3.3.12 \(\int \text {csch}^5(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [212]

Optimal. Leaf size=142 \[ -\frac {3 a^2 (a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d} \]

[Out]

-3/8*a^2*(a+8*b)*arctanh(cosh(d*x+c))/d-b^2*(3*a+b)*cosh(d*x+c)/d+b^2*(a+b)*cosh(d*x+c)^3/d-3/5*b^3*cosh(d*x+c
)^5/d+1/7*b^3*cosh(d*x+c)^7/d+3/8*a^3*coth(d*x+c)*csch(d*x+c)/d-1/4*a^3*coth(d*x+c)*csch(d*x+c)^3/d

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Rubi [A]
time = 0.19, antiderivative size = 142, normalized size of antiderivative = 1.00, 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 = {3294, 1171, 1828, 1824, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {3 a^2 (a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-3*a^2*(a + 8*b)*ArcTanh[Cosh[c + d*x]])/(8*d) - (b^2*(3*a + b)*Cosh[c + d*x])/d + (b^2*(a + b)*Cosh[c + d*x]
^3)/d - (3*b^3*Cosh[c + d*x]^5)/(5*d) + (b^3*Cosh[c + d*x]^7)/(7*d) + (3*a^3*Coth[c + d*x]*Csch[c + d*x])/(8*d
) - (a^3*Coth[c + d*x]*Csch[c + d*x]^3)/(4*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 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 1824

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 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}^5(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 )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {\text {Subst}\left (\int \frac {-3 a^3-12 a^2 b-12 a b^2-4 b^3+4 b \left (3 a^2+9 a b+5 b^2\right ) x^2-4 b^2 (9 a+10 b) x^4+4 b^2 (3 a+10 b) x^6-20 b^3 x^8+4 b^3 x^{10}}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 d}\\ &=\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {\text {Subst}\left (\int \frac {3 a^3+24 a^2 b+24 a b^2+8 b^3-16 b^2 (3 a+2 b) x^2+24 b^2 (a+2 b) x^4-32 b^3 x^6+8 b^3 x^8}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {\text {Subst}\left (\int \left (8 b^2 (3 a+b)-24 b^2 (a+b) x^2+24 b^3 x^4-8 b^3 x^6+\frac {3 \left (a^3+8 a^2 b\right )}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=-\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {\left (3 a^2 (a+8 b)\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=-\frac {3 a^2 (a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 173, normalized size = 1.22 \begin {gather*} \frac {-35 b^2 (144 a+35 b) \cosh (c+d x)+35 b^2 (16 a+7 b) \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))+210 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-35 a^3 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+840 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+6720 a^2 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+210 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+35 a^3 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{2240 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-35*b^2*(144*a + 35*b)*Cosh[c + d*x] + 35*b^2*(16*a + 7*b)*Cosh[3*(c + d*x)] - 49*b^3*Cosh[5*(c + d*x)] + 5*b
^3*Cosh[7*(c + d*x)] + 210*a^3*Csch[(c + d*x)/2]^2 - 35*a^3*Csch[(c + d*x)/2]^4 + 840*a^3*Log[Tanh[(c + d*x)/2
]] + 6720*a^2*b*Log[Tanh[(c + d*x)/2]] + 210*a^3*Sech[(c + d*x)/2]^2 + 35*a^3*Sech[(c + d*x)/2]^4)/(2240*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(335\) vs. \(2(132)=264\).
time = 1.55, size = 336, normalized size = 2.37

