3.3.100 \(\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [300]

Optimal. Leaf size=477 \[ \frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d} \]

[Out]

1/2*e*f*x/b/d+1/4*f^2*x^2/b/d-1/3*(a^2+b^2)*(f*x+e)^3/b^3/f+2*a*f*(f*x+e)*cosh(d*x+c)/b^2/d^2+(a^2+b^2)*(f*x+e
)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d+(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^
3/d+2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2+2*(a^2+b^2)*f*(f*x+e)*polylog(2
,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^2-2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d
^3-2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^3-2*a*f^2*sinh(d*x+c)/b^2/d^3-a*(f*x+e)^
2*sinh(d*x+c)/b^2/d-1/2*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b/d^2+1/4*f^2*sinh(d*x+c)^2/b/d^3+1/2*(f*x+e)^2*sinh
(d*x+c)^2/b/d

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Rubi [A]
time = 0.48, antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5684, 3377, 2717, 5554, 3391, 5680, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {2 f^2 \left (a^2+b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 f^2 \left (a^2+b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {2 f \left (a^2+b^2\right ) (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 f \left (a^2+b^2\right ) (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}-\frac {f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*f*x)/(2*b*d) + (f^2*x^2)/(4*b*d) - ((a^2 + b^2)*(e + f*x)^3)/(3*b^3*f) + (2*a*f*(e + f*x)*Cosh[c + d*x])/(b
^2*d^2) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + ((a^2 + b^2)*(e +
f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E
^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(b^3*d^2) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*
d^3) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a*f^2*Sinh[c +
d*x])/(b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x])/(b^2*d) - (f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d^2)
+ (f^2*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^2*Sinh[c + d*x]^2)/(2*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 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 5554

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

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 5684

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[-a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*
x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 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 ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {a \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}\\ &=-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(2 a f) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d}-\frac {f \int (e+f x) \sinh ^2(c+d x) \, dx}{b d}\\ &=-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac {f \int (e+f x) \, dx}{2 b d}-\frac {\left (2 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (2 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (2 a f^2\right ) \int \cosh (c+d x) \, dx}{b^2 d^2}\\ &=\frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 \left (a^2+b^2\right ) f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (2 \left (a^2+b^2\right ) f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}\\ &=\frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 \left (a^2+b^2\right ) 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^3 d^3}-\frac {\left (2 \left (a^2+b^2\right ) 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^3 d^3}\\ &=\frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1253\) vs. \(2(477)=954\).
time = 12.97, size = 1253, normalized size = 2.63 \begin {gather*} \frac {8 \left (a^2+b^2\right ) x \left (3 e^2+3 e f x+f^2 x^2\right ) \coth (c)-\frac {8 \left (a^2+b^2\right ) \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 )}{-1+e^{2 c}}+\frac {3 b \left (16 a d f (e+f x) \cosh (c+d x)+b \left (f^2+2 d^2 (e+f x)^2\right ) \cosh (2 (c+d x))-4 \left (2 a \left (2 f^2+d^2 (e+f x)^2\right )+b d f (e+f x) \cosh (c+d x)\right ) \sinh (c+d x)\right )}{d^3}}{24 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(a^2 + b^2)*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Coth[c] - (8*(a^2 + b^2)*(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))/Sqrt[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))/Sqrt[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 + S
qrt[(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))/(-1 + E^(2*c)) + (3*b*(16*a*d*f*(e + f*x)*Cosh[c + d*x] + b*(f^2 + 2*d^2*(e + f*x)^2)*Cosh[
2*(c + d*x)] - 4*(2*a*(2*f^2 + d^2*(e + f*x)^2) + b*d*f*(e + f*x)*Cosh[c + d*x])*Sinh[c + d*x]))/d^3)/(24*b^3)

