3.2.74 \(\int \frac {(c+d x)^2}{(a+b \cosh (e+f x))^2} \, dx\) [174]
Optimal. Leaf size=593 \[ -\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 d^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {2 a d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {2 a d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))} \]
[Out]
-(d*x+c)^2/(a^2-b^2)/f+2*d*(d*x+c)*ln(1+b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/f^2+a*(d*x+c)^2*ln(1+b*exp
(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f+2*d*(d*x+c)*ln(1+b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/f^
2-a*(d*x+c)^2*ln(1+b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f+2*d^2*polylog(2,-b*exp(f*x+e)/(a-(a^2-b
^2)^(1/2)))/(a^2-b^2)/f^3+2*a*d*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f^2+2*d^2
*polylog(2,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/f^3-2*a*d*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a+(a^2-b^2)
^(1/2)))/(a^2-b^2)^(3/2)/f^2-2*a*d^2*polylog(3,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f^3+2*a*d^2*
polylog(3,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f^3-b*(d*x+c)^2*sinh(f*x+e)/(a^2-b^2)/f/(a+b*cosh
(f*x+e))
________________________________________________________________________________________
Rubi [A]
time = 0.73, antiderivative size = 593, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3405, 3401,
2296, 2221, 2611, 2320, 6724, 5681, 2317, 2438} \begin {gather*} \frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}-\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac {2 d (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{f^2 \left (a^2-b^2\right )}+\frac {2 d (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{f^2 \left (a^2-b^2\right )}+\frac {a (c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{f \left (a^2-b^2\right )^{3/2}}-\frac {a (c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{f \left (a^2-b^2\right )^{3/2}}-\frac {b (c+d x)^2 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}-\frac {(c+d x)^2}{f \left (a^2-b^2\right )}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )}-\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )^{3/2}}+\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^2/(a + b*Cosh[e + f*x])^2,x]
[Out]
-((c + d*x)^2/((a^2 - b^2)*f)) + (2*d*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f
^2) + (a*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (2*d*(c + d*x)*Lo
g[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) - (a*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a +
Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (2*d^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2
- b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) +
(2*d^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog[2,
-((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) - (2*a*d^2*PolyLog[3, -((b*E^(e + f*x))/(a
- Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^3) + (2*a*d^2*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])
/((a^2 - b^2)^(3/2)*f^3) - (b*(c + d*x)^2*Sinh[e + f*x])/((a^2 - b^2)*f*(a + b*Cosh[e + f*x]))
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 2317
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
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 3401
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*(
(-I)*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Rule 3405
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Rule 5681
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), 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 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 {(c+d x)^2}{(a+b \cosh (e+f x))^2} \, dx &=-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}+\frac {a \int \frac {(c+d x)^2}{a+b \cosh (e+f x)} \, dx}{a^2-b^2}+\frac {(2 b d) \int \frac {(c+d x) \sinh (e+f x)}{a+b \cosh (e+f x)} \, dx}{\left (a^2-b^2\right ) f}\\ &=-\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}+\frac {(2 a) \int \frac {e^{e+f x} (c+d x)^2}{b+2 a e^{e+f x}+b e^{2 (e+f x)}} \, dx}{a^2-b^2}+\frac {(2 b d) \int \frac {e^{e+f x} (c+d x)}{a-\sqrt {a^2-b^2}+b e^{e+f x}} \, dx}{\left (a^2-b^2\right ) f}+\frac {(2 b d) \int \frac {e^{e+f x} (c+d x)}{a+\sqrt {a^2-b^2}+b e^{e+f x}} \, dx}{\left (a^2-b^2\right ) f}\\ &=-\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}+\frac {(2 a b) \int \frac {e^{e+f x} (c+d x)^2}{2 a-2 \sqrt {a^2-b^2}+2 b e^{e+f x}} \, dx}{\left (a^2-b^2\right )^{3/2}}-\frac {(2 a b) \int \frac {e^{e+f x} (c+d x)^2}{2 a+2 \sqrt {a^2-b^2}+2 b e^{e+f x}} \, dx}{\left (a^2-b^2\right )^{3/2}}-\frac {\left (2 d^2\right ) \int \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}-\frac {\left (2 d^2\right ) \int \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}\\ &=-\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2-b^2\right ) f^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2-b^2\right ) f^3}-\frac {(2 a d) \int (c+d x) \log \left (1+\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}+\frac {(2 a d) \int (c+d x) \log \left (1+\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}\\ &=-\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}-\frac {\left (2 a d^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {\left (2 a d^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}\\ &=-\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}-\frac {\left (2 a d^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^{e+f x}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {\left (2 a d^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^{e+f x}\right )}{\left (a^2-b^2\right )^{3/2} f^3}\\ &=-\frac {(c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}-\frac {b (c+d x)^2 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2854\) vs. \(2(593)=1186\).
