3.2.100 \(\int \text {csch}(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\) [200]

Optimal. Leaf size=92 \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b (2 a+b) \cosh (c+d x)}{d}+\frac {b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {3 b^2 \cosh ^5(c+d x)}{5 d}+\frac {b^2 \cosh ^7(c+d x)}{7 d} \]

[Out]

-a^2*arctanh(cosh(d*x+c))/d-b*(2*a+b)*cosh(d*x+c)/d+1/3*b*(2*a+3*b)*cosh(d*x+c)^3/d-3/5*b^2*cosh(d*x+c)^5/d+1/
7*b^2*cosh(d*x+c)^7/d

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Rubi [A]
time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3294, 1167, 212} \begin {gather*} -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {b (2 a+b) \cosh (c+d x)}{d}+\frac {b^2 \cosh ^7(c+d x)}{7 d}-\frac {3 b^2 \cosh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*ArcTanh[Cosh[c + d*x]])/d) - (b*(2*a + b)*Cosh[c + d*x])/d + (b*(2*a + 3*b)*Cosh[c + d*x]^3)/(3*d) - (3
*b^2*Cosh[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]^7)/(7*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 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}(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}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (b (2 a+b)-b (2 a+3 b) x^2+3 b^2 x^4-b^2 x^6+\frac {a^2}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \cosh (c+d x)}{d}+\frac {b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {3 b^2 \cosh ^5(c+d x)}{5 d}+\frac {b^2 \cosh ^7(c+d x)}{7 d}-\frac {a^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b (2 a+b) \cosh (c+d x)}{d}+\frac {b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {3 b^2 \cosh ^5(c+d x)}{5 d}+\frac {b^2 \cosh ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 146, normalized size = 1.59 \begin {gather*} -\frac {3 a b \cosh (c+d x)}{2 d}-\frac {35 b^2 \cosh (c+d x)}{64 d}+\frac {a b \cosh (3 (c+d x))}{6 d}+\frac {7 b^2 \cosh (3 (c+d x))}{64 d}-\frac {7 b^2 \cosh (5 (c+d x))}{320 d}+\frac {b^2 \cosh (7 (c+d x))}{448 d}-\frac {a^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*b*Cosh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x])/(64*d) + (a*b*Cosh[3*(c + d*x)])/(6*d) + (7*b^2*Cosh[3*(
c + d*x)])/(64*d) - (7*b^2*Cosh[5*(c + d*x)])/(320*d) + (b^2*Cosh[7*(c + d*x)])/(448*d) - (a^2*Log[Cosh[c/2 +
(d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(86)=172\).
time = 1.32, size = 234, normalized size = 2.54

