3.212 \(\int \text{csch}^5(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=142 \[ -\frac{3 a^2 (a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a^3 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^3 \coth (c+d x) \text{csch}(c+d x)}{8 d}+\frac{b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac{b^2 (3 a+b) \cosh (c+d x)}{d}+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d} \]
[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)
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Rubi [A] time = 0.272746, antiderivative size = 142, normalized size of antiderivative = 1.,
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 =
{3215, 1157, 1814, 1810, 206} \[ -\frac{3 a^2 (a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a^3 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^3 \coth (c+d x) \text{csch}(c+d x)}{8 d}+\frac{b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac{b^2 (3 a+b) \cosh (c+d x)}{d}+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d} \]
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 3215
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]
Rule 1157
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] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Rule 1814
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 1810
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \text{csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.382854, size = 173, normalized size = 1.22 \[ \frac{6720 a^2 b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-35 a^3 \text{csch}^4\left (\frac{1}{2} (c+d x)\right )+210 a^3 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+35 a^3 \text{sech}^4\left (\frac{1}{2} (c+d x)\right )+210 a^3 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )+840 a^3 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-35 b^2 (144 a+35 b) \cosh (c+d x)+35 b^2 (16 a+7 b) \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))}{2240 d} \]
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 [A] time = 0.046, size = 125, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (dx+c\right )}{8}} \right ){\rm coth} \left (dx+c\right )-{\frac{3\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{4}} \right ) -6\,{a}^{2}b{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +3\,a{b}^{2} \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{b}^{3} \left ( -{\frac{16}{35}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
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)
[Out]
1/d*(a^3*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp(d*x+c)))-6*a^2*b*arctanh(exp(d*x+c)
)+3*a*b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+b^3*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+
c)^2)*cosh(d*x+c))
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Maxima [B] time = 1.05803, size = 459, normalized size = 3.23 \begin{align*} -\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{align*}
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] time = 2.4282, size = 17403, normalized size = 122.56 \begin{align*} \text{result too large to display} \end{align*}
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)*cosh(d*x + c)^13 + 546*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d
*x + c)^11 - 770*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^9 - 420*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*
