3.204 \(\int \text{csch}^5(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\)
Optimal. Leaf size=101 \[ -\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{a (3 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{b^2 \cosh (c+d x)}{d} \]
[Out]
-(a*(3*a + 16*b)*ArcTanh[Cosh[c + d*x]])/(8*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)
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Rubi [A] time = 0.151209, antiderivative size = 101, 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, 1153, 206} \[ -\frac{a^2 \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a^2 \coth (c+d x) \text{csch}(c+d x)}{8 d}-\frac{a (3 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{b^2 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^2,x]
[Out]
-(a*(3*a + 16*b)*ArcTanh[Cosh[c + d*x]])/(8*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 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 1153
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 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 )^2 \, dx &=-\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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)) \operatorname{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*}
Mathematica [A] time = 0.0434006, size = 186, normalized size = 1.84 \[ -\frac{a^2 \text{csch}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a^2 \text{sech}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{3 a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}+\frac{2 a b \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{2 a b \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d} \]
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 [A] time = 0.057, size = 79, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \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 ) -4\,ab{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +{b}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \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)^2,x)
[Out]
1/d*(a^2*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp(d*x+c)))-4*a*b*arctanh(exp(d*x+c))+
b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))
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Maxima [B] time = 1.12611, size = 316, normalized size = 3.13 \begin{align*} \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{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)^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] time = 2.0716, size = 8741, normalized size = 86.54 \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)^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 + 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))
*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*b^2*cosh(d*x + c)^13 - 78*b^2*cosh(d*x + c)^11 + 15*(6*a^2 + 11
*b^2)*cosh(d*x + c)^9 - 12*(22*a^2 + 7*b^2)*cosh(d*x + c)^7 - 9*(22*a^2 + 7*b^2)*cosh(d*x + c)^5 + 6*(6*a^2 +
11*b^2)*cosh(d*x + c)^3 - 13*b^2*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^11 + 11*d*cosh(d*x + c)*sinh(d
*x + c)^10 + d*sinh(d*x + c)^11 - 4*d*cosh(d*x + c)^9 + (55*d*cosh(d*x + c)^2 - 4*d)*sinh(d*x + c)^9 + 3*(55*d
*cosh(d*x + c)^3 - 12*d*cosh(d*x + c))*sinh(d*x + c)^8 + 6*d*cosh(d*x + c)^7 + 6*(55*d*cosh(d*x + c)^4 - 24*d*
cosh(d*x + c)^2 + d)*sinh(d*x + c)^7 + 42*(11*d*cosh(d*x + c)^5 - 8*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(
d*x + c)^6 - 4*d*cosh(d*x + c)^5 + 2*(231*d*cosh(d*x + c)^6 - 252*d*cosh(d*x + c)^4 + 63*d*cosh(d*x + c)^2 - 2
*d)*sinh(d*x + c)^5 + 2*(165*d*cosh(d*x + c)^7 - 252*d*cosh(d*x + c)^5 + 105*d*cosh(d*x + c)^3 - 10*d*cosh(d*x
+ c))*sinh(d*x + c)^4 + d*cosh(d*x + c)^3 + (165*d*cosh(d*x + c)^8 - 336*d*cosh(d*x + c)^6 + 210*d*cosh(d*x +
c)^4 - 40*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + (55*d*cosh(d*x + c)^9 - 144*d*cosh(d*x + c)^7 + 126*d*cosh
(d*x + c)^5 - 40*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (11*d*cosh(d*x + c)^10 - 36*d*cosh(d
*x + c)^8 + 42*d*cosh(d*x + c)^6 - 20*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*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)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.31755, size = 262, normalized size = 2.59 \begin{align*} -\frac{{\left (3 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{16 \, d} + \frac{{\left (3 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{16 \, d} + \frac{b^{2} d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, b^{2} d^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{24 \, d^{3}} + \frac{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 )}}{4 \,{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{2} d} \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)^2,x, algorithm="giac")
[Out]
-1/16*(3*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d + 1/16*(3*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x
- c) - 2)/d + 1/24*(b^2*d^2*(e^(d*x + c) + e^(-d*x - c))^3 - 12*b^2*d^2*(e^(d*x + c) + e^(-d*x - c)))/d^3 + 1
/4*(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)