method	result	size
risch	\(\frac {b^{2} {\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {{\mathrm e}^{3 d x +3 c} a b}{12 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{2}}{128 d}-\frac {3 a b \,{\mathrm e}^{d x +c}}{4 d}-\frac {35 \,{\mathrm e}^{d x +c} b^{2}}{128 d}-\frac {3 \,{\mathrm e}^{-d x -c} a b}{4 d}-\frac {35 \,{\mathrm e}^{-d x -c} b^{2}}{128 d}+\frac {{\mathrm e}^{-3 d x -3 c} a b}{12 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{128 d}-\frac {7 b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b^{2} {\mathrm e}^{-7 d x -7 c}}{896 d}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(229\)
default	\(\frac {b^{2} \left (\frac {\left (\cosh ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-4 b^{2} \left (\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+6 b^{2} \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-4 a b \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-4 b^{2} \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-2 a^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )-4 a b \arctanh \left ({\mathrm e}^{d x +c}\right )-2 b^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )}{d}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^2*(1/7*cosh(d*x+c)^7+1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))-4*b^2*(1/5*
cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))+2*a*b*(1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arct
anh(exp(d*x+c)))+6*b^2*(1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))-4*a*b*(cosh(d*x+c)-2*arctanh(exp(
d*x+c)))-4*b^2*(cosh(d*x+c)-2*arctanh(exp(d*x+c)))-2*a^2*arctanh(exp(d*x+c))-4*a*b*arctanh(exp(d*x+c))-2*b^2*a
rctanh(exp(d*x+c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (86) = 172\).
time = 0.27, size = 177, normalized size = 1.92 \begin {gather*} -\frac {1}{4480} \, b^{2} {\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}{12} \, a b {\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 {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/4480*b^2*((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) + 1/12*a*b*(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) + a^2*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1575 vs. \(2 (86) = 172\).
time = 0.43, size = 1575, normalized size = 17.12 \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)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/13440*(15*b^2*cosh(d*x + c)^14 + 210*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + 15*b^2*sinh(d*x + c)^14 - 147*b^2*
cosh(d*x + c)^12 + 21*(65*b^2*cosh(d*x + c)^2 - 7*b^2)*sinh(d*x + c)^12 + 84*(65*b^2*cosh(d*x + c)^3 - 21*b^2*
cosh(d*x + c))*sinh(d*x + c)^11 + 35*(32*a*b + 21*b^2)*cosh(d*x + c)^10 + 7*(2145*b^2*cosh(d*x + c)^4 - 1386*b
^2*cosh(d*x + c)^2 + 160*a*b + 105*b^2)*sinh(d*x + c)^10 + 70*(429*b^2*cosh(d*x + c)^5 - 462*b^2*cosh(d*x + c)
^3 + 5*(32*a*b + 21*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 105*(96*a*b + 35*b^2)*cosh(d*x + c)^8 + 105*(429*b^2
*cosh(d*x + c)^6 - 693*b^2*cosh(d*x + c)^4 + 15*(32*a*b + 21*b^2)*cosh(d*x + c)^2 - 96*a*b - 35*b^2)*sinh(d*x
+ c)^8 + 24*(2145*b^2*cosh(d*x + c)^7 - 4851*b^2*cosh(d*x + c)^5 + 175*(32*a*b + 21*b^2)*cosh(d*x + c)^3 - 35*
(96*a*b + 35*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 105*(96*a*b + 35*b^2)*cosh(d*x + c)^6 + 21*(2145*b^2*cosh(d
*x + c)^8 - 6468*b^2*cosh(d*x + c)^6 + 350*(32*a*b + 21*b^2)*cosh(d*x + c)^4 - 140*(96*a*b + 35*b^2)*cosh(d*x
+ c)^2 - 480*a*b - 175*b^2)*sinh(d*x + c)^6 + 42*(715*b^2*cosh(d*x + c)^9 - 2772*b^2*cosh(d*x + c)^7 + 210*(32
*a*b + 21*b^2)*cosh(d*x + c)^5 - 140*(96*a*b + 35*b^2)*cosh(d*x + c)^3 - 15*(96*a*b + 35*b^2)*cosh(d*x + c))*s
inh(d*x + c)^5 + 35*(32*a*b + 21*b^2)*cosh(d*x + c)^4 + 35*(429*b^2*cosh(d*x + c)^10 - 2079*b^2*cosh(d*x + c)^
8 + 210*(32*a*b + 21*b^2)*cosh(d*x + c)^6 - 210*(96*a*b + 35*b^2)*cosh(d*x + c)^4 - 45*(96*a*b + 35*b^2)*cosh(
d*x + c)^2 + 32*a*b + 21*b^2)*sinh(d*x + c)^4 - 147*b^2*cosh(d*x + c)^2 + 140*(39*b^2*cosh(d*x + c)^11 - 231*b
^2*cosh(d*x + c)^9 + 30*(32*a*b + 21*b^2)*cosh(d*x + c)^7 - 42*(96*a*b + 35*b^2)*cosh(d*x + c)^5 - 15*(96*a*b
+ 35*b^2)*cosh(d*x + c)^3 + (32*a*b + 21*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 21*(65*b^2*cosh(d*x + c)^12 - 4
62*b^2*cosh(d*x + c)^10 + 75*(32*a*b + 21*b^2)*cosh(d*x + c)^8 - 140*(96*a*b + 35*b^2)*cosh(d*x + c)^6 - 75*(9
6*a*b + 35*b^2)*cosh(d*x + c)^4 + 10*(32*a*b + 21*b^2)*cosh(d*x + c)^2 - 7*b^2)*sinh(d*x + c)^2 + 15*b^2 - 134
40*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)^6*sinh(d*x + c) + 21*a^2*cosh(d*x + c)^5*sinh(d*x + c)^2 + 35*a^
2*cosh(d*x + c)^4*sinh(d*x + c)^3 + 35*a^2*cosh(d*x + c)^3*sinh(d*x + c)^4 + 21*a^2*cosh(d*x + c)^2*sinh(d*x +
c)^5 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 13
440*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)^6*sinh(d*x + c) + 21*a^2*cosh(d*x + c)^5*sinh(d*x + c)^2 + 35*a
^2*cosh(d*x + c)^4*sinh(d*x + c)^3 + 35*a^2*cosh(d*x + c)^3*sinh(d*x + c)^4 + 21*a^2*cosh(d*x + c)^2*sinh(d*x
+ c)^5 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 1
4*(15*b^2*cosh(d*x + c)^13 - 126*b^2*cosh(d*x + c)^11 + 25*(32*a*b + 21*b^2)*cosh(d*x + c)^9 - 60*(96*a*b + 35
*b^2)*cosh(d*x + c)^7 - 45*(96*a*b + 35*b^2)*cosh(d*x + c)^5 + 10*(32*a*b + 21*b^2)*cosh(d*x + c)^3 - 21*b^2*c
osh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)^6*sinh(d*x + c) + 21*d*cosh(d*x + c)^5*sin
h(d*x + c)^2 + 35*d*cosh(d*x + c)^4*sinh(d*x + c)^3 + 35*d*cosh(d*x + c)^3*sinh(d*x + c)^4 + 21*d*cosh(d*x + c
)^2*sinh(d*x + c)^5 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)**4)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (86) = 172\).
time = 0.46, size = 196, normalized size = 2.13 \begin {gather*} \frac {15 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 147 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 1120 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 735 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 10080 \, a b e^{\left (d x + c\right )} - 3675 \, b^{2} e^{\left (d x + c\right )} - 13440 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 13440 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - {\left (10080 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 3675 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 1120 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 735 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 147 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{13440 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/13440*(15*b^2*e^(7*d*x + 7*c) - 147*b^2*e^(5*d*x + 5*c) + 1120*a*b*e^(3*d*x + 3*c) + 735*b^2*e^(3*d*x + 3*c)
- 10080*a*b*e^(d*x + c) - 3675*b^2*e^(d*x + c) - 13440*a^2*log(e^(d*x + c) + 1) + 13440*a^2*log(abs(e^(d*x +
c) - 1)) - (10080*a*b*e^(6*d*x + 6*c) + 3675*b^2*e^(6*d*x + 6*c) - 1120*a*b*e^(4*d*x + 4*c) - 735*b^2*e^(4*d*x
+ 4*c) + 147*b^2*e^(2*d*x + 2*c) - 15*b^2)*e^(-7*d*x - 7*c))/d