time = 18.62, size = 2854, normalized size = 4.81 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^2/(a + b*Cosh[e + f*x])^2,x]
[Out]
-((4*(a^2 - b^2)^2*c*d*((a^2 - b^2)*E^(2*e))^(3/2)*f^2*x + 2*(a^2 - b^2)^2*d^2*((a^2 - b^2)*E^(2*e))^(3/2)*f^2
*x^2 + 4*a^3*Sqrt[a^2 - b^2]*Sqrt[-(a^2 - b^2)^2]*c*d*Sqrt[(a^2 - b^2)*E^(2*e)]*f*ArcTan[(a + b*E^(e + f*x))/S
qrt[-a^2 + b^2]] - 4*a*b^2*Sqrt[a^2 - b^2]*Sqrt[-(a^2 - b^2)^2]*c*d*Sqrt[(a^2 - b^2)*E^(2*e)]*f*ArcTan[(a + b*
E^(e + f*x))/Sqrt[-a^2 + b^2]] + (4*a*b^2*(a^2 - b^2)^(3/2)*c*d*((a^2 - b^2)*E^(2*e))^(3/2)*f*ArcTan[(a + b*E^
(e + f*x))/Sqrt[-a^2 + b^2]])/Sqrt[-(a^2 - b^2)^2] + (4*a^3*Sqrt[-(a^2 - b^2)^2]*c*d*((a^2 - b^2)*E^(2*e))^(3/
2)*f*ArcTan[(a + b*E^(e + f*x))/Sqrt[-a^2 + b^2]])/Sqrt[a^2 - b^2] - 4*a*(a^2 - b^2)^(5/2)*c*d*Sqrt[(a^2 - b^2
)*E^(2*e)]*f*ArcTanh[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] - 4*a*(a^2 - b^2)^(3/2)*c*d*((a^2 - b^2)*E^(2*e))^(3
/2)*f*ArcTanh[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] + 2*a*(a^2 - b^2)^(5/2)*c^2*Sqrt[(a^2 - b^2)*E^(2*e)]*f^2*A
rcTanh[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] + 2*a*(a^2 - b^2)^(3/2)*c^2*((a^2 - b^2)*E^(2*e))^(3/2)*f^2*ArcTan
h[(a + b*E^(e + f*x))/Sqrt[a^2 - b^2]] - 2*(a^2 - b^2)^3*c*d*Sqrt[(a^2 - b^2)*E^(2*e)]*f*Log[b + 2*a*E^(e + f*
x) + b*E^(2*(e + f*x))] - 2*(a^2 - b^2)^2*c*d*((a^2 - b^2)*E^(2*e))^(3/2)*f*Log[b + 2*a*E^(e + f*x) + b*E^(2*(
e + f*x))] - 2*(a^2 - b^2)^3*d^2*Sqrt[(a^2 - b^2)*E^(2*e)]*f*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 -
b^2)*E^(2*e)])] - 2*(a^2 - b^2)^2*d^2*((a^2 - b^2)*E^(2*e))^(3/2)*f*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e - Sqrt[
(a^2 - b^2)*E^(2*e)])] - 2*a*(a^2 - b^2)^3*c*d*E^e*f^2*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E
^(2*e)])] - 2*a*(a^2 - b^2)^3*c*d*E^(3*e)*f^2*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)])]
- a*(a^2 - b^2)^3*d^2*E^e*f^2*x^2*Log[1 + (b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)])] - a*(a^2 - b
^2)^3*d^2*E^(3*e)*f^2*x^2*Log[1 + (b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)])] - 2*(a^2 - b^2)^3*d^2
*Sqrt[(a^2 - b^2)*E^(2*e)]*f*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)])] - 2*(a^2 - b^2)^
2*d^2*((a^2 - b^2)*E^(2*e))^(3/2)*f*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)])] + 2*a*(a^
2 - b^2)^3*c*d*E^e*f^2*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)])] + 2*a*(a^2 - b^2)^3*c*
d*E^(3*e)*f^2*x*Log[1 + (b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)])] + a*(a^2 - b^2)^3*d^2*E^e*f^2*x
