3.3.4 \(\int \text {csch}^5(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\) [204]

Optimal. Leaf size=101 \[ -\frac {a (3 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {b^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d} \]

[Out]

-1/8*a*(3*a+16*b)*arctanh(cosh(d*x+c))/d-b^2*cosh(d*x+c)/d+1/3*b^2*cosh(d*x+c)^3/d+3/8*a^2*coth(d*x+c)*csch(d*
x+c)/d-1/4*a^2*coth(d*x+c)*csch(d*x+c)^3/d

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Rubi [A]
time = 0.10, antiderivative size = 101, 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, 1167, 212} \begin {gather*} -\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a (3 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {b^2 \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

-1/8*(a*(3*a + 16*b)*ArcTanh[Cosh[c + d*x]])/d - (b^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d) + (3*a^2*
Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a^2*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 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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] && IGtQ[q, -2]

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

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Mathematica [A]
time = 0.04, size = 186, normalized size = 1.84 \begin {gather*} -\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d}+\frac {3 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {2 a b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {2 a b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {3 a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(-3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) + (3*a^2*Csch[(c + d*x)/2]^2)/(32*d) - (a^2*Csch
[(c + d*x)/2]^4)/(64*d) - (2*a*b*Log[Cosh[c/2 + (d*x)/2]])/d + (2*a*b*Log[Sinh[c/2 + (d*x)/2]])/d + (3*a^2*Log
[Tanh[(c + d*x)/2]])/(8*d) + (3*a^2*Sech[(c + d*x)/2]^2)/(32*d) + (a^2*Sech[(c + d*x)/2]^4)/(64*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(93)=186\).
time = 1.52, size = 195, normalized size = 1.93

method	result	size
risch	\(\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}+\frac {a^{2} {\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c}-11 \,{\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c}+3\right )}{4 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {2 a b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {2 a b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/24/d*exp(3*d*x+3*c)*b^2-3/8/d*exp(d*x+c)*b^2-3/8/d*exp(-d*x-c)*b^2+1/24/d*exp(-3*d*x-3*c)*b^2+1/4*a^2*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^2/d*ln(exp(d*x+c)+1
)-2*a*b/d*ln(exp(d*x+c)+1)+3/8*a^2/d*ln(exp(d*x+c)-1)+2*a*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. 234 vs. \(2 (93) = 186\).
time = 0.27, size = 234, normalized size = 2.32 \begin {gather*} \frac {1}{24} \, 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^{2} {\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 )} - 2 \, a 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)^2,x, algorithm="maxima")

[Out]

1/24*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^2*(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))) -
2*a*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. 3356 vs. \(2 (93) = 186\).
time = 0.44, size = 3356, normalized size = 33.23 \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)^2,x, algorithm="fricas")

[Out]

