3.3.13 \(\int \text {csch}^7(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [213]

Optimal. Leaf size=156 \[ \frac {a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}+\frac {b^2 (3 a+b) \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]

[Out]

1/16*a^2*(5*a+24*b)*arctanh(cosh(d*x+c))/d+b^2*(3*a+b)*cosh(d*x+c)/d-2/3*b^3*cosh(d*x+c)^3/d+1/5*b^3*cosh(d*x+
c)^5/d-1/16*a^2*(5*a+24*b)*coth(d*x+c)*csch(d*x+c)/d+5/24*a^3*coth(d*x+c)*csch(d*x+c)^3/d-1/6*a^3*coth(d*x+c)*
csch(d*x+c)^5/d

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Rubi [A]
time = 0.21, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, 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}^5(c+d x)}{6 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}+\frac {a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(a^2*(5*a + 24*b)*ArcTanh[Cosh[c + d*x]])/(16*d) + (b^2*(3*a + b)*Cosh[c + d*x])/d - (2*b^3*Cosh[c + d*x]^3)/(
3*d) + (b^3*Cosh[c + d*x]^5)/(5*d) - (a^2*(5*a + 24*b)*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^3*Coth[c + d
*x]*Csch[c + d*x]^3)/(24*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^5)/(6*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}^7(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 )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-5 a^3-18 a^2 b-18 a b^2-6 b^3+6 b \left (3 a^2+9 a b+5 b^2\right ) x^2-6 b^2 (9 a+10 b) x^4+6 b^2 (3 a+10 b) x^6-30 b^3 x^8+6 b^3 x^{10}}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{6 d}\\ &=\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {3 \left (5 a^3+24 a^2 b+24 a b^2+8 b^3\right )-48 b^2 (3 a+2 b) x^2+72 b^2 (a+2 b) x^4-96 b^3 x^6+24 b^3 x^8}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{24 d}\\ &=-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-3 \left (5 a^3+24 a^2 b+48 a b^2+16 b^3\right )+144 b^2 (a+b) x^2-144 b^3 x^4+48 b^3 x^6}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {\text {Subst}\left (\int \left (-48 b^2 (3 a+b)+96 b^3 x^2-48 b^3 x^4-\frac {3 \left (5 a^3+24 a^2 b\right )}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=\frac {b^2 (3 a+b) \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {\left (a^2 (5 a+24 b)\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{16 d}\\ &=\frac {a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}+\frac {b^2 (3 a+b) \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]
time = 0.52, size = 223, normalized size = 1.43 \begin {gather*} -\frac {-240 b^2 (24 a+5 b) \cosh (c+d x)+200 b^3 \cosh (3 (c+d x))-24 b^3 \cosh (5 (c+d x))+150 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+720 a^2 b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-30 a^3 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+5 a^3 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )+600 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+2880 a^2 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+150 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+720 a^2 b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+30 a^3 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+5 a^3 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{1920 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/1920*(-240*b^2*(24*a + 5*b)*Cosh[c + d*x] + 200*b^3*Cosh[3*(c + d*x)] - 24*b^3*Cosh[5*(c + d*x)] + 150*a^3*
Csch[(c + d*x)/2]^2 + 720*a^2*b*Csch[(c + d*x)/2]^2 - 30*a^3*Csch[(c + d*x)/2]^4 + 5*a^3*Csch[(c + d*x)/2]^6 +
600*a^3*Log[Tanh[(c + d*x)/2]] + 2880*a^2*b*Log[Tanh[(c + d*x)/2]] + 150*a^3*Sech[(c + d*x)/2]^2 + 720*a^2*b*
Sech[(c + d*x)/2]^2 + 30*a^3*Sech[(c + d*x)/2]^4 + 5*a^3*Sech[(c + d*x)/2]^6)/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(357\) vs. \(2(144)=288\).
time = 1.52, size = 358, normalized size = 2.29

