3.4.99 \(\int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [399]

Optimal. Leaf size=184 \[ \frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d} \]

[Out]

1/8*(8*a^4+4*a^2*b^2-b^4)*x/b^5-1/3*a*(3*a^2+b^2)*cosh(d*x+c)/b^4/d+1/8*(4*a^2+b^2)*cosh(d*x+c)*sinh(d*x+c)/b^
3/d-1/3*a*cosh(d*x+c)*sinh(d*x+c)^2/b^2/d+1/4*cosh(d*x+c)*sinh(d*x+c)^3/b/d+2*a^3*arctanh((b-a*tanh(1/2*d*x+1/
2*c))/(a^2+b^2)^(1/2))*(a^2+b^2)^(1/2)/b^5/d

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Rubi [A]
time = 0.55, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3129, 3128, 3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 b^3 d}+\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \sinh ^2(c+d x) \cosh (c+d x)}{3 b^2 d}+\frac {\sinh ^3(c+d x) \cosh (c+d x)}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

((8*a^4 + 4*a^2*b^2 - b^4)*x)/(8*b^5) + (2*a^3*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^
2]])/(b^5*d) - (a*(3*a^2 + b^2)*Cosh[c + d*x])/(3*b^4*d) + ((4*a^2 + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^3*
d) - (a*Cosh[c + d*x]*Sinh[c + d*x]^2)/(3*b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*b*d)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3129

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)
*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^
(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e +
f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a,
0] && NeQ[c, 0])))

Rubi steps

\begin {align*} \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\int \frac {\sinh ^3(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx\\ &=\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\int \frac {\sinh ^2(c+d x) \left (-3 a+b \sinh (c+d x)-4 a \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{4 b}\\ &=-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\int \frac {\sinh (c+d x) \left (8 a^2-a b \sinh (c+d x)+3 \left (4 a^2+b^2\right ) \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{12 b^2}\\ &=\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\int \frac {-3 a \left (4 a^2+b^2\right )+b \left (4 a^2-3 b^2\right ) \sinh (c+d x)-8 a \left (3 a^2+b^2\right ) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{24 b^3}\\ &=-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {i \int \frac {3 i a b \left (4 a^2+b^2\right )-3 i \left (8 a^4+4 a^2 b^2-b^4\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{24 b^4}\\ &=\frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{b^5}\\ &=\frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}+\frac {\left (2 i a^3 \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b^5 d}\\ &=\frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}-\frac {\left (4 i a^3 \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b^5 d}\\ &=\frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d}\\ \end {align*}

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Mathematica [A]
time = 1.46, size = 153, normalized size = 0.83 \begin {gather*} \frac {-24 a b \left (4 a^2+b^2\right ) \cosh (c+d x)-8 a b^3 \cosh (3 (c+d x))+3 \left (4 \left (8 a^4+4 a^2 b^2-b^4\right ) (c+d x)+64 a^3 \sqrt {-a^2-b^2} \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+8 a^2 b^2 \sinh (2 (c+d x))+b^4 \sinh (4 (c+d x))\right )}{96 b^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a*b*(4*a^2 + b^2)*Cosh[c + d*x] - 8*a*b^3*Cosh[3*(c + d*x)] + 3*(4*(8*a^4 + 4*a^2*b^2 - b^4)*(c + d*x) +
64*a^3*Sqrt[-a^2 - b^2]*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]] + 8*a^2*b^2*Sinh[2*(c + d*x)] + b^4
*Sinh[4*(c + d*x)]))/(96*b^5*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(354\) vs. \(2(169)=338\).
time = 1.18, size = 355, normalized size = 1.93

