3.1.24 \(\int \text {csch}(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [24]

Optimal. Leaf size=83 \[ -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {(3 a-2 b) b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 83, 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 = {3265, 398, 212} \begin {gather*} -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b^2 (3 a-2 b) \cosh ^3(c+d x)}{3 d}+\frac {b^3 \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]^2)^3,x]

[Out]

-((a^3*ArcTanh[Cosh[c + d*x]])/d) + (b*(3*a^2 - 3*a*b + b^2)*Cosh[c + d*x])/d + ((3*a - 2*b)*b^2*Cosh[c + d*x]
^3)/(3*d) + (b^3*Cosh[c + d*x]^5)/(5*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 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(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 - b*ff^2*x^2)^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 ^2(c+d x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^3}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-b \left (3 a^2-3 a b+b^2\right )-(3 a-2 b) b^2 x^2-b^3 x^4+\frac {a^3}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {(3 a-2 b) b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {(3 a-2 b) b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 83, normalized size = 1.00 \begin {gather*} \frac {30 b \left (24 a^2-18 a b+5 b^2\right ) \cosh (c+d x)+5 (12 a-5 b) b^2 \cosh (3 (c+d x))+3 \left (b^3 \cosh (5 (c+d x))+80 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{240 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(30*b*(24*a^2 - 18*a*b + 5*b^2)*Cosh[c + d*x] + 5*(12*a - 5*b)*b^2*Cosh[3*(c + d*x)] + 3*(b^3*Cosh[5*(c + d*x)
] + 80*a^3*Log[Tanh[(c + d*x)/2]]))/(240*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(223\) vs. \(2(79)=158\).
time = 1.11, size = 224, normalized size = 2.70

method	result	size
default	\(\frac {b^{3} \left (\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-3 b^{3} \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-6 a \,b^{2} \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{3} \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-2 a^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )+6 a^{2} b \arctanh \left ({\mathrm e}^{d x +c}\right )-6 a \,b^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )+2 b^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )}{d}\)	\(224\)
risch	\(\frac {b^{3} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} b^{3}}{96 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}-\frac {9 \,{\mathrm e}^{d x +c} a \,b^{2}}{8 d}+\frac {5 \,{\mathrm e}^{d x +c} b^{3}}{16 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}-\frac {9 \,{\mathrm e}^{-d x -c} a \,b^{2}}{8 d}+\frac {5 \,{\mathrm e}^{-d x -c} b^{3}}{16 d}+\frac {{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{96 d}+\frac {b^{3} {\mathrm e}^{-5 d x -5 c}}{160 d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(b^3*(1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))+3*a*b^2*(1/3*cosh(d*x+c)^3+co
sh(d*x+c)-2*arctanh(exp(d*x+c)))-3*b^3*(1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))+3*a^2*b*(cosh(d*x
+c)-2*arctanh(exp(d*x+c)))-6*a*b^2*(cosh(d*x+c)-2*arctanh(exp(d*x+c)))+3*b^3*(cosh(d*x+c)-2*arctanh(exp(d*x+c)
))-2*a^3*arctanh(exp(d*x+c))+6*a^2*b*arctanh(exp(d*x+c))-6*a*b^2*arctanh(exp(d*x+c))+2*b^3*arctanh(exp(d*x+c))
)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (79) = 158\).
time = 0.28, size = 193, normalized size = 2.33 \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 {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 {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \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)^2)^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) + 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) + 3/2*a^2*b*(e^(d*x + c)/d + e^(-d*x - c)/d) + a^3*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. 1128 vs. \(2 (79) = 158\).
time = 0.42, size = 1128, normalized size = 13.59 \begin {gather*} \frac {3 \, b^{3} \cosh \left (d x + c\right )^{10} + 30 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 3 \, b^{3} \sinh \left (d x + c\right )^{10} + 5 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{8} + 5 \, {\left (27 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 5 \, b^{3}\right )} \sinh \left (d x + c\right )^{8} + 40 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 30 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 10 \, {\left (63 \, b^{3} \cosh \left (d x + c\right )^{4} + 72 \, a^{2} b - 54 \, a b^{2} + 15 \, b^{3} + 14 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (189 \, b^{3} \cosh \left (d x + c\right )^{5} + 70 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 45 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 30 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (63 \, b^{3} \cosh \left (d x + c\right )^{6} + 35 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 72 \, a^{2} b - 54 \, a b^{2} + 15 \, b^{3} + 45 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 40 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{7} + 7 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 15 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b^{3} + 5 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 5 \, {\left (27 \, b^{3} \cosh \left (d x + c\right )^{8} + 28 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 90 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 12 \, a b^{2} - 5 \, b^{3} + 36 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 480 \, {\left (a^{3} \cosh \left (d x + c\right )^{5} + 5 \, a^{3} \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, a^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, a^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a^{3} \sinh \left (d x + c\right )^{5}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 480 \, {\left (a^{3} \cosh \left (d x + c\right )^{5} + 5 \, a^{3} \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, a^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, a^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a^{3} \sinh \left (d x + c\right )^{5}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 10 \, {\left (3 \, b^{3} \cosh \left (d x + c\right )^{9} + 4 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 18 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 12 \, {\left (24 \, a^{2} b - 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{480 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/480*(3*b^3*cosh(d*x + c)^10 + 30*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 3*b^3*sinh(d*x + c)^10 + 5*(12*a*b^2 -
5*b^3)*cosh(d*x + c)^8 + 5*(27*b^3*cosh(d*x + c)^2 + 12*a*b^2 - 5*b^3)*sinh(d*x + c)^8 + 40*(9*b^3*cosh(d*x +
c)^3 + (12*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^6 +
10*(63*b^3*cosh(d*x + c)^4 + 72*a^2*b - 54*a*b^2 + 15*b^3 + 14*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^6 + 4*(189*b^3*cosh(d*x + c)^5 + 70*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + 45*(24*a^2*b - 18*a*b^2 + 5*b^3)*c
osh(d*x + c))*sinh(d*x + c)^5 + 30*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 10*(63*b^3*cosh(d*x + c)^6
+ 35*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + 72*a^2*b - 54*a*b^2 + 15*b^3 + 45*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh
(d*x + c)^2)*sinh(d*x + c)^4 + 40*(9*b^3*cosh(d*x + c)^7 + 7*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 15*(24*a^2*b
- 18*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b^3
+ 5*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^2 + 5*(27*b^3*cosh(d*x + c)^8 + 28*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^6 + 9
0*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 12*a*b^2 - 5*b^3 + 36*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 - 480*(a^3*cosh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^4*sinh(d*x + c) + 10*a^3*cosh(d*x +
c)^3*sinh(d*x + c)^2 + 10*a^3*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*a^3*cosh(d*x + c)*sinh(d*x + c)^4 + a^3*sinh
(d*x + c)^5)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 480*(a^3*cosh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^4*sinh(d*
x + c) + 10*a^3*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*a^3*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*a^3*cosh(d*x + c)
*sinh(d*x + c)^4 + a^3*sinh(d*x + c)^5)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 10*(3*b^3*cosh(d*x + c)^9 + 4
*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^7 + 18*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 12*(24*a^2*b - 18*a*b
^2 + 5*b^3)*cosh(d*x + c)^3 + (12*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d
*x + c)^4*sinh(d*x + c) + 10*d*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*d*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*d*co
sh(d*x + c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5)

