3.3.15 \(\int \text {csch}^{11}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [215]

Optimal. Leaf size=189 \[ \frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}+\frac {b^3 \cosh (c+d x)}{d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d} \]

[Out]

3/256*a*(21*a^2+80*a*b+128*b^2)*arctanh(cosh(d*x+c))/d+b^3*cosh(d*x+c)/d-3/256*a*(21*a^2+80*a*b+128*b^2)*coth(
d*x+c)*csch(d*x+c)/d+1/128*a^2*(21*a+80*b)*coth(d*x+c)*csch(d*x+c)^3/d-1/160*a^2*(21*a+80*b)*coth(d*x+c)*csch(
d*x+c)^5/d+9/80*a^3*coth(d*x+c)*csch(d*x+c)^7/d-1/10*a^3*coth(d*x+c)*csch(d*x+c)^9/d

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Rubi [A]
time = 0.26, antiderivative size = 189, normalized size of antiderivative = 1.00, 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 = {3294, 1171, 1828, 396, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}+\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}+\frac {b^3 \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^11*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(3*a*(21*a^2 + 80*a*b + 128*b^2)*ArcTanh[Cosh[c + d*x]])/(256*d) + (b^3*Cosh[c + d*x])/d - (3*a*(21*a^2 + 80*a
*b + 128*b^2)*Coth[c + d*x]*Csch[c + d*x])/(256*d) + (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^3)/(128*d)
- (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(160*d) + (9*a^3*Coth[c + d*x]*Csch[c + d*x]^7)/(80*d) -
(a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(10*d)

Rule 212

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

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 1171

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] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 3294

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}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}-\frac {\text {Subst}\left (\int \frac {-9 a^3-30 a^2 b-30 a b^2-10 b^3+10 b \left (3 a^2+9 a b+5 b^2\right ) x^2-10 b^2 (9 a+10 b) x^4+10 b^2 (3 a+10 b) x^6-50 b^3 x^8+10 b^3 x^{10}}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{10 d}\\ &=\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {\text {Subst}\left (\int \frac {63 a^3+240 a^2 b+240 a b^2+80 b^3-160 b^2 (3 a+2 b) x^2+240 b^2 (a+2 b) x^4-320 b^3 x^6+80 b^3 x^8}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{80 d}\\ &=-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}-\frac {\text {Subst}\left (\int \frac {-15 \left (21 a^3+80 a^2 b+96 a b^2+32 b^3\right )+1440 b^2 (a+b) x^2-1440 b^3 x^4+480 b^3 x^6}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{480 d}\\ &=\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {\text {Subst}\left (\int \frac {15 \left (63 a^3+240 a^2 b+384 a b^2+128 b^3\right )-3840 b^3 x^2+1920 b^3 x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{1920 d}\\ &=-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}-\frac {\text {Subst}\left (\int \frac {-15 \left (63 a^3+240 a^2 b+384 a b^2+256 b^3\right )+3840 b^3 x^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{3840 d}\\ &=\frac {b^3 \cosh (c+d x)}{d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {\left (3 a \left (21 a^2+80 a b+128 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{256 d}\\ &=\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}+\frac {b^3 \cosh (c+d x)}{d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]
time = 1.72, size = 265, normalized size = 1.40 \begin {gather*} \frac {b^3 \cosh (c+d x)}{d}-\frac {a \left (60 \left (21 a^2+80 a b+128 b^2\right ) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-40 a (7 a+24 b) \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+10 a (7 a+16 b) \text {csch}^6\left (\frac {1}{2} (c+d x)\right )-15 a^2 \text {csch}^8\left (\frac {1}{2} (c+d x)\right )+2 a^2 \text {csch}^{10}\left (\frac {1}{2} (c+d x)\right )+240 \left (21 a^2+80 a b+128 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+60 \left (21 a^2+80 a b+128 b^2\right ) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+40 a (7 a+24 b) \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+10 a (7 a+16 b) \text {sech}^6\left (\frac {1}{2} (c+d x)\right )+15 a^2 \text {sech}^8\left (\frac {1}{2} (c+d x)\right )+2 a^2 \text {sech}^{10}\left (\frac {1}{2} (c+d x)\right )\right )}{20480 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^11*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(b^3*Cosh[c + d*x])/d - (a*(60*(21*a^2 + 80*a*b + 128*b^2)*Csch[(c + d*x)/2]^2 - 40*a*(7*a + 24*b)*Csch[(c + d
*x)/2]^4 + 10*a*(7*a + 16*b)*Csch[(c + d*x)/2]^6 - 15*a^2*Csch[(c + d*x)/2]^8 + 2*a^2*Csch[(c + d*x)/2]^10 + 2
40*(21*a^2 + 80*a*b + 128*b^2)*Log[Tanh[(c + d*x)/2]] + 60*(21*a^2 + 80*a*b + 128*b^2)*Sech[(c + d*x)/2]^2 + 4
0*a*(7*a + 24*b)*Sech[(c + d*x)/2]^4 + 10*a*(7*a + 16*b)*Sech[(c + d*x)/2]^6 + 15*a^2*Sech[(c + d*x)/2]^8 + 2*
a^2*Sech[(c + d*x)/2]^10))/(20480*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(547\) vs. \(2(177)=354\).
time = 1.54, size = 548, normalized size = 2.90

