3.3.14 \(\int \text {csch}^9(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [214]

Optimal. Leaf size=171 \[ -\frac {a \left (35 a^2+144 a b+384 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d} \]

[Out]

-1/128*a*(35*a^2+144*a*b+384*b^2)*arctanh(cosh(d*x+c))/d-b^3*cosh(d*x+c)/d+1/3*b^3*cosh(d*x+c)^3/d+1/128*a^2*(
35*a+144*b)*coth(d*x+c)*csch(d*x+c)/d-1/192*a^2*(35*a+144*b)*coth(d*x+c)*csch(d*x+c)^3/d+7/48*a^3*coth(d*x+c)*
csch(d*x+c)^5/d-1/8*a^3*coth(d*x+c)*csch(d*x+c)^7/d

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Rubi [A]
time = 0.23, antiderivative size = 171, 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, 1167, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a \left (35 a^2+144 a b+384 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {b^3 \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^9*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/128*(a*(35*a^2 + 144*a*b + 384*b^2)*ArcTanh[Cosh[c + d*x]])/d - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^
3)/(3*d) + (a^2*(35*a + 144*b)*Coth[c + d*x]*Csch[c + d*x])/(128*d) - (a^2*(35*a + 144*b)*Coth[c + d*x]*Csch[c
+ d*x]^3)/(192*d) + (7*a^3*Coth[c + d*x]*Csch[c + d*x]^5)/(48*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^7)/(8*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}^9(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 )^5} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {-(a+2 b) \left (7 a^2+10 a b+4 b^2\right )+8 b \left (3 a^2+9 a b+5 b^2\right ) x^2-8 b^2 (9 a+10 b) x^4+8 b^2 (3 a+10 b) x^6-40 b^3 x^8+8 b^3 x^{10}}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \frac {35 a^3+144 a^2 b+144 a b^2+48 b^3-96 b^2 (3 a+2 b) x^2+144 b^2 (a+2 b) x^4-192 b^3 x^6+48 b^3 x^8}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {-3 \left (35 a^3+144 a^2 b+192 a b^2+64 b^3\right )+576 b^2 (a+b) x^2-576 b^3 x^4+192 b^3 x^6}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{192 d}\\ &=\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \frac {3 \left (35 a^3+144 a^2 b+384 a b^2+128 b^3\right )-768 b^3 x^2+384 b^3 x^4}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{384 d}\\ &=\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \left (384 b^3-384 b^3 x^2+\frac {3 \left (35 a^3+144 a^2 b+384 a b^2\right )}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{384 d}\\ &=-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\left (a \left (35 a^2+144 a b+384 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{128 d}\\ &=-\frac {a \left (35 a^2+144 a b+384 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}\\ \end {align*}

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Mathematica [A]
time = 1.27, size = 219, normalized size = 1.28 \begin {gather*} \frac {-4608 b^3 \cosh (c+d x)+512 b^3 \cosh (3 (c+d x))+a \left (12 a (35 a+144 b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-18 a (5 a+16 b) \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+20 a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )-3 a^2 \text {csch}^8\left (\frac {1}{2} (c+d x)\right )+48 \left (35 a^2+144 a b+384 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+12 a (35 a+144 b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+18 a (5 a+16 b) \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+20 a^2 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )+3 a^2 \text {sech}^8\left (\frac {1}{2} (c+d x)\right )\right )}{6144 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^9*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-4608*b^3*Cosh[c + d*x] + 512*b^3*Cosh[3*(c + d*x)] + a*(12*a*(35*a + 144*b)*Csch[(c + d*x)/2]^2 - 18*a*(5*a
+ 16*b)*Csch[(c + d*x)/2]^4 + 20*a^2*Csch[(c + d*x)/2]^6 - 3*a^2*Csch[(c + d*x)/2]^8 + 48*(35*a^2 + 144*a*b +
384*b^2)*Log[Tanh[(c + d*x)/2]] + 12*a*(35*a + 144*b)*Sech[(c + d*x)/2]^2 + 18*a*(5*a + 16*b)*Sech[(c + d*x)/2
]^4 + 20*a^2*Sech[(c + d*x)/2]^6 + 3*a^2*Sech[(c + d*x)/2]^8))/(6144*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(159)=318\).
time = 1.52, size = 375, normalized size = 2.19