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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)**2)**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (79) = 158\).
time = 0.44, size = 202, normalized size = 2.43 \begin {gather*} \frac {3 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 720 \, a^{2} b e^{\left (d x + c\right )} - 540 \, a b^{2} e^{\left (d x + c\right )} + 150 \, b^{3} e^{\left (d x + c\right )} - 480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + {\left (720 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 540 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 25 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, 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)^2)^3,x, algorithm="giac")

[Out]

1/480*(3*b^3*e^(5*d*x + 5*c) + 60*a*b^2*e^(3*d*x + 3*c) - 25*b^3*e^(3*d*x + 3*c) + 720*a^2*b*e^(d*x + c) - 540
*a*b^2*e^(d*x + c) + 150*b^3*e^(d*x + c) - 480*a^3*log(e^(d*x + c) + 1) + 480*a^3*log(abs(e^(d*x + c) - 1)) +
(720*a^2*b*e^(4*d*x + 4*c) - 540*a*b^2*e^(4*d*x + 4*c) + 150*b^3*e^(4*d*x + 4*c) + 60*a*b^2*e^(2*d*x + 2*c) -
25*b^3*e^(2*d*x + 2*c) + 3*b^3)*e^(-5*d*x - 5*c))/d

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Mupad [B]
time = 0.30, size = 184, normalized size = 2.22 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-18\,a\,b^2+5\,b^3\right )}{16\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (24\,a^2\,b-18\,a\,b^2+5\,b^3\right )}{16\,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 {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (12\,a-5\,b\right )}{96\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (12\,a-5\,b\right )}{96\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^2)^3/sinh(c + d*x),x)

[Out]

(exp(c + d*x)*(24*a^2*b - 18*a*b^2 + 5*b^3))/(16*d) - (2*atan((a^3*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^6)^(1/2
)))*(a^6)^(1/2))/(-d^2)^(1/2) + (exp(- c - d*x)*(24*a^2*b - 18*a*b^2 + 5*b^3))/(16*d) + (b^3*exp(- 5*c - 5*d*x
))/(160*d) + (b^3*exp(5*c + 5*d*x))/(160*d) + (b^2*exp(- 3*c - 3*d*x)*(12*a - 5*b))/(96*d) + (b^2*exp(3*c + 3*
d*x)*(12*a - 5*b))/(96*d)

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