method	result	size
risch	\(\frac {x \,a^{4}}{b^{5}}+\frac {x \,a^{2}}{2 b^{3}}-\frac {x}{8 b}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}-\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 b^{2} d}+\frac {a^{2} {\mathrm e}^{2 d x +2 c}}{8 b^{3} d}-\frac {a^{3} {\mathrm e}^{d x +c}}{2 b^{4} d}-\frac {a \,{\mathrm e}^{d x +c}}{8 b^{2} d}-\frac {a^{3} {\mathrm e}^{-d x -c}}{2 b^{4} d}-\frac {a \,{\mathrm e}^{-d x -c}}{8 b^{2} d}-\frac {a^{2} {\mathrm e}^{-2 d x -2 c}}{8 b^{3} d}-\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 b^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}+\frac {\sqrt {a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{5}}-\frac {\sqrt {a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{5}}\)	\(293\)
derivativedivides	\(\frac {\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{5}}-\frac {-8 a^{3}-4 a^{2} b -4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-3 b +2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{4}+4 a^{2} b^{2}-b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{5}}-\frac {8 a^{3}-4 a^{2} b +4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{3} \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{5}}}{d}\)	\(355\)
default	\(\frac {\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{5}}-\frac {-8 a^{3}-4 a^{2} b -4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-3 b +2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{4}+4 a^{2} b^{2}-b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{5}}-\frac {8 a^{3}-4 a^{2} b +4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{3} \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{5}}}{d}\)	\(355\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/4/b/(tanh(1/2*d*x+1/2*c)-1)^4-1/6*(-3*b-2*a)/b^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(-4*a^2-4*a*b-3*b^2)/b^3
/(tanh(1/2*d*x+1/2*c)-1)^2+1/8/b^5*(-8*a^4-4*a^2*b^2+b^4)*ln(tanh(1/2*d*x+1/2*c)-1)-1/8*(-8*a^3-4*a^2*b-4*a*b^
2-b^3)/b^4/(tanh(1/2*d*x+1/2*c)-1)-1/4/b/(tanh(1/2*d*x+1/2*c)+1)^4-1/6*(-3*b+2*a)/b^2/(tanh(1/2*d*x+1/2*c)+1)^
3-1/8*(4*a^2-4*a*b+3*b^2)/b^3/(tanh(1/2*d*x+1/2*c)+1)^2+1/8*(8*a^4+4*a^2*b^2-b^4)/b^5*ln(tanh(1/2*d*x+1/2*c)+1
)-1/8*(8*a^3-4*a^2*b+4*a*b^2-b^3)/b^4/(tanh(1/2*d*x+1/2*c)+1)-2*a^3*(a^2+b^2)^(1/2)/b^5*arctanh(1/2*(2*a*tanh(
1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2)))