method	result	size
risch	\(\frac {b^{3} {\mathrm e}^{d x +c}}{2 d}+\frac {b^{3} {\mathrm e}^{-d x -c}}{2 d}-\frac {a \,{\mathrm e}^{d x +c} \left (1200 a b +315 a^{2}-11600 a b \,{\mathrm e}^{2 d x +2 c}+1920 b^{2}+53280 a b \,{\mathrm e}^{10 d x +10 c}-93120 a b \,{\mathrm e}^{6 d x +6 c}+53280 a b \,{\mathrm e}^{8 d x +8 c}+13188 a^{2} {\mathrm e}^{4 d x +4 c}+1200 a b \,{\mathrm e}^{18 d x +18 c}-11600 a b \,{\mathrm e}^{16 d x +16 c}+50240 a b \,{\mathrm e}^{14 d x +14 c}-93120 a b \,{\mathrm e}^{12 d x +12 c}+50240 a b \,{\mathrm e}^{4 d x +4 c}+315 a^{2} {\mathrm e}^{18 d x +18 c}+1920 b^{2} {\mathrm e}^{18 d x +18 c}-3045 a^{2} {\mathrm e}^{16 d x +16 c}-13440 b^{2} {\mathrm e}^{16 d x +16 c}+13188 a^{2} {\mathrm e}^{14 d x +14 c}+38400 b^{2} {\mathrm e}^{14 d x +14 c}-33660 a^{2} {\mathrm e}^{12 d x +12 c}-33660 a^{2} {\mathrm e}^{6 d x +6 c}+26880 b^{2} {\mathrm e}^{8 d x +8 c}-13440 b^{2} {\mathrm e}^{2 d x +2 c}-53760 b^{2} {\mathrm e}^{6 d x +6 c}+38400 b^{2} {\mathrm e}^{4 d x +4 c}-3045 a^{2} {\mathrm e}^{2 d x +2 c}-53760 b^{2} {\mathrm e}^{12 d x +12 c}+55970 a^{2} {\mathrm e}^{10 d x +10 c}+26880 b^{2} {\mathrm e}^{10 d x +10 c}+55970 a^{2} {\mathrm e}^{8 d x +8 c}\right )}{640 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{10}}-\frac {63 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{256 d}-\frac {15 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{2 d}+\frac {63 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{256 d}+\frac {15 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{2 d}\)	\(548\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*b^3/d*exp(d*x+c)+1/2*b^3/d*exp(-d*x-c)-1/640*a*exp(d*x+c)*(1200*a*b+315*a^2-11600*a*b*exp(2*d*x+2*c)+1920*
b^2+53280*a*b*exp(10*d*x+10*c)-93120*a*b*exp(6*d*x+6*c)+53280*a*b*exp(8*d*x+8*c)+13188*a^2*exp(4*d*x+4*c)+1200
*a*b*exp(18*d*x+18*c)-11600*a*b*exp(16*d*x+16*c)+50240*a*b*exp(14*d*x+14*c)-93120*a*b*exp(12*d*x+12*c)+50240*a
*b*exp(4*d*x+4*c)+315*a^2*exp(18*d*x+18*c)+1920*b^2*exp(18*d*x+18*c)-3045*a^2*exp(16*d*x+16*c)-13440*b^2*exp(1
6*d*x+16*c)+13188*a^2*exp(14*d*x+14*c)+38400*b^2*exp(14*d*x+14*c)-33660*a^2*exp(12*d*x+12*c)-33660*a^2*exp(6*d
*x+6*c)+26880*b^2*exp(8*d*x+8*c)-13440*b^2*exp(2*d*x+2*c)-53760*b^2*exp(6*d*x+6*c)+38400*b^2*exp(4*d*x+4*c)-30
45*a^2*exp(2*d*x+2*c)-53760*b^2*exp(12*d*x+12*c)+55970*a^2*exp(10*d*x+10*c)+26880*b^2*exp(10*d*x+10*c)+55970*a
^2*exp(8*d*x+8*c))/d/(exp(2*d*x+2*c)-1)^10-63/256*a^3/d*ln(exp(d*x+c)-1)-15/16*a^2*b/d*ln(exp(d*x+c)-1)-3/2*a/
d*ln(exp(d*x+c)-1)*b^2+63/256*a^3/d*ln(exp(d*x+c)+1)+15/16*a^2*b/d*ln(exp(d*x+c)+1)+3/2*a/d*ln(exp(d*x+c)+1)*b
^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (177) = 354\).