method	result	size
risch	\(\frac {{\mathrm e}^{5 d x +5 c} b^{3}}{160 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} b^{3}}{96 d}+\frac {3 a \,{\mathrm e}^{d x +c} b^{2}}{2 d}+\frac {5 b^{3} {\mathrm e}^{d x +c}}{16 d}+\frac {3 a \,{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {5 b^{3} {\mathrm e}^{-d x -c}}{16 d}-\frac {5 b^{3} {\mathrm e}^{-3 d x -3 c}}{96 d}+\frac {{\mathrm e}^{-5 d x -5 c} b^{3}}{160 d}-\frac {a^{2} {\mathrm e}^{d x +c} \left (15 a \,{\mathrm e}^{10 d x +10 c}+72 b \,{\mathrm e}^{10 d x +10 c}-85 a \,{\mathrm e}^{8 d x +8 c}-216 b \,{\mathrm e}^{8 d x +8 c}+198 a \,{\mathrm e}^{6 d x +6 c}+144 b \,{\mathrm e}^{6 d x +6 c}+198 a \,{\mathrm e}^{4 d x +4 c}+144 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{2 d x +2 c}-216 b \,{\mathrm e}^{2 d x +2 c}+15 a +72 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}\)	\(358\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^7*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/160/d*exp(5*d*x+5*c)*b^3-5/96/d*exp(3*d*x+3*c)*b^3+3/2*a/d*exp(d*x+c)*b^2+5/16*b^3/d*exp(d*x+c)+3/2*a/d*exp(
-d*x-c)*b^2+5/16*b^3/d*exp(-d*x-c)-5/96*b^3/d*exp(-3*d*x-3*c)+1/160/d*exp(-5*d*x-5*c)*b^3-1/24*a^2*exp(d*x+c)*
(15*a*exp(10*d*x+10*c)+72*b*exp(10*d*x+10*c)-85*a*exp(8*d*x+8*c)-216*b*exp(8*d*x+8*c)+198*a*exp(6*d*x+6*c)+144
*b*exp(6*d*x+6*c)+198*a*exp(4*d*x+4*c)+144*b*exp(4*d*x+4*c)-85*a*exp(2*d*x+2*c)-216*b*exp(2*d*x+2*c)+15*a+72*b
)/d/(exp(2*d*x+2*c)-1)^6-5/16*a^3/d*ln(exp(d*x+c)-1)-3/2*a^2*b/d*ln(exp(d*x+c)-1)+5/16*a^3/d*ln(exp(d*x+c)+1)+
3/2*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. 390 vs. \(2 (144) = 288\).
time = 0.29, size = 390, normalized size = 2.50 \begin {gather*} \frac {1}{480} \, b^{3} {\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 b^{2} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{48} \, a^{3} {\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^{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} + \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)^7*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/480*b^3*(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*b^2*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/48*a^3*(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) + 2
0*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^2*b*(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. 8547 vs. \(2 (144) = 288\).
time = 0.49, size = 8547, normalized size = 54.79 \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)^7*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/480*(3*b^3*cosh(d*x + c)^22 + 66*b^3*cosh(d*x + c)*sinh(d*x + c)^21 + 3*b^3*sinh(d*x + c)^22 - 43*b^3*cosh(d
*x + c)^20 + (693*b^3*cosh(d*x + c)^2 - 43*b^3)*sinh(d*x + c)^20 + 20*(231*b^3*cosh(d*x + c)^3 - 43*b^3*cosh(d
*x + c))*sinh(d*x + c)^19 + 15*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^18 + 5*(4389*b^3*cosh(d*x + c)^4 - 1634*b^3*c
osh(d*x + c)^2 + 144*a*b^2 + 69*b^3)*sinh(d*x + c)^18 + 6*(13167*b^3*cosh(d*x + c)^5 - 8170*b^3*cosh(d*x + c)^
3 + 45*(48*a*b^2 + 23*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - 15*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(
d*x + c)^16 + 3*(74613*b^3*cosh(d*x + c)^6 - 69445*b^3*cosh(d*x + c)^4 - 100*a^3 - 480*a^2*b - 1200*a*b^2 - 39
5*b^3 + 765*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 48*(10659*b^3*cosh(d*x + c)^7 - 13889*b^3*
cosh(d*x + c)^5 + 255*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^3 - 5*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*
x + c))*sinh(d*x + c)^15 + 10*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c)^14 + 10*(95931*b^3*cos
h(d*x + c)^8 - 166668*b^3*cosh(d*x + c)^6 + 4590*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^4 + 170*a^3 + 432*a^2*b + 6
48*a*b^2 + 187*b^3 - 180*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 20*(7461
3*b^3*cosh(d*x + c)^9 - 166668*b^3*cosh(d*x + c)^7 + 6426*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^5 - 420*(20*a^3 +
96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^3 + 7*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c))*
sinh(d*x + c)^13 - 90*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^12 + 2*(969969*b^3*cosh(d*x + c)^1
0 - 2708355*b^3*cosh(d*x + c)^8 + 139230*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^6 - 13650*(20*a^3 + 96*a^2*b + 240*
a*b^2 + 79*b^3)*cosh(d*x + c)^4 - 1980*a^3 - 1440*a^2*b - 1800*a*b^2 - 495*b^3 + 455*(170*a^3 + 432*a^2*b + 64
8*a*b^2 + 187*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 8*(264537*b^3*cosh(d*x + c)^11 - 902785*b^3*cosh(d*x +
c)^9 + 59670*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^7 - 8190*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)
^5 + 455*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c)^3 - 135*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*
b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 90*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^10 + 2*(969969
*b^3*cosh(d*x + c)^12 - 3972254*b^3*cosh(d*x + c)^10 + 328185*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^8 - 60060*(20*
a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^6 + 5005*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d
*x + c)^4 - 1980*a^3 - 1440*a^2*b - 1800*a*b^2 - 495*b^3 - 2970*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^10 + 20*(74613*b^3*cosh(d*x + c)^13 - 361114*b^3*cosh(d*x + c)^11 + 36465*(48*a*b^2 +
23*b^3)*cosh(d*x + c)^9 - 8580*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^7 + 1001*(170*a^3 + 432
*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c)^5 - 990*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^3 -
45*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 10*(170*a^3 + 432*a^2*b + 648*a*b^
2 + 187*b^3)*cosh(d*x + c)^8 + 10*(95931*b^3*cosh(d*x + c)^14 - 541671*b^3*cosh(d*x + c)^12 + 65637*(48*a*b^2
+ 23*b^3)*cosh(d*x + c)^10 - 19305*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^8 + 3003*(170*a^3 +
432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c)^6 - 4455*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^
4 + 170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3 - 405*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*s
inh(d*x + c)^8 + 16*(31977*b^3*cosh(d*x + c)^15 - 208335*b^3*cosh(d*x + c)^13 + 29835*(48*a*b^2 + 23*b^3)*cosh
(d*x + c)^11 - 10725*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^9 + 2145*(170*a^3 + 432*a^2*b + 64
8*a*b^2 + 187*b^3)*cosh(d*x + c)^7 - 4455*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^5 - 675*(44*a^
3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^3 + 5*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c
))*sinh(d*x + c)^7 - 15*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^6 + (223839*b^3*cosh(d*x + c)^1
6 - 1666680*b^3*cosh(d*x + c)^14 + 278460*(48*a*b^2 + 23*b^3)*cosh(d*x + c)^12 - 120120*(20*a^3 + 96*a^2*b + 2
40*a*b^2 + 79*b^3)*cosh(d*x + c)^10 + 30030*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c)^8 - 8316
0*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^6 - 18900*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh
(d*x + c)^4 - 300*a^3 - 1440*a^2*b - 3600*a*b^2 - 1185*b^3 + 280*(170*a^3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(39501*b^3*cosh(d*x + c)^17 - 333336*b^3*cosh(d*x + c)^15 + 64260*(48*a*b^
2 + 23*b^3)*cosh(d*x + c)^13 - 32760*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c)^11 + 10010*(170*a^
3 + 432*a^2*b + 648*a*b^2 + 187*b^3)*cosh(d*x + c)^9 - 35640*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x
+ c)^7 - 11340*(44*a^3 + 32*a^2*b + 40*a*b^2 + 11*b^3)*cosh(d*x + c)^5 + 280*(170*a^3 + 432*a^2*b + 648*a*b^2
+ 187*b^3)*cosh(d*x + c)^3 - 45*(20*a^3 + 96*a^2*b + 240*a*b^2 + 79*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 43*b
^3*cosh(d*x + c)^2 + 15*(48*a*b^2 + 23*b^3)*cos...