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Maple [F]
time = 1.56, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\cosh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-1/8*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 8*(a^2 + b^2)*(d*x + c)/(b^3*d) - (4*a*e^(-d*x - c) + b
*e^(-2*d*x - 2*c))/(b^2*d) - 8*(a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^3*d))*e^2 + 1/48
*(16*(a^2*d^3*f^2*e^(2*c) + b^2*d^3*f^2*e^(2*c))*x^3 + 48*(a^2*d^3*f*e^(2*c) + b^2*d^3*f*e^(2*c))*x^2*e + 3*(2
*b^2*d^2*f^2*x^2*e^(4*c) + b^2*f^2*e^(4*c) - 2*b^2*d*f*e^(4*c + 1) - 2*(b^2*d*f^2*e^(4*c) - 2*b^2*d^2*f*e^(4*c
+ 1))*x)*e^(2*d*x) - 24*(a*b*d^2*f^2*x^2*e^(3*c) + 2*a*b*f^2*e^(3*c) - 2*a*b*d*f*e^(3*c + 1) - 2*(a*b*d*f^2*e
^(3*c) - a*b*d^2*f*e^(3*c + 1))*x)*e^(d*x) + 24*(a*b*d^2*f^2*x^2*e^c + 2*a*b*d*f*e^(c + 1) + 2*a*b*f^2*e^c + 2
*(a*b*d^2*f*e^(c + 1) + a*b*d*f^2*e^c)*x)*e^(-d*x) + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d*f*e + b^2*f^2 + 2*(2*b^2*d
^2*f*e + b^2*d*f^2)*x)*e^(-2*d*x))*e^(-2*c)/(b^3*d^3) - integrate(-2*((a^2*b*f^2 + b^3*f^2)*x^2 + 2*(a^2*b*f +
b^3*f)*x*e - ((a^3*f^2*e^c + a*b^2*f^2*e^c)*x^2 + 2*(a^3*f*e^c + a*b^2*f*e^c)*x*e)*e^(d*x))/(b^4*e^(2*d*x + 2
*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4069 vs. \(2 (455) = 910\).
time = 0.44, size = 4069, normalized size = 8.53 \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)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(6*b^2*d^2*f^2*x^2 + 6*b^2*d*f^2*x + 6*b^2*d^2*cosh(1)^2 + 6*b^2*d^2*sinh(1)^2 + 3*(2*b^2*d^2*f^2*x^2 - 2
*b^2*d*f^2*x + 2*b^2*d^2*cosh(1)^2 + 2*b^2*d^2*sinh(1)^2 + b^2*f^2 + 2*(2*b^2*d^2*f*x - b^2*d*f)*cosh(1) + 2*(
2*b^2*d^2*f*x + 2*b^2*d^2*cosh(1) - b^2*d*f)*sinh(1))*cosh(d*x + c)^4 + 3*(2*b^2*d^2*f^2*x^2 - 2*b^2*d*f^2*x +
2*b^2*d^2*cosh(1)^2 + 2*b^2*d^2*sinh(1)^2 + b^2*f^2 + 2*(2*b^2*d^2*f*x - b^2*d*f)*cosh(1) + 2*(2*b^2*d^2*f*x
+ 2*b^2*d^2*cosh(1) - b^2*d*f)*sinh(1))*sinh(d*x + c)^4 + 3*b^2*f^2 - 24*(a*b*d^2*f^2*x^2 - 2*a*b*d*f^2*x + a*
b*d^2*cosh(1)^2 + a*b*d^2*sinh(1)^2 + 2*a*b*f^2 + 2*(a*b*d^2*f*x - a*b*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*d^2
*cosh(1) - a*b*d*f)*sinh(1))*cosh(d*x + c)^3 - 12*(2*a*b*d^2*f^2*x^2 - 4*a*b*d*f^2*x + 2*a*b*d^2*cosh(1)^2 + 2
*a*b*d^2*sinh(1)^2 + 4*a*b*f^2 + 4*(a*b*d^2*f*x - a*b*d*f)*cosh(1) - (2*b^2*d^2*f^2*x^2 - 2*b^2*d*f^2*x + 2*b^
2*d^2*cosh(1)^2 + 2*b^2*d^2*sinh(1)^2 + b^2*f^2 + 2*(2*b^2*d^2*f*x - b^2*d*f)*cosh(1) + 2*(2*b^2*d^2*f*x + 2*b
^2*d^2*cosh(1) - b^2*d*f)*sinh(1))*cosh(d*x + c) + 4*(a*b*d^2*f*x + a*b*d^2*cosh(1) - a*b*d*f)*sinh(1))*sinh(d
*x + c)^3 - 16*((a^2 + b^2)*d^3*f^2*x^3 + 2*(a^2 + b^2)*c^3*f^2 + 3*((a^2 + b^2)*d^3*x + 2*(a^2 + b^2)*c*d^2)*
cosh(1)^2 + 3*((a^2 + b^2)*d^3*x + 2*(a^2 + b^2)*c*d^2)*sinh(1)^2 + 3*((a^2 + b^2)*d^3*f*x^2 - 2*(a^2 + b^2)*c
^2*d*f)*cosh(1) + 3*((a^2 + b^2)*d^3*f*x^2 - 2*(a^2 + b^2)*c^2*d*f + 2*((a^2 + b^2)*d^3*x + 2*(a^2 + b^2)*c*d^
2)*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(8*(a^2 + b^2)*d^3*f^2*x^3 + 16*(a^2 + b^2)*c^3*f^2 + 24*((a^2 + b^2)
*d^3*x + 2*(a^2 + b^2)*c*d^2)*cosh(1)^2 - 9*(2*b^2*d^2*f^2*x^2 - 2*b^2*d*f^2*x + 2*b^2*d^2*cosh(1)^2 + 2*b^2*d
^2*sinh(1)^2 + b^2*f^2 + 2*(2*b^2*d^2*f*x - b^2*d*f)*cosh(1) + 2*(2*b^2*d^2*f*x + 2*b^2*d^2*cosh(1) - b^2*d*f)
*sinh(1))*cosh(d*x + c)^2 + 24*((a^2 + b^2)*d^3*x + 2*(a^2 + b^2)*c*d^2)*sinh(1)^2 + 24*((a^2 + b^2)*d^3*f*x^2