time = 0.29, size = 573, normalized size = 3.03 \begin {gather*} \frac {1}{2} \, b^{3} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{1280} \, a^{3} {\left (\frac {315 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {315 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (315 \, e^{\left (-d x - c\right )} - 3045 \, e^{\left (-3 \, d x - 3 \, c\right )} + 13188 \, e^{\left (-5 \, d x - 5 \, c\right )} - 33660 \, e^{\left (-7 \, d x - 7 \, c\right )} + 55970 \, e^{\left (-9 \, d x - 9 \, c\right )} + 55970 \, e^{\left (-11 \, d x - 11 \, c\right )} - 33660 \, e^{\left (-13 \, d x - 13 \, c\right )} + 13188 \, e^{\left (-15 \, d x - 15 \, c\right )} - 3045 \, e^{\left (-17 \, d x - 17 \, c\right )} + 315 \, e^{\left (-19 \, d x - 19 \, c\right )}\right )}}{d {\left (10 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} + 120 \, e^{\left (-6 \, d x - 6 \, c\right )} - 210 \, e^{\left (-8 \, d x - 8 \, c\right )} + 252 \, e^{\left (-10 \, d x - 10 \, c\right )} - 210 \, e^{\left (-12 \, d x - 12 \, c\right )} + 120 \, e^{\left (-14 \, d x - 14 \, c\right )} - 45 \, e^{\left (-16 \, d x - 16 \, c\right )} + 10 \, e^{\left (-18 \, d x - 18 \, c\right )} - e^{\left (-20 \, d x - 20 \, c\right )} - 1\right )}}\right )} + \frac {1}{16} \, a^{2} b {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + \frac {3}{2} \, a b^{2} {\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/2*b^3*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/1280*a^3*(315*log(e^(-d*x - c) + 1)/d - 315*log(e^(-d*x - c) - 1)
/d + 2*(315*e^(-d*x - c) - 3045*e^(-3*d*x - 3*c) + 13188*e^(-5*d*x - 5*c) - 33660*e^(-7*d*x - 7*c) + 55970*e^(
-9*d*x - 9*c) + 55970*e^(-11*d*x - 11*c) - 33660*e^(-13*d*x - 13*c) + 13188*e^(-15*d*x - 15*c) - 3045*e^(-17*d
*x - 17*c) + 315*e^(-19*d*x - 19*c))/(d*(10*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) + 120*e^(-6*d*x - 6*c) - 21
0*e^(-8*d*x - 8*c) + 252*e^(-10*d*x - 10*c) - 210*e^(-12*d*x - 12*c) + 120*e^(-14*d*x - 14*c) - 45*e^(-16*d*x
- 16*c) + 10*e^(-18*d*x - 18*c) - e^(-20*d*x - 20*c) - 1))) + 1/16*a^2*b*(15*log(e^(-d*x - c) + 1)/d - 15*log(
e^(-d*x - c) - 1)/d + 2*(15*e^(-d*x - c) - 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) -
85*e^(-9*d*x - 9*c) + 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*
c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + 3/2*a*b^2*(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 13503 vs. \(2 (177) = 354\).
time = 0.48, size = 13503, normalized size = 71.44 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/1280*(640*b^3*cosh(d*x + c)^22 + 14080*b^3*cosh(d*x + c)*sinh(d*x + c)^21 + 640*b^3*sinh(d*x + c)^22 - 30*(2
1*a^3 + 80*a^2*b + 128*a*b^2 + 192*b^3)*cosh(d*x + c)^20 + 30*(4928*b^3*cosh(d*x + c)^2 - 21*a^3 - 80*a^2*b -
128*a*b^2 - 192*b^3)*sinh(d*x + c)^20 + 200*(4928*b^3*cosh(d*x + c)^3 - 3*(21*a^3 + 80*a^2*b + 128*a*b^2 + 192
*b^3)*cosh(d*x + c))*sinh(d*x + c)^19 + 10*(609*a^3 + 2320*a^2*b + 2688*a*b^2 + 2240*b^3)*cosh(d*x + c)^18 + 1