x + c)^7 + 84*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^5 - 5*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^3 + (
560*a*b^2 + 471*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*b^3 + 3*(385*b^3*cosh(d*x + c)^20 - 4370*b^3*cosh(d*x
+ c)^18 + 51*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^16 - 40*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^14 + 182*(560*a
^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^12 - 308*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^10 - 210*(880*
a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^8 + 56*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^6 - 5*(7280*a*b
^2 + 2519*b^3)*cosh(d*x + c)^4 - 23*b^3 + 2*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 1680*((a^
3 + 8*a^2*b)*cosh(d*x + c)^15 + 15*(a^3 + 8*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^14 + (a^3 + 8*a^2*b)*sinh(d*x +
c)^15 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c)^13 - (4*a^3 + 32*a^2*b - 105*(a^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*
x + c)^13 + 13*(35*(a^3 + 8*a^2*b)*cosh(d*x + c)^3 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^12 + 6*(a^
3 + 8*a^2*b)*cosh(d*x + c)^11 + 3*(455*(a^3 + 8*a^2*b)*cosh(d*x + c)^4 + 2*a^3 + 16*a^2*b - 104*(a^3 + 8*a^2*b
)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 11*(273*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 - 104*(a^3 + 8*a^2*b)*cosh(d*x +
c)^3 + 6*(a^3 + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^10 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c)^9 + (5005*(a^3 + 8
*a^2*b)*cosh(d*x + c)^6 - 2860*(a^3 + 8*a^2*b)*cosh(d*x + c)^4 - 4*a^3 - 32*a^2*b + 330*(a^3 + 8*a^2*b)*cosh(d
*x + c)^2)*sinh(d*x + c)^9 + 9*(715*(a^3 + 8*a^2*b)*cosh(d*x + c)^7 - 572*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 + 11
0*(a^3 + 8*a^2*b)*cosh(d*x + c)^3 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^8 + (a^3 + 8*a^2*b)*cosh(d*
x + c)^7 + (6435*(a^3 + 8*a^2*b)*cosh(d*x + c)^8 - 6864*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 + 1980*(a^3 + 8*a^2*b)
*cosh(d*x + c)^4 + a^3 + 8*a^2*b - 144*(a^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + (5005*(a^3 + 8*a^2*b
)*cosh(d*x + c)^9 - 6864*(a^3 + 8*a^2*b)*cosh(d*x + c)^7 + 2772*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 - 336*(a^3 + 8
*a^2*b)*cosh(d*x + c)^3 + 7*(a^3 + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^6 + 3*(1001*(a^3 + 8*a^2*b)*cosh(d*x
+ c)^10 - 1716*(a^3 + 8*a^2*b)*cosh(d*x + c)^8 + 924*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 - 168*(a^3 + 8*a^2*b)*cos
h(d*x + c)^4 + 7*(a^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + (1365*(a^3 + 8*a^2*b)*cosh(d*x + c)^11 - 2
860*(a^3 + 8*a^2*b)*cosh(d*x + c)^9 + 1980*(a^3 + 8*a^2*b)*cosh(d*x + c)^7 - 504*(a^3 + 8*a^2*b)*cosh(d*x + c)
^5 + 35*(a^3 + 8*a^2*b)*cosh(d*x + c)^3)*sinh(d*x + c)^4 + (455*(a^3 + 8*a^2*b)*cosh(d*x + c)^12 - 1144*(a^3 +
8*a^2*b)*cosh(d*x + c)^10 + 990*(a^3 + 8*a^2*b)*cosh(d*x + c)^8 - 336*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 + 35*(a
^3 + 8*a^2*b)*cosh(d*x + c)^4)*sinh(d*x + c)^3 + 3*(35*(a^3 + 8*a^2*b)*cosh(d*x + c)^13 - 104*(a^3 + 8*a^2*b)*
cosh(d*x + c)^11 + 110*(a^3 + 8*a^2*b)*cosh(d*x + c)^9 - 48*(a^3 + 8*a^2*b)*cosh(d*x + c)^7 + 7*(a^3 + 8*a^2*b
)*cosh(d*x + c)^5)*sinh(d*x + c)^2 + (15*(a^3 + 8*a^2*b)*cosh(d*x + c)^14 - 52*(a^3 + 8*a^2*b)*cosh(d*x + c)^1
