3.328 \(\int \frac{(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx\)
Optimal. Leaf size=306 \[ \frac{a f^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}-\frac{a f^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}+\frac{a f (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{a f (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{f (e+f x) \cosh (c+d x)}{d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac{f^2 \log (a+b \sinh (c+d x))}{b d^3 \left (a^2+b^2\right )}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2} \]
[Out]
(a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) - (a*f*(e + f*x)*Log[
1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) + (f^2*Log[a + b*Sinh[c + d*x]])/(b*(a^2
+ b^2)*d^3) + (a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (a*f^2
*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (e + f*x)^2/(2*b*d*(a + b*S
inh[c + d*x])^2) - (f*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 0.519948, antiderivative size = 306, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used
= {5464, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{a f^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}-\frac{a f^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}+\frac{a f (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{a f (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{f (e+f x) \cosh (c+d x)}{d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac{f^2 \log (a+b \sinh (c+d x))}{b d^3 \left (a^2+b^2\right )}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
Int[((e + f*x)^2*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3,x]
[Out]
(a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) - (a*f*(e + f*x)*Log[
1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) + (f^2*Log[a + b*Sinh[c + d*x]])/(b*(a^2
+ b^2)*d^3) + (a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (a*f^2
*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (e + f*x)^2/(2*b*d*(a + b*S
inh[c + d*x])^2) - (f*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))
Rule 5464
Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)])^(n_.), x_Symbo
l] :> Simp[((e + f*x)^m*(a + b*Sinh[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e +
f*x)^(m - 1)*(a + b*Sinh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n,
-1]
Rule 3324
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 3322
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 2264
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 2190
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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279
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 2391
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 2668
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx &=-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}+\frac{f \int \frac{e+f x}{(a+b \sinh (c+d x))^2} \, dx}{b d}\\ &=-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac{f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac{(a f) \int \frac{e+f x}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right ) d}+\frac{f^2 \int \frac{\cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac{f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac{(2 a f) \int \frac{e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right ) d}+\frac{f^2 \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b \left (a^2+b^2\right ) d^3}\\ &=\frac{f^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac{f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac{(2 a f) \int \frac{e^{c+d x} (e+f x)}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac{(2 a f) \int \frac{e^{c+d x} (e+f x)}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2} d}\\ &=\frac{a f (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a f (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{f^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac{f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac{\left (a f^2\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{\left (a f^2\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^2}\\ &=\frac{a f (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a f (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{f^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac{f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac{\left (a f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac{\left (a f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}\\ &=\frac{a f (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a f (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{f^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}+\frac{a f^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{a f^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2}-\frac{f (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}\\ \end{align*}