0*(468160*b^3*cosh(d*x + c)^4 + 609*a^3 + 2320*a^2*b + 2688*a*b^2 + 2240*b^3 - 570*(21*a^3 + 80*a^2*b + 128*a*
b^2 + 192*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^18 + 180*(93632*b^3*cosh(d*x + c)^5 - 190*(21*a^3 + 80*a^2*b + 1
28*a*b^2 + 192*b^3)*cosh(d*x + c)^3 + (609*a^3 + 2320*a^2*b + 2688*a*b^2 + 2240*b^3)*cosh(d*x + c))*sinh(d*x +
c)^17 - 8*(3297*a^3 + 12560*a^2*b + 9600*a*b^2 + 6000*b^3)*cosh(d*x + c)^16 + 2*(23876160*b^3*cosh(d*x + c)^6
- 72675*(21*a^3 + 80*a^2*b + 128*a*b^2 + 192*b^3)*cosh(d*x + c)^4 - 13188*a^3 - 50240*a^2*b - 38400*a*b^2 - 2
4000*b^3 + ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**11*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (177) = 354\).
time = 0.64, size = 477, normalized size = 2.52 \begin {gather*} \frac {1280 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, {\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (315 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1200 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1920 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 5880 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 22400 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 30720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 43008 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 163840 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 184320 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 151680 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 542720 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 491520 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 247040 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 675840 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 491520 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{5}}}{2560 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^11*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/2560*(1280*b^3*(e^(d*x + c) + e^(-d*x - c)) + 15*(21*a^3 + 80*a^2*b + 128*a*b^2)*log(e^(d*x + c) + e^(-d*x -
c) + 2) - 15*(21*a^3 + 80*a^2*b + 128*a*b^2)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(315*a^3*(e^(d*x + c) +
e^(-d*x - c))^9 + 1200*a^2*b*(e^(d*x + c) + e^(-d*x - c))^9 + 1920*a*b^2*(e^(d*x + c) + e^(-d*x - c))^9 - 5880
*a^3*(e^(d*x + c) + e^(-d*x - c))^7 - 22400*a^2*b*(e^(d*x + c) + e^(-d*x - c))^7 - 30720*a*b^2*(e^(d*x + c) +
e^(-d*x - c))^7 + 43008*a^3*(e^(d*x + c) + e^(-d*x - c))^5 + 163840*a^2*b*(e^(d*x + c) + e^(-d*x - c))^5 + 184
320*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 - 151680*a^3*(e^(d*x + c) + e^(-d*x - c))^3 - 542720*a^2*b*(e^(d*x +
c) + e^(-d*x - c))^3 - 491520*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3 + 247040*a^3*(e^(d*x + c) + e^(-d*x - c)) +
675840*a^2*b*(e^(d*x + c) + e^(-d*x - c)) + 491520*a*b^2*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*
x - c))^2 - 4)^5)/d