method	result	size
risch	\(\frac {b^{3} {\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 \,{\mathrm e}^{5 d x +5 c} b^{3}}{640 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{3}}{128 d}+\frac {a \,b^{2} {\mathrm e}^{3 d x +3 c}}{8 d}-\frac {9 a \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {35 b^{3} {\mathrm e}^{d x +c}}{128 d}-\frac {9 a \,{\mathrm e}^{-d x -c} b^{2}}{8 d}-\frac {35 b^{3} {\mathrm e}^{-d x -c}}{128 d}+\frac {7 b^{3} {\mathrm e}^{-3 d x -3 c}}{128 d}+\frac {a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{8 d}-\frac {7 \,{\mathrm e}^{-5 d x -5 c} b^{3}}{640 d}+\frac {b^{3} {\mathrm e}^{-7 d x -7 c}}{896 d}+\frac {a^{3} {\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c}-11 \,{\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c}+3\right )}{4 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(336\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/896*b^3/d*exp(7*d*x+7*c)-7/640/d*exp(5*d*x+5*c)*b^3+7/128/d*exp(3*d*x+3*c)*b^3+1/8*a*b^2/d*exp(3*d*x+3*c)-9/
8*a/d*exp(d*x+c)*b^2-35/128*b^3/d*exp(d*x+c)-9/8*a/d*exp(-d*x-c)*b^2-35/128*b^3/d*exp(-d*x-c)+7/128*b^3/d*exp(
-3*d*x-3*c)+1/8*a*b^2/d*exp(-3*d*x-3*c)-7/640/d*exp(-5*d*x-5*c)*b^3+1/896*b^3/d*exp(-7*d*x-7*c)+1/4*a^3*exp(d*
x+c)*(3*exp(6*d*x+6*c)-11*exp(4*d*x+4*c)-11*exp(2*d*x+2*c)+3)/d/(exp(2*d*x+2*c)-1)^4+3/8*a^3/d*ln(exp(d*x+c)-1
)+3*a^2*b/d*ln(exp(d*x+c)-1)-3/8*a^3/d*ln(exp(d*x+c)+1)-3*a^2*b/d*ln(exp(d*x+c)+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (132) = 264\).
time = 0.28, size = 340, normalized size = 2.39 \begin {gather*} -\frac {1}{4480} \, b^{3} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{8} \, a b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{3} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 3 \, a^{2} b {\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}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/4480*b^3*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (122
5*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/8*a*b^2*(e^(3*d*x + 3
*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) - 1/8*a^3*(3*log(e^(-d*x - c) + 1)/d - 3*log(
e^(-d*x - c) - 1)/d + 2*(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(
4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) - 3*a^2*b*(log(e^(-d*x
- c) + 1)/d - log(e^(-d*x - c) - 1)/d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6441 vs. \(2 (132) = 264\).
time = 0.46, size = 6441, normalized size = 45.36 \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)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/4480*(5*b^3*cosh(d*x + c)^22 + 110*b^3*cosh(d*x + c)*sinh(d*x + c)^21 + 5*b^3*sinh(d*x + c)^22 - 69*b^3*cosh
(d*x + c)^20 + 3*(385*b^3*cosh(d*x + c)^2 - 23*b^3)*sinh(d*x + c)^20 + 20*(385*b^3*cosh(d*x + c)^3 - 69*b^3*co
sh(d*x + c))*sinh(d*x + c)^19 + (560*a*b^2 + 471*b^3)*cosh(d*x + c)^18 + (36575*b^3*cosh(d*x + c)^4 - 13110*b^
3*cosh(d*x + c)^2 + 560*a*b^2 + 471*b^3)*sinh(d*x + c)^18 + 18*(7315*b^3*cosh(d*x + c)^5 - 4370*b^3*cosh(d*x +
c)^3 + (560*a*b^2 + 471*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - (7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^16 + (37
3065*b^3*cosh(d*x + c)^6 - 334305*b^3*cosh(d*x + c)^4 - 7280*a*b^2 - 2519*b^3 + 153*(560*a*b^2 + 471*b^3)*cosh
(d*x + c)^2)*sinh(d*x + c)^16 + 16*(53295*b^3*cosh(d*x + c)^7 - 66861*b^3*cosh(d*x + c)^5 + 51*(560*a*b^2 + 47
1*b^3)*cosh(d*x + c)^3 - (7280*a*b^2 + 2519*b^3)*cosh(d*x + c))*sinh(d*x + c)^15 + 6*(560*a^3 + 3080*a*b^2 + 8
91*b^3)*cosh(d*x + c)^14 + 6*(266475*b^3*cosh(d*x + c)^8 - 445740*b^3*cosh(d*x + c)^6 + 510*(560*a*b^2 + 471*b
^3)*cosh(d*x + c)^4 + 560*a^3 + 3080*a*b^2 + 891*b^3 - 20*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^14 + 4*(621775*b^3*cosh(d*x + c)^9 - 1337220*b^3*cosh(d*x + c)^7 + 2142*(560*a*b^2 + 471*b^3)*cosh(d*x + c)
^5 - 140*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^3 + 21*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^13 - 14*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^12 + 2*(1616615*b^3*cosh(d*x + c)^10 - 4345965*b^3
*cosh(d*x + c)^8 + 9282*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^6 - 910*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^4 -
6160*a^3 - 5880*a*b^2 - 1617*b^3 + 273*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 24
*(146965*b^3*cosh(d*x + c)^11 - 482885*b^3*cosh(d*x + c)^9 + 1326*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^7 - 182*
(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^5 + 91*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^3 - 7*(880*a^3 + 8
40*a*b^2 + 231*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 14*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^10 + 2*