method	result	size
risch	\(\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}-\frac {3 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 b^{3} {\mathrm e}^{-d x -c}}{8 d}+\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {a^{2} {\mathrm e}^{d x +c} \left (105 a \,{\mathrm e}^{14 d x +14 c}+432 b \,{\mathrm e}^{14 d x +14 c}-805 a \,{\mathrm e}^{12 d x +12 c}-3312 b \,{\mathrm e}^{12 d x +12 c}+2681 a \,{\mathrm e}^{10 d x +10 c}+7344 b \,{\mathrm e}^{10 d x +10 c}-5053 a \,{\mathrm e}^{8 d x +8 c}-4464 b \,{\mathrm e}^{8 d x +8 c}-5053 a \,{\mathrm e}^{6 d x +6 c}-4464 b \,{\mathrm e}^{6 d x +6 c}+2681 a \,{\mathrm e}^{4 d x +4 c}+7344 b \,{\mathrm e}^{4 d x +4 c}-805 a \,{\mathrm e}^{2 d x +2 c}-3312 b \,{\mathrm e}^{2 d x +2 c}+105 a +432 b \right )}{192 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{8}}+\frac {35 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{128 d}+\frac {9 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}-\frac {35 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{128 d}-\frac {9 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}\)	\(375\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^9*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/24/d*exp(3*d*x+3*c)*b^3-3/8*b^3/d*exp(d*x+c)-3/8*b^3/d*exp(-d*x-c)+1/24*b^3/d*exp(-3*d*x-3*c)+1/192*a^2*exp(
d*x+c)*(105*a*exp(14*d*x+14*c)+432*b*exp(14*d*x+14*c)-805*a*exp(12*d*x+12*c)-3312*b*exp(12*d*x+12*c)+2681*a*ex
p(10*d*x+10*c)+7344*b*exp(10*d*x+10*c)-5053*a*exp(8*d*x+8*c)-4464*b*exp(8*d*x+8*c)-5053*a*exp(6*d*x+6*c)-4464*
b*exp(6*d*x+6*c)+2681*a*exp(4*d*x+4*c)+7344*b*exp(4*d*x+4*c)-805*a*exp(2*d*x+2*c)-3312*b*exp(2*d*x+2*c)+105*a+
432*b)/d/(exp(2*d*x+2*c)-1)^8+35/128*a^3/d*ln(exp(d*x+c)-1)+9/8*a^2*b/d*ln(exp(d*x+c)-1)+3*a/d*ln(exp(d*x+c)-1
)*b^2-35/128*a^3/d*ln(exp(d*x+c)+1)-9/8*a^2*b/d*ln(exp(d*x+c)+1)-3*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. 463 vs. \(2 (159) = 318\).
time = 0.30, size = 463, normalized size = 2.71 \begin {gather*} \frac {1}{24} \, b^{3} {\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}{384} \, a^{3} {\left (\frac {105 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {105 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (105 \, e^{\left (-d x - c\right )} - 805 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2681 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5053 \, e^{\left (-7 \, d x - 7 \, c\right )} - 5053 \, e^{\left (-9 \, d x - 9 \, c\right )} + 2681 \, e^{\left (-11 \, d x - 11 \, c\right )} - 805 \, e^{\left (-13 \, d x - 13 \, c\right )} + 105 \, e^{\left (-15 \, d x - 15 \, c\right )}\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} - 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} - 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} - 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} - 1\right )}}\right )} - \frac {3}{8} \, a^{2} b {\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 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}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^9*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/24*b^3*(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) - 1/384*a^3*(105*log(e^
(-d*x - c) + 1)/d - 105*log(e^(-d*x - c) - 1)/d + 2*(105*e^(-d*x - c) - 805*e^(-3*d*x - 3*c) + 2681*e^(-5*d*x
- 5*c) - 5053*e^(-7*d*x - 7*c) - 5053*e^(-9*d*x - 9*c) + 2681*e^(-11*d*x - 11*c) - 805*e^(-13*d*x - 13*c) + 10
5*e^(-15*d*x - 15*c))/(d*(8*e^(-2*d*x - 2*c) - 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) - 70*e^(-8*d*x - 8*c)
+ 56*e^(-10*d*x - 10*c) - 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) - e^(-16*d*x - 16*c) - 1))) - 3/8*a^2*
b*(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*b^2*(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. 10848 vs. \(2 (159) = 318\).
time = 0.48, size = 10848, normalized size = 63.44 \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)^9*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/384*(16*b^3*cosh(d*x + c)^22 + 352*b^3*cosh(d*x + c)*sinh(d*x + c)^21 + 16*b^3*sinh(d*x + c)^22 - 272*b^3*co
sh(d*x + c)^20 + 16*(231*b^3*cosh(d*x + c)^2 - 17*b^3)*sinh(d*x + c)^20 + 320*(77*b^3*cosh(d*x + c)^3 - 17*b^3
*cosh(d*x + c))*sinh(d*x + c)^19 + 2*(105*a^3 + 432*a^2*b + 728*b^3)*cosh(d*x + c)^18 + 2*(58520*b^3*cosh(d*x
+ c)^4 - 25840*b^3*cosh(d*x + c)^2 + 105*a^3 + 432*a^2*b + 728*b^3)*sinh(d*x + c)^18 + 12*(35112*b^3*cosh(d*x
+ c)^5 - 25840*b^3*cosh(d*x + c)^3 + 3*(105*a^3 + 432*a^2*b + 728*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - 2*(80
5*a^3 + 3312*a^2*b + 1880*b^3)*cosh(d*x + c)^16 + 2*(596904*b^3*cosh(d*x + c)^6 - 658920*b^3*cosh(d*x + c)^4 -
805*a^3 - 3312*a^2*b - 1880*b^3 + 153*(105*a^3 + 432*a^2*b + 728*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 32*
(85272*b^3*cosh(d*x + c)^7 - 131784*b^3*cosh(d*x + c)^5 + 51*(105*a^3 + 432*a^2*b + 728*b^3)*cosh(d*x + c)^3 -
(805*a^3 + 3312*a^2*b + 1880*b^3)*cosh(d*x + c))*sinh(d*x + c)^15 + 2*(2681*a^3 + 7344*a^2*b + 2512*b^3)*cosh
(d*x + c)^ ...