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Maxima [A]
time = 0.50, size = 257, normalized size = 1.40 \begin {gather*} -\frac {\sqrt {a^{2} + b^{2}} a^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{5} d} - \frac {{\left (8 \, a b^{2} e^{\left (-d x - c\right )} - 24 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, b^{3} + 24 \, {\left (4 \, a^{3} + a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{8 \, b^{5} d} - \frac {24 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} + 8 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + 24 \, {\left (4 \, a^{3} + a b^{2}\right )} e^{\left (-d x - c\right )}}{192 \, b^{4} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-sqrt(a^2 + b^2)*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^5*d
) - 1/192*(8*a*b^2*e^(-d*x - c) - 24*a^2*b*e^(-2*d*x - 2*c) - 3*b^3 + 24*(4*a^3 + a*b^2)*e^(-3*d*x - 3*c))*e^(
4*d*x + 4*c)/(b^4*d) + 1/8*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5*d) - 1/192*(24*a^2*b*e^(-2*d*x - 2*c) + 8*
a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(-4*d*x - 4*c) + 24*(4*a^3 + a*b^2)*e^(-d*x - c))/(b^4*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (171) = 342\).
time = 0.41, size = 1134, normalized size = 6.16 \begin {gather*} \frac {3 \, b^{4} \cosh \left (d x + c\right )^{8} + 3 \, b^{4} \sinh \left (d x + c\right )^{8} - 8 \, a b^{3} \cosh \left (d x + c\right )^{7} + 24 \, a^{2} b^{2} \cosh \left (d x + c\right )^{6} + 8 \, {\left (3 \, b^{4} \cosh \left (d x + c\right ) - a b^{3}\right )} \sinh \left (d x + c\right )^{7} + 24 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} d x \cosh \left (d x + c\right )^{4} + 4 \, {\left (21 \, b^{4} \cosh \left (d x + c\right )^{2} - 14 \, a b^{3} \cosh \left (d x + c\right ) + 6 \, a^{2} b^{2}\right )} \sinh \left (d x + c\right )^{6} - 24 \, a^{2} b^{2} \cosh \left (d x + c\right )^{2} - 24 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{5} + 24 \, {\left (7 \, b^{4} \cosh \left (d x + c\right )^{3} - 7 \, a b^{3} \cosh \left (d x + c\right )^{2} + 6 \, a^{2} b^{2} \cosh \left (d x + c\right ) - 4 \, a^{3} b - a b^{3}\right )} \sinh \left (d x + c\right )^{5} - 8 \, a b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (105 \, b^{4} \cosh \left (d x + c\right )^{4} - 140 \, a b^{3} \cosh \left (d x + c\right )^{3} + 180 \, a^{2} b^{2} \cosh \left (d x + c\right )^{2} + 12 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} d x - 60 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 3 \, b^{4} - 24 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{3} + 8 \, {\left (21 \, b^{4} \cosh \left (d x + c\right )^{5} - 35 \, a b^{3} \cosh \left (d x + c\right )^{4} + 60 \, a^{2} b^{2} \cosh \left (d x + c\right )^{3} - 12 \, a^{3} b - 3 \, a b^{3} + 12 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} d x \cosh \left (d x + c\right ) - 30 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 12 \, {\left (7 \, b^{4} \cosh \left (d x + c\right )^{6} - 14 \, a b^{3} \cosh \left (d x + c\right )^{5} + 30 \, a^{2} b^{2} \cosh \left (d x + c\right )^{4} + 12 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} d x \cosh \left (d x + c\right )^{2} - 2 \, a^{2} b^{2} - 20 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{3} - 6 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 192 \, {\left (a^{3} \cosh \left (d x + c\right )^{4} + 4 \, a^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, a^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} \sinh \left (d x + c\right )^{4}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + 8 \, {\left (3 \, b^{4} \cosh \left (d x + c\right )^{7} - 7 \, a b^{3} \cosh \left (d x + c\right )^{6} + 18 \, a^{2} b^{2} \cosh \left (d x + c\right )^{5} + 12 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} d x \cosh \left (d x + c\right )^{3} - 6 \, a^{2} b^{2} \cosh \left (d x + c\right ) - 15 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{4} - a b^{3} - 9 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{192 \, {\left (b^{5} d \cosh \left (d x + c\right )^{4} + 4 \, b^{5} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b^{5} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b^{5} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{5} d \sinh \left (d x + c\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*b^4*cosh(d*x + c)^8 + 3*b^4*sinh(d*x + c)^8 - 8*a*b^3*cosh(d*x + c)^7 + 24*a^2*b^2*cosh(d*x + c)^6 +
8*(3*b^4*cosh(d*x + c) - a*b^3)*sinh(d*x + c)^7 + 24*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c)^4 + 4*(21*b^4
*cosh(d*x + c)^2 - 14*a*b^3*cosh(d*x + c) + 6*a^2*b^2)*sinh(d*x + c)^6 - 24*a^2*b^2*cosh(d*x + c)^2 - 24*(4*a^
3*b + a*b^3)*cosh(d*x + c)^5 + 24*(7*b^4*cosh(d*x + c)^3 - 7*a*b^3*cosh(d*x + c)^2 + 6*a^2*b^2*cosh(d*x + c) -
4*a^3*b - a*b^3)*sinh(d*x + c)^5 - 8*a*b^3*cosh(d*x + c) + 2*(105*b^4*cosh(d*x + c)^4 - 140*a*b^3*cosh(d*x +
c)^3 + 180*a^2*b^2*cosh(d*x + c)^2 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x - 60*(4*a^3*b + a*b^3)*cosh(d*x + c))*si
nh(d*x + c)^4 - 3*b^4 - 24*(4*a^3*b + a*b^3)*cosh(d*x + c)^3 + 8*(21*b^4*cosh(d*x + c)^5 - 35*a*b^3*cosh(d*x +
c)^4 + 60*a^2*b^2*cosh(d*x + c)^3 - 12*a^3*b - 3*a*b^3 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c) - 30*
(4*a^3*b + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 12*(7*b^4*cosh(d*x + c)^6 - 14*a*b^3*cosh(d*x + c)^5 + 30
*a^2*b^2*cosh(d*x + c)^4 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c)^2 - 2*a^2*b^2 - 20*(4*a^3*b + a*b^3)
*cosh(d*x + c)^3 - 6*(4*a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 192*(a^3*cosh(d*x + c)^4 + 4*a^3*cosh(
d*x + c)^3*sinh(d*x + c) + 6*a^3*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*s
inh(d*x + c)^4)*sqrt(a^2 + b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + 2*a^2 +
b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/
(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) - b)) + 8*(
3*b^4*cosh(d*x + c)^7 - 7*a*b^3*cosh(d*x + c)^6 + 18*a^2*b^2*cosh(d*x + c)^5 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*
x*cosh(d*x + c)^3 - 6*a^2*b^2*cosh(d*x + c) - 15*(4*a^3*b + a*b^3)*cosh(d*x + c)^4 - a*b^3 - 9*(4*a^3*b + a*b^
3)*cosh(d*x + c)^2)*sinh(d*x + c))/(b^5*d*cosh(d*x + c)^4 + 4*b^5*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*d*co
sh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*d*sinh(d*x + c)^4)