(1616615*b^3*cosh(d*x + c)^12 - 6374082*b^3*cosh(d*x + c)^10 + 21879*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^8 - 4
004*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^6 + 3003*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^4 - 6160*a^3
- 5880*a*b^2 - 1617*b^3 - 462*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 4*(621775*b
^3*cosh(d*x + c)^13 - 2897310*b^3*cosh(d*x + c)^11 + 12155*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^9 - 2860*(7280*
a*b^2 + 2519*b^3)*cosh(d*x + c)^7 + 3003*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^5 - 770*(880*a^3 + 840
*a*b^2 + 231*b^3)*cosh(d*x + c)^3 - 35*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(560
*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^8 + 6*(266475*b^3*cosh(d*x + c)^14 - 1448655*b^3*cosh(d*x + c)^12 +
7293*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^10 - 2145*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^8 + 3003*(560*a^3 +
3080*a*b^2 + 891*b^3)*cosh(d*x + c)^6 - 1155*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^4 + 560*a^3 + 3080*
a*b^2 + 891*b^3 - 105*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(53295*b^3*cosh(d*
x + c)^15 - 334305*b^3*cosh(d*x + c)^13 + 1989*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^11 - 715*(7280*a*b^2 + 2519
*b^3)*cosh(d*x + c)^9 + 1287*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^7 - 693*(880*a^3 + 840*a*b^2 + 231
*b^3)*cosh(d*x + c)^5 - 105*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^3 + 3*(560*a^3 + 3080*a*b^2 + 891*b^
3)*cosh(d*x + c))*sinh(d*x + c)^7 - (7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^6 + (373065*b^3*cosh(d*x + c)^16 - 2
674440*b^3*cosh(d*x + c)^14 + 18564*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^12 - 8008*(7280*a*b^2 + 2519*b^3)*cosh
(d*x + c)^10 + 18018*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^8 - 12936*(880*a^3 + 840*a*b^2 + 231*b^3)*
cosh(d*x + c)^6 - 2940*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^4 - 7280*a*b^2 - 2519*b^3 + 168*(560*a^3
+ 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 6*(21945*b^3*cosh(d*x + c)^17 - 178296*b^3*cosh(d*x
+ c)^15 + 1428*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^13 - 728*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^11 + 2002*(
560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^9 - 1848*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^7 - 588*(
880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^5 + 56*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^3 - (7280*a
*b^2 + 2519*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 69*b^3*cosh(d*x + c)^2 + (560*a*b^2 + 471*b^3)*cosh(d*x + c)
^4 + (36575*b^3*cosh(d*x + c)^18 - 334305*b^3*cosh(d*x + c)^16 + 3060*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^14 -
1820*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^12 + 6006*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^10 - 6930
*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^8 - 2940*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^6 + 420*
(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^4 + 560*a*b^2 + 471*b^3 - 15*(7280*a*b^2 + 2519*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 4*(1925*b^3*cosh(d*x + c)^19 - 19665*b^3*cosh(d*x + c)^17 + 204*(560*a*b^2 + 471*b^3)
*cosh(d*x + c)^15 - 140*(7280*a*b^2 + 2519*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)**5*(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. 271 vs. \(2 (132) = 264\).
time = 0.62, size = 271, normalized size = 1.91 \begin {gather*} \frac {5 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 84 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 560 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 560 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 6720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 2240 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 840 \, {\left (a^{3} + 8 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 840 \, {\left (a^{3} + 8 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) + \frac {1120 \, {\left (3 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 20 \, a^{3} {\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 )}^{2}}}{4480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/4480*(5*b^3*(e^(d*x + c) + e^(-d*x - c))^7 - 84*b^3*(e^(d*x + c) + e^(-d*x - c))^5 + 560*a*b^2*(e^(d*x + c)
+ e^(-d*x - c))^3 + 560*b^3*(e^(d*x + c) + e^(-d*x - c))^3 - 6720*a*b^2*(e^(d*x + c) + e^(-d*x - c)) - 2240*b^
3*(e^(d*x + c) + e^(-d*x - c)) - 840*(a^3 + 8*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) + 840*(a^3 + 8*a^2*b)
*log(e^(d*x + c) + e^(-d*x - c) - 2) + 1120*(3*a^3*(e^(d*x + c) + e^(-d*x - c))^3 - 20*a^3*(e^(d*x + c) + e^(-
d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4)^2)/d