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Mupad [B]
time = 1.11, size = 1194, normalized size = 6.32 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {\frac {24\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{5\,d}-\frac {48\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{5\,d}-\frac {48\,{\mathrm {e}}^{11\,c+11\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{13\,c+13\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{9\,c+9\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{5\,d}-\frac {48\,a\,b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{5\,d}-\frac {48\,a\,b^2\,{\mathrm {e}}^{15\,c+15\,d\,x}}{5\,d}+\frac {6\,a\,b^2\,{\mathrm {e}}^{17\,c+17\,d\,x}}{5\,d}+\frac {6\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d}}{45\,{\mathrm {e}}^{4\,c+4\,d\,x}-10\,{\mathrm {e}}^{2\,c+2\,d\,x}-120\,{\mathrm {e}}^{6\,c+6\,d\,x}+210\,{\mathrm {e}}^{8\,c+8\,d\,x}-252\,{\mathrm {e}}^{10\,c+10\,d\,x}+210\,{\mathrm {e}}^{12\,c+12\,d\,x}-120\,{\mathrm {e}}^{14\,c+14\,d\,x}+45\,{\mathrm {e}}^{16\,c+16\,d\,x}-10\,{\mathrm {e}}^{18\,c+18\,d\,x}+{\mathrm {e}}^{20\,c+20\,d\,x}+1}+\frac {b^3\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (21\,a^3\,\sqrt {-d^2}+128\,a\,b^2\,\sqrt {-d^2}+80\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {441\,a^6+3360\,a^5\,b+11776\,a^4\,b^2+20480\,a^3\,b^3+16384\,a^2\,b^4}}\right )\,\sqrt {441\,a^6+3360\,a^5\,b+11776\,a^4\,b^2+20480\,a^3\,b^3+16384\,a^2\,b^4}}{128\,\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+208\,b\,a^2\right )}{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 {{\mathrm {e}}^{c+d\,x}\,\left (21\,a^3+80\,b\,a^2\right )}{80\,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 {3\,{\mathrm {e}}^{c+d\,x}\,\left (464\,a^2\,b-3\,a^3\right )}{40\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (21\,a^3+80\,a^2\,b+128\,a\,b^2\right )}{128\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {1032\,a^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (105\,a^3+400\,a^2\,b-1536\,a\,b^2\right )}{320\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (209\,a^3+32\,b\,a^2\right )}{5\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {176\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (28\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,{\mathrm {e}}^{2\,c+2\,d\,x}-56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}-56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}-8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {256\,a^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (9\,{\mathrm {e}}^{2\,c+2\,d\,x}-36\,{\mathrm {e}}^{4\,c+4\,d\,x}+84\,{\mathrm {e}}^{6\,c+6\,d\,x}-126\,{\mathrm {e}}^{8\,c+8\,d\,x}+126\,{\mathrm {e}}^{10\,c+10\,d\,x}-84\,{\mathrm {e}}^{12\,c+12\,d\,x}+36\,{\mathrm {e}}^{14\,c+14\,d\,x}-9\,{\mathrm {e}}^{16\,c+16\,d\,x}+{\mathrm {e}}^{18\,c+18\,d\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^11,x)