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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)**7*(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. 321 vs. \(2 (144) = 288\).
time = 0.59, size = 321, normalized size = 2.06 \begin {gather*} \frac {3 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 40 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 240 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, {\left (5 \, a^{3} + 24 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (5 \, a^{3} + 24 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {20 \, {\left (15 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 72 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 160 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 576 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 1152 \, a^{2} b {\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 )}^{3}}}{480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/480*(3*b^3*(e^(d*x + c) + e^(-d*x - c))^5 - 40*b^3*(e^(d*x + c) + e^(-d*x - c))^3 + 720*a*b^2*(e^(d*x + c) +
e^(-d*x - c)) + 240*b^3*(e^(d*x + c) + e^(-d*x - c)) + 15*(5*a^3 + 24*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) +
2) - 15*(5*a^3 + 24*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 20*(15*a^3*(e^(d*x + c) + e^(-d*x - c))^5 +
72*a^2*b*(e^(d*x + c) + e^(-d*x - c))^5 - 160*a^3*(e^(d*x + c) + e^(-d*x - c))^3 - 576*a^2*b*(e^(d*x + c) + e^
(-d*x - c))^3 + 528*a^3*(e^(d*x + c) + e^(-d*x - c)) + 1152*a^2*b*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c)
+ e^(-d*x - c))^2 - 4)^3)/d