- 2*(a^2 + b^2)*c^2*d*f)*cosh(1) + 36*(a*b*d^2*f^2*x^2 - 2*a*b*d*f^2*x + a*b*d^2*cosh(1)^2 + a*b*d^2*sinh(1)^
2 + 2*a*b*f^2 + 2*(a*b*d^2*f*x - a*b*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*d^2*cosh(1) - a*b*d*f)*sinh(1))*cosh(
d*x + c) + 24*((a^2 + b^2)*d^3*f*x^2 - 2*(a^2 + b^2)*c^2*d*f + 2*((a^2 + b^2)*d^3*x + 2*(a^2 + b^2)*c*d^2)*cos
h(1))*sinh(1))*sinh(d*x + c)^2 + 6*(2*b^2*d^2*f*x + b^2*d*f)*cosh(1) + 24*(a*b*d^2*f^2*x^2 + 2*a*b*d*f^2*x + a
*b*d^2*cosh(1)^2 + a*b*d^2*sinh(1)^2 + 2*a*b*f^2 + 2*(a*b*d^2*f*x + a*b*d*f)*cosh(1) + 2*(a*b*d^2*f*x + a*b*d^
2*cosh(1) + a*b*d*f)*sinh(1))*cosh(d*x + c) + 96*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)
*d*f*sinh(1))*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f*sinh(1))*co
sh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f*sinh(1))*sinh(d*x
+ c)^2)*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) + 96*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f*sinh(1))*cosh(d*x + c)^2 +
2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + ((a^
2 + b^2)*d*f^2*x + (a^2 + b^2)*d*f*cosh(1) + (a^2 + b^2)*d*f*sinh(1))*sinh(d*x + c)^2)*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) + 48*(((a^2 + b^2)*c
^2*f^2 - 2*(a^2 + b^2)*c*d*f*cosh(1) + (a^2 + b^2)*d^2*cosh(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 - 2*((a^2 + b^2)*
c*d*f - (a^2 + b^2)*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*((a^2 + b^2)*c^2*f^2 - 2*(a^2 + b^2)*c*d*f*cosh(
1) + (a^2 + b^2)*d^2*cosh(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 - 2*((a^2 + b^2)*c*d*f - (a^2 + b^2)*d^2*cosh(1))*s
inh(1))*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*c^2*f^2 - 2*(a^2 + b^2)*c*d*f*cosh(1) + (a^2 + b^2)*d^2*cos
h(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 - 2*((a^2 + b^2)*c*d*f - (a^2 + b^2)*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^2)
*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*(((a^2 + b^2)*c^2*f^2 - 2*(
a^2 + b^2)*c*d*f*cosh(1) + (a^2 + b^2)*d^2*cosh(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 - 2*((a^2 + b^2)*c*d*f - (a^2
+ b^2)*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*((a^2 + b^2)*c^2*f^2 - 2*(a^2 + b^2)*c*d*f*cosh(1) + (a^2 +
b^2)*d^2*cosh(1)^2 + (a^2 + b^2)*d^2*sinh(1)^2 - 2*((a^2 + b^2)*c*d*f - (a^2 + b^2)*d^2*cosh(1))*sinh(1))*cosh
(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*c^2*f^2 - 2*(a^2 + b^2)*c*d*f*cosh(1) + (a^2 + b^2)*d^2*cosh(1)^2 + (a^
2 + b^2)*d^2*sinh(1)^2 - 2*((a^2 + b^2)*c*d*f - (a^2 + b^2)*d^2*cosh(1))*sinh(1))*sinh(d*x + c)^2)*log(2*b*cos
h(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*(((a^2 + b^2)*d^2*f^2*x^2 - (a^2 + b^2)
*c^2*f^2 + 2*((a^2 + b^2)*d^2*f*x + (a^2 + b^2)*c*d*f)*cosh(1) + 2*((a^2 + b^2)*d^2*f*x + (a^2 + b^2)*c*d*f)*s
inh(1))*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*f^2*x^2 - (a^2 + b^2)*c^2*f^2 + 2*((a^2 + b^2)*d^2*f*x + (a^2 + b
^2)*c*d*f)*cosh(1) + 2*((a^2 + b^2)*d^2*f*x + (a^2 + b^2)*c*d*f)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + ((a^2
+ b^2)*d^2*f^2*x^2 - (a^2 + b^2)*c^2*f^2 + 2*((...

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

[Out]

Timed out

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\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)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)

[Out]

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

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