[Out]

(b^3*exp(c + d*x))/(2*d) - ((24*exp(5*c + 5*d*x)*(7*a*b^2 + 4*a^2*b))/(5*d) - (48*exp(7*c + 7*d*x)*(7*a*b^2 +
8*a^2*b))/(5*d) - (48*exp(11*c + 11*d*x)*(7*a*b^2 + 8*a^2*b))/(5*d) + (24*exp(13*c + 13*d*x)*(7*a*b^2 + 4*a^2*
b))/(5*d) + (4*exp(9*c + 9*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(5*d) - (48*a*b^2*exp(3*c + 3*d*x))/(5*d) -
(48*a*b^2*exp(15*c + 15*d*x))/(5*d) + (6*a*b^2*exp(17*c + 17*d*x))/(5*d) + (6*a*b^2*exp(c + d*x))/(5*d))/(45*
exp(4*c + 4*d*x) - 10*exp(2*c + 2*d*x) - 120*exp(6*c + 6*d*x) + 210*exp(8*c + 8*d*x) - 252*exp(10*c + 10*d*x)
+ 210*exp(12*c + 12*d*x) - 120*exp(14*c + 14*d*x) + 45*exp(16*c + 16*d*x) - 10*exp(18*c + 18*d*x) + exp(20*c +
20*d*x) + 1) + (b^3*exp(- c - d*x))/(2*d) + (3*atan((exp(d*x)*exp(c)*(21*a^3*(-d^2)^(1/2) + 128*a*b^2*(-d^2)^
(1/2) + 80*a^2*b*(-d^2)^(1/2)))/(d*(3360*a^5*b + 441*a^6 + 16384*a^2*b^4 + 20480*a^3*b^3 + 11776*a^4*b^2)^(1/2
)))*(3360*a^5*b + 441*a^6 + 16384*a^2*b^4 + 20480*a^3*b^3 + 11776*a^4*b^2)^(1/2))/(128*(-d^2)^(1/2)) - (exp(c
+ d*x)*(208*a^2*b + a^3))/(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)) - (exp(c + d*x)*(80*a^2*b + 21*a^3))/(80*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c +
4*d*x) + exp(6*c + 6*d*x) - 1)) - (3*exp(c + d*x)*(464*a^2*b - 3*a^3))/(40*d*(6*exp(4*c + 4*d*x) - 4*exp(2*c +
2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (3*exp(c + d*x)*(128*a*b^2 + 80*a^2*b + 21*a^3))/(128*
d*(exp(2*c + 2*d*x) - 1)) - (1032*a^3*exp(c + d*x))/(5*d*(7*exp(2*c + 2*d*x) - 21*exp(4*c + 4*d*x) + 35*exp(6*
c + 6*d*x) - 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) - 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) - 1)) + (
exp(c + d*x)*(400*a^2*b - 1536*a*b^2 + 105*a^3))/(320*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (2*exp(
c + d*x)*(32*a^2*b + 209*a^3))/(5*d*(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8
*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) - (176*a^3*exp(c + d*x))/(d*(28*exp(4*c + 4*d*x)
- 8*exp(2*c + 2*d*x) - 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) - 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d
*x) - 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1)) - (256*a^3*exp(c + d*x))/(5*d*(9*exp(2*c + 2*d*x) - 36*e
xp(4*c + 4*d*x) + 84*exp(6*c + 6*d*x) - 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) - 84*exp(12*c + 12*d*x)
+ 36*exp(14*c + 14*d*x) - 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) - 1))

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