^2*Log[1 + (b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)])] + a*(a^2 - b^2)^3*d^2*E^(3*e)*f^2*x^2*Log[1
+ (b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)])] - 2*(a^2 - b^2)^3*d^2*Sqrt[(a^2 - b^2)*E^(2*e)]*PolyL
og[2, -((b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*(a^2 - b^2)^2*d^2*((a^2 - b^2)*E^(2*e))^(3
/2)*PolyLog[2, -((b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*a*(a^2 - b^2)^3*c*d*E^e*f*PolyLog
[2, -((b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*a*(a^2 - b^2)^3*c*d*E^(3*e)*f*PolyLog[2, -((
b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*a*(a^2 - b^2)^3*d^2*E^e*f*x*PolyLog[2, -((b*E^(2*e
+ f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*a*(a^2 - b^2)^3*d^2*E^(3*e)*f*x*PolyLog[2, -((b*E^(2*e + f*x
))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*(a^2 - b^2)^3*d^2*Sqrt[(a^2 - b^2)*E^(2*e)]*PolyLog[2, -((b*E^(2*
e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*(a^2 - b^2)^2*d^2*((a^2 - b^2)*E^(2*e))^(3/2)*PolyLog[2, -
((b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)]))] + 2*a*(a^2 - b^2)^3*c*d*E^e*f*PolyLog[2, -((b*E^(2*e
+ f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)]))] + 2*a*(a^2 - b^2)^3*c*d*E^(3*e)*f*PolyLog[2, -((b*E^(2*e + f*x))
/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)]))] + 2*a*(a^2 - b^2)^3*d^2*E^e*f*x*PolyLog[2, -((b*E^(2*e + f*x))/(a*E^e +
Sqrt[(a^2 - b^2)*E^(2*e)]))] + 2*a*(a^2 - b^2)^3*d^2*E^(3*e)*f*x*PolyLog[2, -((b*E^(2*e + f*x))/(a*E^e + Sqrt
[(a^2 - b^2)*E^(2*e)]))] + 2*a*(a^2 - b^2)^3*d^2*E^e*PolyLog[3, -((b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*
E^(2*e)]))] + 2*a*(a^2 - b^2)^3*d^2*E^(3*e)*PolyLog[3, -((b*E^(2*e + f*x))/(a*E^e - Sqrt[(a^2 - b^2)*E^(2*e)])
)] - 2*a*(a^2 - b^2)^3*d^2*E^e*PolyLog[3, -((b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)]))] - 2*a*(a^2
- b^2)^3*d^2*E^(3*e)*PolyLog[3, -((b*E^(2*e + f*x))/(a*E^e + Sqrt[(a^2 - b^2)*E^(2*e)]))])/((a^2 - b^2)^4*Sqr
t[(a^2 - b^2)*E^(2*e)]*(1 + E^(2*e))*f^3)) + (Sech[e]*(a*c^2*Sinh[e] + 2*a*c*d*x*Sinh[e] + a*d^2*x^2*Sinh[e] -
b*c^2*Sinh[f*x] - 2*b*c*d*x*Sinh[f*x] - b*d^2*x^2*Sinh[f*x]))/((a - b)*(a + b)*f*(a + b*Cosh[e + f*x]))
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Maple [F]
time = 0.85, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{2}}{\left (a +b \cosh \left (f x +e \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^2/(a+b*cosh(f*x+e))^2,x)
[Out]
int((d*x+c)^2/(a+b*cosh(f*x+e))^2,x)
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^2/(a+b*cosh(f*x+e))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6143 vs.