2 + 66*(a^3 + 8*a^2*b)*cosh(d*x + c)^10 - 36*(a^3 + 8*a^2*b)*cosh(d*x + c)^8 + 7*(a^3 + 8*a^2*b)*cosh(d*x + c)
^6)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 1680*((a^3 + 8*a^2*b)*cosh(d*x + c)^15 + 15*(a^3 +
8*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^14 + (a^3 + 8*a^2*b)*sinh(d*x + c)^15 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c)^
13 - (4*a^3 + 32*a^2*b - 105*(a^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 13*(35*(a^3 + 8*a^2*b)*cosh(d
*x + c)^3 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^12 + 6*(a^3 + 8*a^2*b)*cosh(d*x + c)^11 + 3*(455*(a
^3 + 8*a^2*b)*cosh(d*x + c)^4 + 2*a^3 + 16*a^2*b - 104*(a^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 11*
(273*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 - 104*(a^3 + 8*a^2*b)*cosh(d*x + c)^3 + 6*(a^3 + 8*a^2*b)*cosh(d*x + c))*
sinh(d*x + c)^10 - 4*(a^3 + 8*a^2*b)*cosh(d*x + c)^9 + (5005*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 - 2860*(a^3 + 8*a
^2*b)*cosh(d*x + c)^4 - 4*a^3 - 32*a^2*b + 330*(a^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*(715*(a^3
+ 8*a^2*b)*cosh(d*x + c)^7 - 572*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 + 110*(a^3 + 8*a^2*b)*cosh(d*x + c)^3 - 4*(a^
3 + 8*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^8 + (a^3 + 8*a^2*b)*cosh(d*x + c)^7 + (6435*(a^3 + 8*a^2*b)*cosh(d*x
+ c)^8 - 6864*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 + 1980*(a^3 + 8*a^2*b)*cosh(d*x + c)^4 + a^3 + 8*a^2*b - 144*(a
^3 + 8*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + (5005*(a^3 + 8*a^2*b)*cosh(d*x + c)^9 - 6864*(a^3 + 8*a^2*b)*
cosh(d*x + c)^7 + 2772*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 - 336*(a^3 + 8*a^2*b)*cosh(d*x + c)^3 + 7*(a^3 + 8*a^2*
b)*cosh(d*x + c))*sinh(d*x + c)^6 + 3*(1001*(a^3 + 8*a^2*b)*cosh(d*x + c)^10 - 1716*(a^3 + 8*a^2*b)*cosh(d*x +
c)^8 + 924*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 - 168*(a^3 + 8*a^2*b)*cosh(d*x + c)^4 + 7*(a^3 + 8*a^2*b)*cosh(d*x
+ c)^2)*sinh(d*x + c)^5 + (1365*(a^3 + 8*a^2*b)*cosh(d*x + c)^11 - 2860*(a^3 + 8*a^2*b)*cosh(d*x + c)^9 + 198
0*(a^3 + 8*a^2*b)*cosh(d*x + c)^7 - 504*(a^3 + 8*a^2*b)*cosh(d*x + c)^5 + 35*(a^3 + 8*a^2*b)*cosh(d*x + c)^3)*
sinh(d*x + c)^4 + (455*(a^3 + 8*a^2*b)*cosh(d*x + c)^12 - 1144*(a^3 + 8*a^2*b)*cosh(d*x + c)^10 + 990*(a^3 + 8
*a^2*b)*cosh(d*x + c)^8 - 336*(a^3 + 8*a^2*b)*cosh(d*x + c)^6 + 35*(a^3 + 8*a^2*b)*cosh(d*x + c)^4)*sinh(d*x +
c)^3 + 3*(35*(a^3 + 8*a^2*b)*cosh(d*x + c)^13 - 104*(a^3 + 8*a^2*b)*cosh(d*x + c)^11 + 110*(a^3 + 8*a^2*b)*co
sh(d*x + c)^9 - 48*(a^3 + 8*a^2*b)*cosh(d*x + c)^7 + 7*(a^3 + 8*a^2*b)*cosh(d*x + c)^5)*sinh(d*x + c)^2 + (15*
(a^3 + 8*a^2*b)*cosh(d*x + c)^14 - 52*(a^3 + 8*a^2*b)*cosh(d*x + c)^12 + 66*(a^3 + 8*a^2*b)*cosh(d*x + c)^10 -
36*(a^3 + 8*a^2*b)*cosh(d*x + c)^8 + 7*(a^3 + 8*a^2*b)*cosh(d*x + c)^6)*sinh(d*x + c))*log(cosh(d*x + c) + si
nh(d*x + c) - 1) + 2*(55*b^3*cosh(d*x + c)^21 - 690*b^3*cosh(d*x + c)^19 + 9*(560*a*b^2 + 471*b^3)*cosh(d*x +
c)^17 - 8*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^15 + 42*(560*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^13 - 84
*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^11 - 70*(880*a^3 + 840*a*b^2 + 231*b^3)*cosh(d*x + c)^9 + 24*(5
60*a^3 + 3080*a*b^2 + 891*b^3)*cosh(d*x + c)^7 - 3*(7280*a*b^2 + 2519*b^3)*cosh(d*x + c)^5 - 69*b^3*cosh(d*x +
