3.4.34 \(\int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [334]

Optimal. Leaf size=330 \[ \frac {a (e+f x)^3}{3 b^2 f}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {2 a f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 a f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {2 a f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 a f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d} \]

[Out]

1/3*a*(f*x+e)^3/b^2/f-2*f*(f*x+e)*cosh(d*x+c)/b/d^2-a*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/d-a
*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d-2*a*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/
2)))/b^2/d^2-2*a*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d^2+2*a*f^2*polylog(3,-b*exp(d*x+c
)/(a-(a^2+b^2)^(1/2)))/b^2/d^3+2*a*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/d^3+2*f^2*sinh(d*x+c)/
b/d^3+(f*x+e)^2*sinh(d*x+c)/b/d

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5698, 3377, 2717, 5680, 2221, 2611, 2320, 6724} \begin {gather*} \frac {2 a f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 a f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {a (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d}+\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \sinh (c+d x)}{b d^3}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}+\frac {(e+f x)^2 \sinh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(e + f*x)^3)/(3*b^2*f) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2])])/(b^2*d) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) - (2*a*f
*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (2*a*f*(e + f*x)*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))
])/(b^2*d^3) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^3) + (2*f^2*Sinh[c + d*x]
)/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*d)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5698

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \cosh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {a \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b}-\frac {a \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b}-\frac {(2 f) \int (e+f x) \sinh (c+d x) \, dx}{b d}\\ &=\frac {a (e+f x)^3}{3 b^2 f}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}+\frac {(2 a f) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 d}+\frac {(2 a f) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 d}+\frac {\left (2 f^2\right ) \int \cosh (c+d x) \, dx}{b d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}+\frac {\left (2 a f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^2}+\frac {\left (2 a f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}+\frac {\left (2 a f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}+\frac {\left (2 a f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}\\ &=\frac {a (e+f x)^3}{3 b^2 f}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 a f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {2 a f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 a f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1301\) vs. \(2(330)=660\).
time = 13.17, size = 1301, normalized size = 3.94 \begin {gather*} \frac {1}{2} \left (\frac {2 a \left (6 e^2 e^{2 c} x+6 e e^{2 c} f x^2+2 e^{2 c} f^2 x^3+\frac {6 a \sqrt {a^2+b^2} e^2 \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {3 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}-\frac {3 e^2 e^{2 c} \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}+\frac {6 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 e e^{2 c} f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {3 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {3 e^{2 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {6 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 e e^{2 c} f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {3 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {3 e^{2 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}-\frac {6 f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}\right )}{3 b^2 \left (-1+e^{2 c}\right )}-\frac {a x \left (3 e^2+3 e f x+f^2 x^2\right ) \cosh (c) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right )}{3 b^2}+\frac {2 \cosh (d x) \left (-2 d e f \cosh (c)-2 d f^2 x \cosh (c)+d^2 e^2 \sinh (c)+2 f^2 \sinh (c)+2 d^2 e f x \sinh (c)+d^2 f^2 x^2 \sinh (c)\right )}{b d^3}+\frac {2 \left (d^2 e^2 \cosh (c)+2 f^2 \cosh (c)+2 d^2 e f x \cosh (c)+d^2 f^2 x^2 \cosh (c)-2 d e f \sinh (c)-2 d f^2 x \sinh (c)\right ) \sinh (d x)}{b d^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c +
d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c
+ d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/S
qrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sq
rt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^
(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a
^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d +
(3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*
E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2
+ b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (
3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^
(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c +
d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x)
)/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)
*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 -
(6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3,
-((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*b^2*(-1 + E^(2*c))) - (a*x*(3*e^2 + 3*e*f*
x + f^2*x^2)*Cosh[c]*Csch[c/2]*Sech[c/2])/(3*b^2) + (2*Cosh[d*x]*(-2*d*e*f*Cosh[c] - 2*d*f^2*x*Cosh[c] + d^2*e
^2*Sinh[c] + 2*f^2*Sinh[c] + 2*d^2*e*f*x*Sinh[c] + d^2*f^2*x^2*Sinh[c]))/(b*d^3) + (2*(d^2*e^2*Cosh[c] + 2*f^2
*Cosh[c] + 2*d^2*e*f*x*Cosh[c] + d^2*f^2*x^2*Cosh[c] - 2*d*e*f*Sinh[c] - 2*d*f^2*x*Sinh[c])*Sinh[d*x])/(b*d^3)
)/2