\(2 (559) = 1118\).
time = 0.68, size = 6143, normalized size = 10.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^2/(a+b*cosh(f*x+e))^2,x, algorithm="fricas")
[Out]
(2*(a^2*b - b^3)*c^2*f^2 - 4*(a^2*b - b^3)*c*d*f*cosh(1) + 2*(a^2*b - b^3)*d^2*cosh(1)^2 + 2*(a^2*b - b^3)*d^2
*sinh(1)^2 - 2*((a^2*b - b^3)*d^2*f^2*x^2 + 2*(a^2*b - b^3)*c*d*f^2*x + 2*(a^2*b - b^3)*c*d*f*cosh(1) - (a^2*b
- b^3)*d^2*cosh(1)^2 - (a^2*b - b^3)*d^2*sinh(1)^2 + 2*((a^2*b - b^3)*c*d*f - (a^2*b - b^3)*d^2*cosh(1))*sinh
(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 2*((a^2*b - b^3)*d^2*f^2*x^2 + 2*(a^2*b - b^3)*c*d*f^2*x + 2*(a^2*b - b
^3)*c*d*f*cosh(1) - (a^2*b - b^3)*d^2*cosh(1)^2 - (a^2*b - b^3)*d^2*sinh(1)^2 + 2*((a^2*b - b^3)*c*d*f - (a^2*
b - b^3)*d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(a*b^2*d^2*cosh(f*x + cosh(1) + sinh(1))^2
+ a*b^2*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*a^2*b*d^2*cosh(f*x + cosh(1) + sinh(1)) + a*b^2*d^2 + 2*(a*b^2
*d^2*cosh(f*x + cosh(1) + sinh(1)) + a^2*b*d^2)*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2)*polylog(3
, -(a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) + (b*cosh(f*x + cosh(1) + sinh(1)) + b*s
inh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2))/b) + 2*(a*b^2*d^2*cosh(f*x + cosh(1) + sinh(1))^2 + a*b^2
*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*a^2*b*d^2*cosh(f*x + cosh(1) + sinh(1)) + a*b^2*d^2 + 2*(a*b^2*d^2*co
sh(f*x + cosh(1) + sinh(1)) + a^2*b*d^2)*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2)*polylog(3, -(a*c
osh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) - (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x
+ cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2))/b) - 2*((a^3 - a*b^2)*d^2*f^2*x^2 + 2*(a^3 - a*b^2)*c*d*f^2*x -
(a^3 - a*b^2)*c^2*f^2 + 4*(a^3 - a*b^2)*c*d*f*cosh(1) - 2*(a^3 - a*b^2)*d^2*cosh(1)^2 - 2*(a^3 - a*b^2)*d^2*si
nh(1)^2 + 4*((a^3 - a*b^2)*c*d*f - (a^3 - a*b^2)*d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1)) + 2*((a^2
*b - b^3)*d^2*cosh(f*x + cosh(1) + sinh(1))^2 + (a^2*b - b^3)*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^3 - a
*b^2)*d^2*cosh(f*x + cosh(1) + sinh(1)) + (a^2*b - b^3)*d^2 + 2*((a^2*b - b^3)*d^2*cosh(f*x + cosh(1) + sinh(1
)) + (a^3 - a*b^2)*d^2)*sinh(f*x + cosh(1) + sinh(1)) + (a*b^2*d^2*f*x + a*b^2*c*d*f + (a*b^2*d^2*f*x + a*b^2*
c*d*f)*cosh(f*x + cosh(1) + sinh(1))^2 + (a*b^2*d^2*f*x + a*b^2*c*d*f)*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^
2*b*d^2*f*x + a^2*b*c*d*f)*cosh(f*x + cosh(1) + sinh(1)) + 2*(a^2*b*d^2*f*x + a^2*b*c*d*f + (a*b^2*d^2*f*x + a
*b^2*c*d*f)*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2))*dilog(-(a*cos
h(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) + (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x +
cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + 2*((a^2*b - b^3)*d^2*cosh(f*x + cosh(1) + sinh(1))^2
+ (a^2*b - b^3)*d^2*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^3 - a*b^2)*d^2*cosh(f*x + cosh(1) + sinh(1)) + (a^2
*b - b^3)*d^2 + 2*((a^2*b - b^3)*d^2*cosh(f*x + cosh(1) + sinh(1)) + (a^3 - a*b^2)*d^2)*sinh(f*x + cosh(1) + s
inh(1)) - (a*b^2*d^2*f*x + a*b^2*c*d*f + (a*b^2*d^2*f*x + a*b^2*c*d*f)*cosh(f*x + cosh(1) + sinh(1))^2 + (a*b^
2*d^2*f*x + a*b^2*c*d*f)*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^2*b*d^2*f*x + a^2*b*c*d*f)*cosh(f*x + cosh(1)
+ sinh(1)) + 2*(a^2*b*d^2*f*x + a^2*b*c*d*f + (a*b^2*d^2*f*x + a*b^2*c*d*f)*cosh(f*x + cosh(1) + sinh(1)))*sin
h(f*x + cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2))*dilog(-(a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh
(1) + sinh(1)) - (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 - b^2)/b^2) + b
)/b + 1) + (2*(a^2*b - b^3)*c*d*f - 2*(a^2*b - b^3)*d^2*cosh(1) - 2*(a^2*b - b^3)*d^2*sinh(1) + 2*((a^2*b - b^
3)*c*d*f - (a^2*b - b^3)*d^2*cosh(1) - (a^2*b - b^3)*d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*((a^2*b
- b^3)*c*d*f - (a^2*b - b^3)*d^2*cosh(1) - (a^2*b - b^3)*d^2*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 4*((a^
3 - a*b^2)*c*d*f - (a^3 - a*b^2)*d^2*cosh(1) - (a^3 - a*b^2)*d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1)) + 4*((
a^3 - a*b^2)*c*d*f - (a^3 - a*b^2)*d^2*cosh(1) - (a^3 - a*b^2)*d^2*sinh(1) + ((a^2*b - b^3)*c*d*f - (a^2*b - b
^3)*d^2*cosh(1) - (a^2*b - b^3)*d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)) - (a
*b^2*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 + (a*b^2*c^2*f^2 - 2*a*b^2*c*
d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1))*cosh(f*
x + cosh(1) + sinh(1))^2 + (a*b^2*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2
- 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(a^2*b*c^2*f^2 - 2*a^2*b*c*
d*f*cosh(1) + a^2*b*d^2*cosh(1)^2 + a^2*b*d^2*sinh(1)^2 - 2*(a^2*b*c*d*f - a^2*b*d^2*cosh(1))*sinh(1))*cosh(f*
x + cosh(1) + sinh(1)) - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1) + 2*(a^2*b*c^2*f^2 - 2*a^2*b*c*d*f*cosh(1
) + a^2*b*d^2*cosh(1)^2 + a^2*b*d^2*sinh(1)^2 + (a*b^2*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 +
a*b^2*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1)) - 2*(a^2*b*c
*d*f - a^2*b*d^2*cosh(1))*sinh(1))*sinh(f*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((d*x+c)**2/(a+b*cosh(f*x+e))**2,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((d*x+c)^2/(a+b*cosh(f*x+e))^2,x, algorithm="giac")
[Out]
integrate((d*x + c)^2/(b*cosh(f*x + e) + a)^2, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^2/(a + b*cosh(e + f*x))^2,x)
[Out]
int((c + d*x)^2/(a + b*cosh(e + f*x))^2, x)
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