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Mupad [B]
time = 1.15, size = 421, normalized size = 2.96 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{896\,d}-\frac {7\,b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{640\,d}-\frac {7\,b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{640\,d}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {-d^2}+8\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+16\,a^5\,b+64\,a^4\,b^2}}\right )\,\sqrt {a^6+16\,a^5\,b+64\,a^4\,b^2}}{4\,\sqrt {-d^2}}+\frac {b^3\,{\mathrm {e}}^{7\,c+7\,d\,x}}{896\,d}-\frac {6\,a^3\,{\mathrm {e}}^{c+d\,x}}{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 {b^2\,{\mathrm {e}}^{c+d\,x}\,\left (144\,a+35\,b\right )}{128\,d}-\frac {4\,a^3\,{\mathrm {e}}^{c+d\,x}}{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 {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (16\,a+7\,b\right )}{128\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (16\,a+7\,b\right )}{128\,d}-\frac {b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (144\,a+35\,b\right )}{128\,d}+\frac {3\,a^3\,{\mathrm {e}}^{c+d\,x}}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,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)^5,x)

[Out]

(b^3*exp(- 7*c - 7*d*x))/(896*d) - (7*b^3*exp(- 5*c - 5*d*x))/(640*d) - (7*b^3*exp(5*c + 5*d*x))/(640*d) - (3*
atan((exp(d*x)*exp(c)*(a^3*(-d^2)^(1/2) + 8*a^2*b*(-d^2)^(1/2)))/(d*(16*a^5*b + a^6 + 64*a^4*b^2)^(1/2)))*(16*
a^5*b + a^6 + 64*a^4*b^2)^(1/2))/(4*(-d^2)^(1/2)) + (b^3*exp(7*c + 7*d*x))/(896*d) - (6*a^3*exp(c + d*x))/(d*(
3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (b^2*exp(c + d*x)*(144*a + 35*b))/(128*d) -
(4*a^3*exp(c + d*x))/(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)
) + (b^2*exp(- 3*c - 3*d*x)*(16*a + 7*b))/(128*d) + (b^2*exp(3*c + 3*d*x)*(16*a + 7*b))/(128*d) - (b^2*exp(- c
- d*x)*(144*a + 35*b))/(128*d) + (3*a^3*exp(c + d*x))/(4*d*(exp(2*c + 2*d*x) - 1)) - (a^3*exp(c + d*x))/(2*d*
(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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