1/24*(b^2*cosh(d*x + c)^14 + 14*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + b^2*sinh(d*x + c)^14 - 13*b^2*cosh(d*x +
c)^12 + 13*(7*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^12 + 52*(7*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*s
inh(d*x + c)^11 + 3*(6*a^2 + 11*b^2)*cosh(d*x + c)^10 + (1001*b^2*cosh(d*x + c)^4 - 858*b^2*cosh(d*x + c)^2 +
18*a^2 + 33*b^2)*sinh(d*x + c)^10 + 2*(1001*b^2*cosh(d*x + c)^5 - 1430*b^2*cosh(d*x + c)^3 + 15*(6*a^2 + 11*b^
2)*cosh(d*x + c))*sinh(d*x + c)^9 - 3*(22*a^2 + 7*b^2)*cosh(d*x + c)^8 + 3*(1001*b^2*cosh(d*x + c)^6 - 2145*b^
2*cosh(d*x + c)^4 + 45*(6*a^2 + 11*b^2)*cosh(d*x + c)^2 - 22*a^2 - 7*b^2)*sinh(d*x + c)^8 + 24*(143*b^2*cosh(d
*x + c)^7 - 429*b^2*cosh(d*x + c)^5 + 15*(6*a^2 + 11*b^2)*cosh(d*x + c)^3 - (22*a^2 + 7*b^2)*cosh(d*x + c))*si
nh(d*x + c)^7 - 3*(22*a^2 + 7*b^2)*cosh(d*x + c)^6 + 3*(1001*b^2*cosh(d*x + c)^8 - 4004*b^2*cosh(d*x + c)^6 +
210*(6*a^2 + 11*b^2)*cosh(d*x + c)^4 - 28*(22*a^2 + 7*b^2)*cosh(d*x + c)^2 - 22*a^2 - 7*b^2)*sinh(d*x + c)^6 +
2*(1001*b^2*cosh(d*x + c)^9 - 5148*b^2*cosh(d*x + c)^7 + 378*(6*a^2 + 11*b^2)*cosh(d*x + c)^5 - 84*(22*a^2 +
7*b^2)*cosh(d*x + c)^3 - 9*(22*a^2 + 7*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(6*a^2 + 11*b^2)*cosh(d*x + c)^
4 + (1001*b^2*cosh(d*x + c)^10 - 6435*b^2*cosh(d*x + c)^8 + 630*(6*a^2 + 11*b^2)*cosh(d*x + c)^6 - 210*(22*a^2
+ 7*b^2)*cosh(d*x + c)^4 - 45*(22*a^2 + 7*b^2)*cosh(d*x + c)^2 + 18*a^2 + 33*b^2)*sinh(d*x + c)^4 - 13*b^2*co
sh(d*x + c)^2 + 4*(91*b^2*cosh(d*x + c)^11 - 715*b^2*cosh(d*x + c)^9 + 90*(6*a^2 + 11*b^2)*cosh(d*x + c)^7 - 4
2*(22*a^2 + 7*b^2)*cosh(d*x + c)^5 - 15*(22*a^2 + 7*b^2)*cosh(d*x + c)^3 + 3*(6*a^2 + 11*b^2)*cosh(d*x + c))*s
inh(d*x + c)^3 + (91*b^2*cosh(d*x + c)^12 - 858*b^2*cosh(d*x + c)^10 + 135*(6*a^2 + 11*b^2)*cosh(d*x + c)^8 -
84*(22*a^2 + 7*b^2)*cosh(d*x + c)^6 - 45*(22*a^2 + 7*b^2)*cosh(d*x + c)^4 + 18*(6*a^2 + 11*b^2)*cosh(d*x + c)^
2 - 13*b^2)*sinh(d*x + c)^2 + b^2 - 3*((3*a^2 + 16*a*b)*cosh(d*x + c)^11 + 11*(3*a^2 + 16*a*b)*cosh(d*x + c)*s
inh(d*x + c)^10 + (3*a^2 + 16*a*b)*sinh(d*x + c)^11 - 4*(3*a^2 + 16*a*b)*cosh(d*x + c)^9 + (55*(3*a^2 + 16*a*b
)*cosh(d*x + c)^2 - 12*a^2 - 64*a*b)*sinh(d*x + c)^9 + 3*(55*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 - 12*(3*a^2 + 16
*a*b)*cosh(d*x + c))*sinh(d*x + c)^8 + 6*(3*a^2 + 16*a*b)*cosh(d*x + c)^7 + 6*(55*(3*a^2 + 16*a*b)*cosh(d*x +
c)^4 - 24*(3*a^2 + 16*a*b)*cosh(d*x + c)^2 + 3*a^2 + 16*a*b)*sinh(d*x + c)^7 + 42*(11*(3*a^2 + 16*a*b)*cosh(d*
x + c)^5 - 8*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 + (3*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 - 4*(3*a^2 + 1
6*a*b)*cosh(d*x + c)^5 + 2*(231*(3*a^2 + 16*a*b)*cosh(d*x + c)^6 - 252*(3*a^2 + 16*a*b)*cosh(d*x + c)^4 + 63*(
3*a^2 + 16*a*b)*cosh(d*x + c)^2 - 6*a^2 - 32*a*b)*sinh(d*x + c)^5 + 2*(165*(3*a^2 + 16*a*b)*cosh(d*x + c)^7 -
252*(3*a^2 + 16*a*b)*cosh(d*x + c)^5 + 105*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 - 10*(3*a^2 + 16*a*b)*cosh(d*x + c
))*sinh(d*x + c)^4 + (3*a^2 + 16*a*b)*cosh(d*x + c)^3 + (165*(3*a^2 + 16*a*b)*cosh(d*x + c)^8 - 336*(3*a^2 + 1
6*a*b)*cosh(d*x + c)^6 + 210*(3*a^2 + 16*a*b)*cosh(d*x + c)^4 - 40*(3*a^2 + 16*a*b)*cosh(d*x + c)^2 + 3*a^2 +
16*a*b)*sinh(d*x + c)^3 + (55*(3*a^2 + 16*a*b)*cosh(d*x + c)^9 - 144*(3*a^2 + 16*a*b)*cosh(d*x + c)^7 + 126*(3
*a^2 + 16*a*b)*cosh(d*x + c)^5 - 40*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 + 3*(3*a^2 + 16*a*b)*cosh(d*x + c))*sinh(
d*x + c)^2 + (11*(3*a^2 + 16*a*b)*cosh(d*x + c)^10 - 36*(3*a^2 + 16*a*b)*cosh(d*x + c)^8 + 42*(3*a^2 + 16*a*b)
*cosh(d*x + c)^6 - 20*(3*a^2 + 16*a*b)*cosh(d*x + c)^4 + 3*(3*a^2 + 16*a*b)*cosh(d*x + c)^2)*sinh(d*x + c))*lo
g(cosh(d*x + c) + sinh(d*x + c) + 1) + 3*((3*a^2 + 16*a*b)*cosh(d*x + c)^11 + 11*(3*a^2 + 16*a*b)*cosh(d*x + c
)*sinh(d*x + c)^10 + (3*a^2 + 16*a*b)*sinh(d*x + c)^11 - 4*(3*a^2 + 16*a*b)*cosh(d*x + c)^9 + (55*(3*a^2 + 16*
a*b)*cosh(d*x + c)^2 - 12*a^2 - 64*a*b)*sinh(d*x + c)^9 + 3*(55*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 - 12*(3*a^2 +
16*a*b)*cosh(d*x + c))*sinh(d*x + c)^8 + 6*(3*a^2 + 16*a*b)*cosh(d*x + c)^7 + 6*(55*(3*a^2 + 16*a*b)*cosh(d*x
+ c)^4 - 24*(3*a^2 + 16*a*b)*cosh(d*x + c)^2 + 3*a^2 + 16*a*b)*sinh(d*x + c)^7 + 42*(11*(3*a^2 + 16*a*b)*cosh
(d*x + c)^5 - 8*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 + (3*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 - 4*(3*a^2
+ 16*a*b)*cosh(d*x + c)^5 + 2*(231*(3*a^2 + 16*a*b)*cosh(d*x + c)^6 - 252*(3*a^2 + 16*a*b)*cosh(d*x + c)^4 + 6
3*(3*a^2 + 16*a*b)*cosh(d*x + c)^2 - 6*a^2 - 32*a*b)*sinh(d*x + c)^5 + 2*(165*(3*a^2 + 16*a*b)*cosh(d*x + c)^7
- 252*(3*a^2 + 16*a*b)*cosh(d*x + c)^5 + 105*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 - 10*(3*a^2 + 16*a*b)*cosh(d*x
+ c))*sinh(d*x + c)^4 + (3*a^2 + 16*a*b)*cosh(d*x + c)^3 + (165*(3*a^2 + 16*a*b)*cosh(d*x + c)^8 - 336*(3*a^2
+ 16*a*b)*cosh(d*x + c)^6 + 210*(3*a^2 + 16*a*b)*cosh(d*x + c)^4 - 40*(3*a^2 + 16*a*b)*cosh(d*x + c)^2 + 3*a^2
+ 16*a*b)*sinh(d*x + c)^3 + (55*(3*a^2 + 16*a*b)*cosh(d*x + c)^9 - 144*(3*a^2 + 16*a*b)*cosh(d*x + c)^7 + 126
*(3*a^2 + 16*a*b)*cosh(d*x + c)^5 - 40*(3*a^2 + 16*a*b)*cosh(d*x + c)^3 + 3*(3*a^2 + 16*a*b)*cosh(d*x + c))*si
nh(d*x + c)^2 + (11*(3*a^2 + 16*a*b)*cosh(d*x +...