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Mupad [B]
time = 1.14, size = 633, normalized size = 4.06 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^3\,\sqrt {-d^2}+24\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^6+240\,a^5\,b+576\,a^4\,b^2}}\right )\,\sqrt {25\,a^6+240\,a^5\,b+576\,a^4\,b^2}}{8\,\sqrt {-d^2}}-\frac {\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{3\,d}-\frac {8\,a^2\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d}-\frac {8\,a^2\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{d}+\frac {2\,a^2\,b\,{\mathrm {e}}^{9\,c+9\,d\,x}}{d}+\frac {2\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{d}}{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}-\frac {5\,b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{96\,d}-\frac {5\,b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{96\,d}+\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+24\,b\,a^2\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 (24\,a+5\,b\right )}{16\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (48\,a^2\,b-5\,a^3\right )}{12\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {22\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 {16\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 {b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (24\,a+5\,b\right )}{16\,d} \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)^7,x)

[Out]

(atan((exp(d*x)*exp(c)*(5*a^3*(-d^2)^(1/2) + 24*a^2*b*(-d^2)^(1/2)))/(d*(240*a^5*b + 25*a^6 + 576*a^4*b^2)^(1/
2)))*(240*a^5*b + 25*a^6 + 576*a^4*b^2)^(1/2))/(8*(-d^2)^(1/2)) - ((4*exp(5*c + 5*d*x)*(9*a^2*b + 8*a^3))/(3*d
) - (8*a^2*b*exp(3*c + 3*d*x))/d - (8*a^2*b*exp(7*c + 7*d*x))/d + (2*a^2*b*exp(9*c + 9*d*x))/d + (2*a^2*b*exp(
c + d*x))/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) - (5*b^3*exp(- 3*c - 3*d*x))/(96*d) - (5*b^3*exp(3*c + 3*d*x))/(96*d) +
(b^3*exp(- 5*c - 5*d*x))/(160*d) + (b^3*exp(5*c + 5*d*x))/(160*d) - (exp(c + d*x)*(24*a^2*b + 5*a^3))/(8*d*(ex
p(2*c + 2*d*x) - 1)) - (a^3*exp(c + d*x))/(3*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)*(24*a + 5*b))/(16*d) - (exp(c + d*x)*(48*a^2*b - 5*a^3))/(12*d*(exp(4*c + 4*d*x) - 2*ex
p(2*c + 2*d*x) + 1)) - (22*a^3*exp(c + d*x))/(3*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)) - (16*a^3*exp(c + d*x))/(3*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)) + (b^2*exp(- c - d*x)*(24*a + 5*b))/(16*d)

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