3.3.95 \(\int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [295]

Optimal. Leaf size=389 \[ -\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \cosh (c+d x)}{b d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {2 \sqrt {a^2+b^2} f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 \sqrt {a^2+b^2} f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {2 \sqrt {a^2+b^2} f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {2 \sqrt {a^2+b^2} f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{b d^2} \]

[Out]

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

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Rubi [A]
time = 0.57, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5684, 32, 3377, 2718, 3403, 2296, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {2 f^2 \sqrt {a^2+b^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 \sqrt {a^2+b^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 \sqrt {a^2+b^2} (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 \sqrt {a^2+b^2} (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 {\sqrt {a^2+b^2} (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (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 \cosh (c+d x)}{b d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{b d^2}+\frac {(e+f x)^2 \cosh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 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 ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {a \int (e+f x)^2 \, dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x) \, dx}{b}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{b^2}\\ &=-\frac {a (e+f x)^3}{3 b^2 f}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {\left (2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^2}-\frac {(2 f) \int (e+f x) \cosh (c+d x) \, dx}{b d}\\ &=-\frac {a (e+f x)^3}{3 b^2 f}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}-\frac {2 f (e+f x) \sinh (c+d x)}{b d^2}+\frac {\left (2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b}-\frac {\left (2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b}+\frac {\left (2 f^2\right ) \int \sinh (c+d x) \, dx}{b d^2}\\ &=-\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \cosh (c+d x)}{b d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (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 f (e+f x) \sinh (c+d x)}{b d^2}-\frac {\left (2 \sqrt {a^2+b^2} f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d}+\frac {\left (2 \sqrt {a^2+b^2} f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d}\\ &=-\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \cosh (c+d x)}{b d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (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 \sqrt {a^2+b^2} 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 \sqrt {a^2+b^2} 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 (e+f x) \sinh (c+d x)}{b d^2}-\frac {\left (2 \sqrt {a^2+b^2} f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^2}+\frac {\left (2 \sqrt {a^2+b^2} f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^2}\\ &=-\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \cosh (c+d x)}{b d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (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 \sqrt {a^2+b^2} 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 \sqrt {a^2+b^2} 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 (e+f x) \sinh (c+d x)}{b d^2}-\frac {\left (2 \sqrt {a^2+b^2} 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 \sqrt {a^2+b^2} 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^2 \cosh (c+d x)}{b d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (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 \sqrt {a^2+b^2} 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 \sqrt {a^2+b^2} 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 \sqrt {a^2+b^2} 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 \sqrt {a^2+b^2} 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 (e+f x) \sinh (c+d x)}{b d^2}\\ \end {align*}

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Mathematica [A]
time = 4.72, size = 723, normalized size = 1.86 \begin {gather*} \frac {-a x \left (3 e^2+3 e f x+f^2 x^2\right )+\frac {3 \left (a^2+b^2\right ) \left (\frac {2 d^2 e^2 \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {2 d^2 e e^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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {d^2 e^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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 d^2 e e^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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {d^2 e^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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {2 d e^c 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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 d e^c 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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}-\frac {2 e^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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {2 e^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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {3 b \cosh (d x) \left (\left (2 f^2+d^2 (e+f x)^2\right ) \cosh (c)-2 d f (e+f x) \sinh (c)\right )}{d^3}+\frac {3 b \left (-2 d f (e+f x) \cosh (c)+\left (2 f^2+d^2 (e+f x)^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 2.08, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\cosh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1741 vs. \(2 (362) = 724\).
time = 0.40, size = 1741, normalized size = 4.48 \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)^2/(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*c
osh(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 + 1
2*((b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*cosh(d*x + c) + (b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*sin
h(d*x + c))*sqrt((a^2 + b^2)/b^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) - 12*((b*d*f^2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*cosh(d*x + c) + (b*d*f^
2*x + b*d*f*cosh(1) + b*d*f*sinh(1))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^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) - 6*((b*c^2*f^2 - 2*b*c*d*f*cosh(
1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*cosh(d*x + c) + (b*c^2*f^2 - 2*b
*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*sinh(d*x + c))*sqrt(
(a^2 + b^2)/b^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) + 6*((b*c^2*f^2
- 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*cosh(d*x + c) +
(b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*sin
h(d*x + c))*sqrt((a^2 + b^2)/b^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)
+ 6*((b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x + b*c*d*f)*sinh(1))*cosh(d*x
+ c) + (b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x + b*c*d*f)*sinh(1))*sinh(d
*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*s
qrt((a^2 + b^2)/b^2) - b)/b) - 6*((b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*x
+ b*c*d*f)*sinh(1))*cosh(d*x + c) + (b*d^2*f^2*x^2 - b*c^2*f^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(1) + 2*(b*d^2*f*
x + b*c*d*f)*sinh(1))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*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*(b*f^2*cosh(d*x + c) + b*f^2*sinh(d*x + c))*sqrt
((a^2 + b^2)/b^2)*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*(b*f^2*cosh(d*x + c) + b*f^2*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*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*
f)*cosh(1) - 2*(a*d^3*f^2*x^3 + 3*a*d^3*f*x^2*cosh(1) + 3*a*d^3*x*cosh(1)^2 + 3*a*d^3*x*sinh(1)^2 + 3*(a*d^3*f
*x^2 + 2*a*d^3*x*cosh(1))*sinh(1))*cosh(d*x + c) + 6*(b*d^2*f*x + b*d^2*cosh(1) + b*d*f)*sinh(1) - 2*(a*d^3*f^
2*x^3 + 3*a*d^3*f*x^2*cosh(1) + 3*a*d^3*x*cosh(1)^2 + 3*a*d^3*x*sinh(1)^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) + 3*(a*d^3*f*x^2 + 2*a*d^3*x*cosh(1))*sinh(1))*sinh(d*x + c))/(b^2*d^3*cosh(d*x +
c) + b^2*d^3*sinh(d*x + c))

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

[Out]

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

[Out]

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

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