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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)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.59, size = 179, normalized size = 1.77 \begin {gather*} \frac {2 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 24 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 3 \, {\left (3 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 3 \, {\left (3 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) + \frac {12 \, {\left (3 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 20 \, a^{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 )}^{2}}}{48 \, 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)^2,x, algorithm="giac")

[Out]

1/48*(2*b^2*(e^(d*x + c) + e^(-d*x - c))^3 - 24*b^2*(e^(d*x + c) + e^(-d*x - c)) - 3*(3*a^2 + 16*a*b)*log(e^(d
*x + c) + e^(-d*x - c) + 2) + 3*(3*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) + 12*(3*a^2*(e^(d*x + c)
+ e^(-d*x - c))^3 - 20*a^2*(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 = 0.22, size = 328, normalized size = 3.25 \begin {gather*} \frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}}{8\,d}-\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}}{8\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a^2\,\sqrt {-d^2}+16\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {9\,a^4+96\,a^3\,b+256\,a^2\,b^2}}\right )\,\sqrt {9\,a^4+96\,a^3\,b+256\,a^2\,b^2}}{4\,\sqrt {-d^2}}-\frac {6\,a^2\,{\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 {4\,a^2\,{\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 {3\,a^2\,{\mathrm {e}}^{c+d\,x}}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {a^2\,{\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)^2/sinh(c + d*x)^5,x)

[Out]

(b^2*exp(- 3*c - 3*d*x))/(24*d) - (3*b^2*exp(- c - d*x))/(8*d) - (3*b^2*exp(c + d*x))/(8*d) + (b^2*exp(3*c + 3
*d*x))/(24*d) - (atan((exp(d*x)*exp(c)*(3*a^2*(-d^2)^(1/2) + 16*a*b*(-d^2)^(1/2)))/(d*(96*a^3*b + 9*a^4 + 256*
a^2*b^2)^(1/2)))*(96*a^3*b + 9*a^4 + 256*a^2*b^2)^(1/2))/(4*(-d^2)^(1/2)) - (6*a^2*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)) - (4*a^2*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)) + (3*a^2*exp(c + d*x))/(4*d*(exp(2*c + 2*d*x) - 1))
- (a^2*exp(c + d*x))/(2*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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