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

[Out]

Timed out

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Giac [A]
time = 0.46, size = 258, normalized size = 1.40 \begin {gather*} \frac {\frac {24 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{b^{5}} + \frac {3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 96 \, a^{3} e^{\left (d x + c\right )} - 24 \, a b^{2} e^{\left (d x + c\right )}}{b^{4}} - \frac {{\left (24 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{3} e^{\left (d x + c\right )} + 3 \, b^{4} + 24 \, {\left (4 \, a^{3} b + a b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{b^{5}} - \frac {192 \, {\left (a^{5} + a^{3} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{5}}}{192 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

1/192*(24*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/b^5 + (3*b^3*e^(4*d*x + 4*c) - 8*a*b^2*e^(3*d*x + 3*c) + 24*a^2*
b*e^(2*d*x + 2*c) - 96*a^3*e^(d*x + c) - 24*a*b^2*e^(d*x + c))/b^4 - (24*a^2*b^2*e^(2*d*x + 2*c) + 8*a*b^3*e^(
d*x + c) + 3*b^4 + 24*(4*a^3*b + a*b^3)*e^(3*d*x + 3*c))*e^(-4*d*x - 4*c)/b^5 - 192*(a^5 + a^3*b^2)*log(abs(2*
b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^(d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^5)
)/d

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Mupad [B]
time = 0.73, size = 330, normalized size = 1.79 \begin {gather*} \frac {x\,\left (8\,a^4+4\,a^2\,b^2-b^4\right )}{8\,b^5}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}-\frac {a\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b^2\,d}-\frac {a\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b^2\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^3+a\,b^2\right )}{8\,b^4\,d}-\frac {a^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^3\,d}+\frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^3\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^3+a\,b^2\right )}{8\,b^4\,d}-\frac {a^3\,\ln \left (\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^6}-\frac {2\,a^3\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^6}\right )\,\sqrt {a^2+b^2}}{b^5\,d}+\frac {a^3\,\ln \left (\frac {2\,a^3\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^6}+\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^6}\right )\,\sqrt {a^2+b^2}}{b^5\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(8*a^4 - b^4 + 4*a^2*b^2))/(8*b^5) - exp(- 4*c - 4*d*x)/(64*b*d) + exp(4*c + 4*d*x)/(64*b*d) - (a*exp(- 3*c
- 3*d*x))/(24*b^2*d) - (a*exp(3*c + 3*d*x))/(24*b^2*d) - (exp(c + d*x)*(a*b^2 + 4*a^3))/(8*b^4*d) - (a^2*exp(
- 2*c - 2*d*x))/(8*b^3*d) + (a^2*exp(2*c + 2*d*x))/(8*b^3*d) - (exp(- c - d*x)*(a*b^2 + 4*a^3))/(8*b^4*d) - (a
^3*log((2*a^3*exp(c + d*x)*(a^2 + b^2))/b^6 - (2*a^3*(a^2 + b^2)^(1/2)*(b - a*exp(c + d*x)))/b^6)*(a^2 + b^2)^
(1/2))/(b^5*d) + (a^3*log((2*a^3*(a^2 + b^2)^(1/2)*(b - a*exp(c + d*x)))/b^6 + (2*a^3*exp(c + d*x)*(a^2 + b^2)
)/b^6)*(a^2 + b^2)^(1/2))/(b^5*d)

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