^2*b + 1624*a*b^2 + 473*b^3)*cosh(d*x + c)^7 - 27*(3472*a*b^2 + 1529*b
^3)*cosh(d*x + c)^5 - 475*b^3*cosh(d*x + c) + 2*(3024*a*b^2 + 3113*b^3)*cosh(d*x + c)^3)*sinh(d*x + c))/(d*cos
h(d*x + c)^13 + 13*d*cosh(d*x + c)*sinh(d*x + c)^12 + d*sinh(d*x + c)^13 - 2*d*cosh(d*x + c)^11 + 2*(39*d*cosh
(d*x + c)^2 - d)*sinh(d*x + c)^11 + 22*(13*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^10 + d*cosh(d*x
+ c)^9 + (715*d*cosh(d*x + c)^4 - 110*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^9 + 3*(429*d*cosh(d*x + c)^5 - 110*
d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^8 + 12*(143*d*cosh(d*x + c)^6 - 55*d*cosh(d*x + c)^4 + 3*
d*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 12*(143*d*cosh(d*x + c)^7 - 77*d*cosh(d*x + c)^5 + 7*d*cosh(d*x + c)^3)*s
inh(d*x + c)^6 + 3*(429*d*cosh(d*x + c)^8 - 308*d*cosh(d*x + c)^6 + 42*d*cosh(d*x + c)^4)*sinh(d*x + c)^5 + (7
15*d*cosh(d*x + c)^9 - 660*d*cosh(d*x + c)^7 + 126*d*cosh(d*x + c)^5)*sinh(d*x + c)^4 + 2*(143*d*cosh(d*x + c)
^10 - 165*d*cosh(d*x + c)^8 + 42*d*cosh(d*x + c)^6)*sinh(d*x + c)^3 + 2*(39*d*cosh(d*x + c)^11 - 55*d*cosh(d*x
+ c)^9 + 18*d*cosh(d*x + c)^7)*sinh(d*x + c)^2 + (13*d*cosh(d*x + c)^12 - 22*d*cosh(d*x + c)^10 + 9*d*cosh(d*
x + c)^8)*sinh(d*x + c))
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giac [B] time = 0.44, size = 300, normalized size = 2.03 \[ \frac {35 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 720 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 3024 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 6048 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 40320 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 26880 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 241920 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 241920 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 80640 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 40320 \, a^{3} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 40320 \, a^{3} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {161280 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4}}{161280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
1/161280*(35*b^3*(e^(d*x + c) + e^(-d*x - c))^9 - 720*b^3*(e^(d*x + c) + e^(-d*x - c))^7 + 3024*a*b^2*(e^(d*x
+ c) + e^(-d*x - c))^5 + 6048*b^3*(e^(d*x + c) + e^(-d*x - c))^5 - 40320*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3
- 26880*b^3*(e^(d*x + c) + e^(-d*x - c))^3 + 241920*a^2*b*(e^(d*x + c) + e^(-d*x - c)) + 241920*a*b^2*(e^(d*x
+ c) + e^(-d*x - c)) + 80640*b^3*(e^(d*x + c) + e^(-d*x - c)) + 40320*a^3*log(e^(d*x + c) + e^(-d*x - c) + 2)
- 40320*a^3*log(e^(d*x + c) + e^(-d*x - c) - 2) - 161280*a^3*(e^(d*x + c) + e^(-d*x - c))/((e^(d*x + c) + e^(-
d*x - c))^2 - 4))/d
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maple [A] time = 0.10, size = 130, normalized size = 0.88 \[ \frac {a^{3} \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \cosh \left (d x +c \right )+3 a \,b^{2} \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {128}{315}+\frac {\left (\sinh ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sinh ^{6}\left (d x +c \right )\right )}{63}+\frac {16 \left (\sinh ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sinh ^{2}\left (d x +c \right )\right )}{315}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a^2*b*cosh(d*x+c)+3*a*b^2*(8/15+1/5*sinh(d*x+c)^
4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+b^3*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*x+c)^6+16/105*sinh(d*x+c)^4-64/31
5*sinh(d*x+c)^2)*cosh(d*x+c))
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maxima [B] time = 0.33, size = 334, normalized size = 2.26 \[ -\frac {1}{161280} \, b^{3} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} + \frac {1}{160} \, a b^{2} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
-1/161280*b^3*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c)
- 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x
- 7*c) + 35*e^(-9*d*x - 9*c))/d) + 1/160*a*b^2*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d
+ 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d) + 3/2*a^2*b*(e^(d*x + c)/d + e^(-d*x - c
)/d) + 1/2*a^3*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*
e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))
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mupad [B] time = 1.19, size = 326, normalized size = 2.20 \[ \frac {\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}-\frac {9\,b^3\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{3584\,d}-\frac {9\,b^3\,{\mathrm {e}}^{7\,c+7\,d\,x}}{3584\,d}+\frac {b^3\,{\mathrm {e}}^{-9\,c-9\,d\,x}}{4608\,d}+\frac {b^3\,{\mathrm {e}}^{9\,c+9\,d\,x}}{4608\,d}+\frac {3\,b\,{\mathrm {e}}^{-c-d\,x}\,\left (128\,a^2+80\,a\,b+21\,b^2\right )}{256\,d}+\frac {3\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}\,\left (4\,a+3\,b\right )}{640\,d}+\frac {3\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (4\,a+3\,b\right )}{640\,d}-\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (20\,a+7\,b\right )}{128\,d}-\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (20\,a+7\,b\right )}{128\,d}+\frac {3\,b\,{\mathrm {e}}^{c+d\,x}\,\left (128\,a^2+80\,a\,b+21\,b^2\right )}{256\,d}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^3,x)
[Out]
(atan((a^3*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/2) - (9*b^3*exp(- 7*c - 7*d*x
))/(3584*d) - (9*b^3*exp(7*c + 7*d*x))/(3584*d) + (b^3*exp(- 9*c - 9*d*x))/(4608*d) + (b^3*exp(9*c + 9*d*x))/(
4608*d) + (3*b*exp(- c - d*x)*(80*a*b + 128*a^2 + 21*b^2))/(256*d) + (3*b^2*exp(- 5*c - 5*d*x)*(4*a + 3*b))/(6
40*d) + (3*b^2*exp(5*c + 5*d*x)*(4*a + 3*b))/(640*d) - (b^2*exp(- 3*c - 3*d*x)*(20*a + 7*b))/(128*d) - (b^2*ex
p(3*c + 3*d*x)*(20*a + 7*b))/(128*d) + (3*b*exp(c + d*x)*(80*a*b + 128*a^2 + 21*b^2))/(256*d) - (a^3*exp(c + d
*x))/(d*(exp(2*c + 2*d*x) - 1)) - (2*a^3*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
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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(csch(d*x+c)**3*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
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