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Mupad [B]
time = 0.34, size = 198, normalized size = 2.15 \begin {gather*} \frac {b^2\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{896\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (35\,b^2+96\,a\,b\right )}{128\,d}-\frac {7\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{640\,d}-\frac {7\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}}{640\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (35\,b^2+96\,a\,b\right )}{128\,d}+\frac {b^2\,{\mathrm {e}}^{7\,c+7\,d\,x}}{896\,d}+\frac {b\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (32\,a+21\,b\right )}{384\,d}+\frac {b\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (32\,a+21\,b\right )}{384\,d} \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),x)

[Out]

(b^2*exp(- 7*c - 7*d*x))/(896*d) - (2*atan((a^2*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^4)^(1/2)))*(a^4)^(1/2))/(-
d^2)^(1/2) - (exp(- c - d*x)*(96*a*b + 35*b^2))/(128*d) - (7*b^2*exp(- 5*c - 5*d*x))/(640*d) - (7*b^2*exp(5*c
+ 5*d*x))/(640*d) - (exp(c + d*x)*(96*a*b + 35*b^2))/(128*d) + (b^2*exp(7*c + 7*d*x))/(896*d) + (b*exp(- 3*c -
3*d*x)*(32*a + 21*b))/(384*d) + (b*exp(3*c + 3*d*x)*(32*a + 21*b))/(384*d)

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