________________________________________________________________________________________

Maple [F]
time = 0.90, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d) + 2*a*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x
- 2*c) - b)/(b^2*d))*e^2 - 1/6*(2*a*d^3*f^2*x^3*e^c + 6*a*d^3*f*x^2*e^(c + 1) - 3*(b*d^2*f^2*x^2*e^(2*c) + 2*
b*f^2*e^(2*c) - 2*b*d*f*e^(2*c + 1) - 2*(b*d*f^2*e^(2*c) - b*d^2*f*e^(2*c + 1))*x)*e^(d*x) + 3*(b*d^2*f^2*x^2
+ 2*b*d*f*e + 2*b*f^2 + 2*(b*d^2*f*e + b*d*f^2)*x)*e^(-d*x))*e^(-c)/(b^2*d^3) + integrate(-2*(a*b*f^2*x^2 + 2*
a*b*f*x*e - (a^2*f^2*x^2*e^c + 2*a^2*f*x*e^(c + 1))*e^(d*x))/(b^3*e^(2*d*x + 2*c) + 2*a*b^2*e^(d*x + c) - b^3)
, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1745 vs. \(2 (315) = 630\).
time = 0.38, size = 1745, normalized size = 5.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*b*d^2*f^2*x^2 + 6*b*d*f^2*x + 3*b*d^2*cosh(1)^2 + 3*b*d^2*sinh(1)^2 + 6*b*f^2 - 3*(b*d^2*f^2*x^2 - 2*b
*d*f^2*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 + 2*b*f^2 + 2*(b*d^2*f*x - b*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*
cosh(1) - b*d*f)*sinh(1))*cosh(d*x + c)^2 - 3*(b*d^2*f^2*x^2 - 2*b*d*f^2*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2
+ 2*b*f^2 + 2*(b*d^2*f*x - b*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*cosh(1) - b*d*f)*sinh(1))*sinh(d*x + c)^2 +
6*(b*d^2*f*x + b*d*f)*cosh(1) - 2*(a*d^3*f^2*x^3 + 2*a*c^3*f^2 + 3*(a*d^3*x + 2*a*c*d^2)*cosh(1)^2 + 3*(a*d^3*
x + 2*a*c*d^2)*sinh(1)^2 + 3*(a*d^3*f*x^2 - 2*a*c^2*d*f)*cosh(1) + 3*(a*d^3*f*x^2 - 2*a*c^2*d*f + 2*(a*d^3*x +
2*a*c*d^2)*cosh(1))*sinh(1))*cosh(d*x + c) + 12*((a*d*f^2*x + a*d*f*cosh(1) + a*d*f*sinh(1))*cosh(d*x + c) +
(a*d*f^2*x + a*d*f*cosh(1) + a*d*f*sinh(1))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(
d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 12*((a*d*f^2*x + a*d*f*cosh(1) + a*d*f*sinh(1)
)*cosh(d*x + c) + (a*d*f^2*x + a*d*f*cosh(1) + a*d*f*sinh(1))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d
*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 6*((a*c^2*f^2 - 2*a*c*d*f*co
sh(1) + a*d^2*cosh(1)^2 + a*d^2*sinh(1)^2 - 2*(a*c*d*f - a*d^2*cosh(1))*sinh(1))*cosh(d*x + c) + (a*c^2*f^2 -
2*a*c*d*f*cosh(1) + a*d^2*cosh(1)^2 + a*d^2*sinh(1)^2 - 2*(a*c*d*f - a*d^2*cosh(1))*sinh(1))*sinh(d*x + c))*lo
g(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 6*((a*c^2*f^2 - 2*a*c*d*f*cosh(1)
+ a*d^2*cosh(1)^2 + a*d^2*sinh(1)^2 - 2*(a*c*d*f - a*d^2*cosh(1))*sinh(1))*cosh(d*x + c) + (a*c^2*f^2 - 2*a*c
*d*f*cosh(1) + a*d^2*cosh(1)^2 + a*d^2*sinh(1)^2 - 2*(a*c*d*f - a*d^2*cosh(1))*sinh(1))*sinh(d*x + c))*log(2*b
*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 6*((a*d^2*f^2*x^2 - a*c^2*f^2 + 2*(a*d
^2*f*x + a*c*d*f)*cosh(1) + 2*(a*d^2*f*x + a*c*d*f)*sinh(1))*cosh(d*x + c) + (a*d^2*f^2*x^2 - a*c^2*f^2 + 2*(a
*d^2*f*x + a*c*d*f)*cosh(1) + 2*(a*d^2*f*x + a*c*d*f)*sinh(1))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d
*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 6*((a*d^2*f^2*x^2 - a*c^2*f^2 +
2*(a*d^2*f*x + a*c*d*f)*cosh(1) + 2*(a*d^2*f*x + a*c*d*f)*sinh(1))*cosh(d*x + c) + (a*d^2*f^2*x^2 - a*c^2*f^2
+ 2*(a*d^2*f*x + a*c*d*f)*cosh(1) + 2*(a*d^2*f*x + a*c*d*f)*sinh(1))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*
sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 12*(a*f^2*cosh(d*x + c) +
a*f^2*sinh(d*x + c))*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt(
(a^2 + b^2)/b^2))/b) - 12*(a*f^2*cosh(d*x + c) + a*f^2*sinh(d*x + c))*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 6*(b*d^2*f*x + b*d^2*cosh(1) + b*d*f)*
sinh(1) - 2*(a*d^3*f^2*x^3 + 2*a*c^3*f^2 + 3*(a*d^3*x + 2*a*c*d^2)*cosh(1)^2 + 3*(a*d^3*x + 2*a*c*d^2)*sinh(1)
^2 + 3*(a*d^3*f*x^2 - 2*a*c^2*d*f)*cosh(1) + 3*(b*d^2*f^2*x^2 - 2*b*d*f^2*x + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^
2 + 2*b*f^2 + 2*(b*d^2*f*x - b*d*f)*cosh(1) + 2*(b*d^2*f*x + b*d^2*cosh(1) - b*d*f)*sinh(1))*cosh(d*x + c) + 3
*(a*d^3*f*x^2 - 2*a*c^2*d*f + 2*(a*d^3*x + 2*a*c*d^2)*cosh(1))*sinh(1))*sinh(d*x + c))/(b^2*d^3*cosh(d*x + c)
+ b^2*d^3*sinh(d*x + c))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*cosh(d*x+c)*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**2*sinh(c + d*x)*cosh(c + d*x)/(a + b*sinh(c + d*x)), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x + e)^2*cosh(d*x + c)*sinh(d*x + c)/(b*sinh(d*x + c) + a), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________