c) + 2*(560*a*b^2 + 471*b^3)*cosh(d*x + c)^3)*sinh(d*x + c))/(d*cosh(d*x + c)^15 + 15*d*cosh(d*x + c)*sinh(d*
x + c)^14 + d*sinh(d*x + c)^15 - 4*d*cosh(d*x + c)^13 + (105*d*cosh(d*x + c)^2 - 4*d)*sinh(d*x + c)^13 + 13*(3
5*d*cosh(d*x + c)^3 - 4*d*cosh(d*x + c))*sinh(d*x + c)^12 + 6*d*cosh(d*x + c)^11 + 3*(455*d*cosh(d*x + c)^4 -
104*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^11 + 11*(273*d*cosh(d*x + c)^5 - 104*d*cosh(d*x + c)^3 + 6*d*cosh(d
*x + c))*sinh(d*x + c)^10 - 4*d*cosh(d*x + c)^9 + (5005*d*cosh(d*x + c)^6 - 2860*d*cosh(d*x + c)^4 + 330*d*cos
h(d*x + c)^2 - 4*d)*sinh(d*x + c)^9 + 9*(715*d*cosh(d*x + c)^7 - 572*d*cosh(d*x + c)^5 + 110*d*cosh(d*x + c)^3
- 4*d*cosh(d*x + c))*sinh(d*x + c)^8 + d*cosh(d*x + c)^7 + (6435*d*cosh(d*x + c)^8 - 6864*d*cosh(d*x + c)^6 +
1980*d*cosh(d*x + c)^4 - 144*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^7 + (5005*d*cosh(d*x + c)^9 - 6864*d*cosh(d
*x + c)^7 + 2772*d*cosh(d*x + c)^5 - 336*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x + c)^6 + 3*(1001*d*co
sh(d*x + c)^10 - 1716*d*cosh(d*x + c)^8 + 924*d*cosh(d*x + c)^6 - 168*d*cosh(d*x + c)^4 + 7*d*cosh(d*x + c)^2)
*sinh(d*x + c)^5 + (1365*d*cosh(d*x + c)^11 - 2860*d*cosh(d*x + c)^9 + 1980*d*cosh(d*x + c)^7 - 504*d*cosh(d*x
+ c)^5 + 35*d*cosh(d*x + c)^3)*sinh(d*x + c)^4 + (455*d*cosh(d*x + c)^12 - 1144*d*cosh(d*x + c)^10 + 990*d*co
sh(d*x + c)^8 - 336*d*cosh(d*x + c)^6 + 35*d*cosh(d*x + c)^4)*sinh(d*x + c)^3 + 3*(35*d*cosh(d*x + c)^13 - 104
*d*cosh(d*x + c)^11 + 110*d*cosh(d*x + c)^9 - 48*d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)^5)*sinh(d*x + c)^2 + (1
5*d*cosh(d*x + c)^14 - 52*d*cosh(d*x + c)^12 + 66*d*cosh(d*x + c)^10 - 36*d*cosh(d*x + c)^8 + 7*d*cosh(d*x + c
)^6)*sinh(d*x + c))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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
________________________________________________________________________________________
Giac [B] time = 1.74344, size = 404, normalized size = 2.85 \begin{align*} -\frac{3 \,{\left (a^{3} + 8 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{16 \, d} + \frac{3 \,{\left (a^{3} + 8 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{16 \, d} + \frac{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 )}}{4 \,{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{2} d} + \frac{5 \, b^{3} d^{6}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 84 \, b^{3} d^{6}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 560 \, a b^{2} d^{6}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 560 \, b^{3} d^{6}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 6720 \, a b^{2} d^{6}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 2240 \, b^{3} d^{6}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{4480 \, d^{7}} \end{align*}
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]
-3/16*(a^3 + 8*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d + 3/16*(a^3 + 8*a^2*b)*log(e^(d*x + c) + e^(-d*x -
c) - 2)/d + 1/4*(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) + 1/4480*(5*b^3*d^6*(e^(d*x + c) + e^(-d*x - c))^7 - 84*b^3*d^6*(e^(d*x + c) + e^(-
d*x - c))^5 + 560*a*b^2*d^6*(e^(d*x + c) + e^(-d*x - c))^3 + 560*b^3*d^6*(e^(d*x + c) + e^(-d*x - c))^3 - 6720
*a*b^2*d^6*(e^(d*x + c) + e^(-d*x - c)) - 2240*b^3*d^6*(e^(d*x + c) + e^(-d*x - c)))/d^7