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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)**9*(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. 335 vs. \(2 (159) = 318\).
time = 0.60, size = 335, normalized size = 1.96 \begin {gather*} \frac {32 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 384 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 3 \, {\left (35 \, a^{3} + 144 \, a^{2} b + 384 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 3 \, {\left (35 \, a^{3} + 144 \, a^{2} b + 384 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) + \frac {4 \, {\left (105 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 432 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 1540 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 6336 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 8176 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 29952 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 17856 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 46080 \, 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 )}^{4}}}{768 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^9*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/768*(32*b^3*(e^(d*x + c) + e^(-d*x - c))^3 - 384*b^3*(e^(d*x + c) + e^(-d*x - c)) - 3*(35*a^3 + 144*a^2*b +
384*a*b^2)*log(e^(d*x + c) + e^(-d*x - c) + 2) + 3*(35*a^3 + 144*a^2*b + 384*a*b^2)*log(e^(d*x + c) + e^(-d*x
- c) - 2) + 4*(105*a^3*(e^(d*x + c) + e^(-d*x - c))^7 + 432*a^2*b*(e^(d*x + c) + e^(-d*x - c))^7 - 1540*a^3*(e
^(d*x + c) + e^(-d*x - c))^5 - 6336*a^2*b*(e^(d*x + c) + e^(-d*x - c))^5 + 8176*a^3*(e^(d*x + c) + e^(-d*x - c
))^3 + 29952*a^2*b*(e^(d*x + c) + e^(-d*x - c))^3 - 17856*a^3*(e^(d*x + c) + e^(-d*x - c)) - 46080*a^2*b*(e^(d
*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4)^4)/d

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Mupad [B]
time = 1.12, size = 759, normalized size = 4.44 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b^3\,{\mathrm {e}}^{-c-d\,x}}{8\,d}-\frac {3\,b^3\,{\mathrm {e}}^{c+d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (35\,a^3\,\sqrt {-d^2}+384\,a\,b^2\,\sqrt {-d^2}+144\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {1225\,a^6+10080\,a^5\,b+47616\,a^4\,b^2+110592\,a^3\,b^3+147456\,a^2\,b^4}}\right )\,\sqrt {1225\,a^6+10080\,a^5\,b+47616\,a^4\,b^2+110592\,a^3\,b^3+147456\,a^2\,b^4}}{64\,\sqrt {-d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (35\,a^3+144\,b\,a^2\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+48\,b\,a^2\right )}{4\,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 {{\mathrm {e}}^{c+d\,x}\,\left (35\,a^3+144\,b\,a^2\right )}{96\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (432\,a^2\,b-7\,a^3\right )}{24\,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 {170\,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 {404\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 {112\,a^3\,{\mathrm {e}}^{c+d\,x}}{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 {32\,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 )} \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)^9,x)

[Out]

(b^3*exp(- 3*c - 3*d*x))/(24*d) - (3*b^3*exp(- c - d*x))/(8*d) - (3*b^3*exp(c + d*x))/(8*d) + (b^3*exp(3*c + 3
*d*x))/(24*d) - (atan((exp(d*x)*exp(c)*(35*a^3*(-d^2)^(1/2) + 384*a*b^2*(-d^2)^(1/2) + 144*a^2*b*(-d^2)^(1/2))
)/(d*(10080*a^5*b + 1225*a^6 + 147456*a^2*b^4 + 110592*a^3*b^3 + 47616*a^4*b^2)^(1/2)))*(10080*a^5*b + 1225*a^
6 + 147456*a^2*b^4 + 110592*a^3*b^3 + 47616*a^4*b^2)^(1/2))/(64*(-d^2)^(1/2)) + (exp(c + d*x)*(144*a^2*b + 35*
a^3))/(64*d*(exp(2*c + 2*d*x) - 1)) - (exp(c + d*x)*(48*a^2*b + a^3))/(4*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)) - (exp(c + d*x)*(144*a^2*b + 35*a^3))/(96*d*(exp(4*c + 4*d
*x) - 2*exp(2*c + 2*d*x) + 1)) - (exp(c + d*x)*(432*a^2*b - 7*a^3))/(24*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*
d*x) + exp(6*c + 6*d*x) - 1)) - (170*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)) - (404*a^3*exp(c + d*x))/(3*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)) - (112*a^3*exp(c + d*x))/(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)) - (32*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))

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