Mathematica [B] time = 16.4123, size = 623, normalized size = 2.04 \[ \frac{2 e^c f \left (\frac{a \left (e^{2 c}-1\right ) f \left (\text{PolyLog}\left (2,-\frac{b e^{2 c+d x}}{a e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )-\text{PolyLog}\left (2,-\frac{b e^{2 c+d x}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+a e^c}\right )+d x \left (\log \left (\frac{b e^{2 c+d x}}{a e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}+1\right )-\log \left (\frac{b e^{2 c+d x}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+a e^c}+1\right )\right )\right )}{2 d \sqrt{e^{2 c} \left (a^2+b^2\right )}}-\frac{a e^{-c} \left (e^{2 c}-1\right ) e \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}+\frac{a e^{-c} \left (e^{2 c}-1\right ) f \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )}{d \sqrt{a^2+b^2}}+\frac{1}{2} e^{-c} \left (e^{2 c}-1\right ) f \left (\frac{2 a \tan ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{-a^2-b^2}}\right )}{d \sqrt{-a^2-b^2}}+\frac{\log \left (2 a e^{c+d x}+b \left (e^{2 (c+d x)}-1\right )\right )}{d}-2 x\right )-e^c f x+e^{-c} \left (e^{2 c}-1\right ) f x\right )}{b \left (e^{2 c}-1\right ) d^2 \left (a^2+b^2\right )}+\frac{\text{csch}\left (\frac{c}{2}\right ) \text{sech}\left (\frac{c}{2}\right ) \left (a e f \cosh (c)+a f^2 x \cosh (c)+b e f \sinh (d x)+b f^2 x \sinh (d x)\right )}{2 b d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac{f^2 x \coth (c)}{b d^2 \left (a^2+b^2\right )}-\frac{f^2 x \cosh (c) \text{csch}\left (\frac{c}{2}\right ) \text{sech}\left (\frac{c}{2}\right )}{2 b d^2 \left (a^2+b^2\right )}-\frac{(e+f x)^2}{2 b d (a+b \sinh (c+d x))^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((e + f*x)^2*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3,x]
[Out]
(f^2*x*Coth[c])/(b*(a^2 + b^2)*d^2) + (2*E^c*f*(-(E^c*f*x) + ((-1 + E^(2*c))*f*x)/E^c - (a*e*(-1 + E^(2*c))*Ar
cTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*E^c) + (a*(-1 + E^(2*c))*f*ArcTanh[(a + b*E^(c +
d*x))/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d*E^c) + ((-1 + E^(2*c))*f*(-2*x + (2*a*ArcTan[(a + b*E^(c + d*x))/Sq
rt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d) + Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))]/d))/(2*E^c) + (a*(-1 +
E^(2*c))*f*(d*x*(Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - Log[1 + (b*E^(2*c + d*x))/(
a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])]) + PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] -
PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/(2*d*Sqrt[(a^2 + b^2)*E^(2*c)])))/(b*(a
^2 + b^2)*d^2*(-1 + E^(2*c))) - (f^2*x*Cosh[c]*Csch[c/2]*Sech[c/2])/(2*b*(a^2 + b^2)*d^2) - (e + f*x)^2/(2*b*d
*(a + b*Sinh[c + d*x])^2) + (Csch[c/2]*Sech[c/2]*(a*e*f*Cosh[c] + a*f^2*x*Cosh[c] + b*e*f*Sinh[d*x] + b*f^2*x*
Sinh[d*x]))/(2*b*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))
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Maple [B] time = 0.3, size = 805, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x)
[Out]
-2/b*(a^2*d*f^2*x^2*exp(2*d*x+2*c)+b^2*d*f^2*x^2*exp(2*d*x+2*c)+2*a^2*d*e*f*x*exp(2*d*x+2*c)-a*b*f^2*x*exp(3*d
*x+3*c)+2*b^2*d*e*f*x*exp(2*d*x+2*c)+a^2*d*e^2*exp(2*d*x+2*c)-2*a^2*f^2*x*exp(2*d*x+2*c)-a*b*e*f*exp(3*d*x+3*c
)+b^2*d*e^2*exp(2*d*x+2*c)+b^2*f^2*x*exp(2*d*x+2*c)-2*a^2*e*f*exp(2*d*x+2*c)+3*a*b*f^2*x*exp(d*x+c)+b^2*e*f*ex
p(2*d*x+2*c)+3*a*b*e*f*exp(d*x+c)-b^2*f^2*x-b^2*e*f)/d^2/(a^2+b^2)/(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)^2+1/b*f
^2/d^3/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/b*f^2/d^3/(a^2+b^2)*ln(exp(d*x+c))-2/b*f/d^2/(a^2+b^2
)^(3/2)*a*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/b*f^2/d^2/(a^2+b^2)^(3/2)*a*ln((-b*exp(d*x+c)+
(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/b*f^2/d^3/(a^2+b^2)^(3/2)*a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/
(-a+(a^2+b^2)^(1/2)))*c-1/b*f^2/d^2/(a^2+b^2)^(3/2)*a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))
*x-1/b*f^2/d^3/(a^2+b^2)^(3/2)*a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/b*f^2/d^3/(a^2+b
^2)^(3/2)*a*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/b*f^2/d^3/(a^2+b^2)^(3/2)*a*dilog(
(b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/b*f^2/d^3/(a^2+b^2)^(3/2)*a*c*arctanh(1/2*(2*b*exp(d*x
+c)+2*a)/(a^2+b^2)^(1/2))
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.17122, size = 11416, normalized size = 37.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")
[Out]
-(2*((a^2*b^2 + b^4)*d*f^2*x + (a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)^4 + 2*((a^2*b^2 + b^4)*d*f^2*x + (a^2*b^2
+ b^4)*c*f^2)*sinh(d*x + c)^4 - 2*(a^2*b^2 + b^4)*d*e*f + 2*(a^2*b^2 + b^4)*c*f^2 + 2*(3*(a^3*b + a*b^3)*d*f^2
*x - (a^3*b + a*b^3)*d*e*f + 4*(a^3*b + a*b^3)*c*f^2)*cosh(d*x + c)^3 + 2*(3*(a^3*b + a*b^3)*d*f^2*x - (a^3*b
+ a*b^3)*d*e*f + 4*(a^3*b + a*b^3)*c*f^2 + 4*((a^2*b^2 + b^4)*d*f^2*x + (a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c))*
sinh(d*x + c)^3 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f^2*x^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2 - (2*a^4 + a^2*b^2
- b^4)*d*e*f + 2*(2*a^4 + a^2*b^2 - b^4)*c*f^2 + (2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f + (2*a^4 + a^2*b^2 - b^4)*
d*f^2)*x)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f^2*x^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2 - (2*a^4
+ a^2*b^2 - b^4)*d*e*f + 2*(2*a^4 + a^2*b^2 - b^4)*c*f^2 + 6*((a^2*b^2 + b^4)*d*f^2*x + (a^2*b^2 + b^4)*c*f^2)
*cosh(d*x + c)^2 + (2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f + (2*a^4 + a^2*b^2 - b^4)*d*f^2)*x + 3*(3*(a^3*b + a*b^3
)*d*f^2*x - (a^3*b + a*b^3)*d*e*f + 4*(a^3*b + a*b^3)*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^2 - (a*b^3*f^2*cosh(
d*x + c)^4 + a*b^3*f^2*sinh(d*x + c)^4 + 4*a^2*b^2*f^2*cosh(d*x + c)^3 - 4*a^2*b^2*f^2*cosh(d*x + c) + a*b^3*f
^2 + 2*(2*a^3*b - a*b^3)*f^2*cosh(d*x + c)^2 + 4*(a*b^3*f^2*cosh(d*x + c) + a^2*b^2*f^2)*sinh(d*x + c)^3 + 2*(
3*a*b^3*f^2*cosh(d*x + c)^2 + 6*a^2*b^2*f^2*cosh(d*x + c) + (2*a^3*b - a*b^3)*f^2)*sinh(d*x + c)^2 + 4*(a*b^3*
f^2*cosh(d*x + c)^3 + 3*a^2*b^2*f^2*cosh(d*x + c)^2 - a^2*b^2*f^2 + (2*a^3*b - a*b^3)*f^2*cosh(d*x + c))*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) + (a*b^3*f^2*cosh(d*x + c)^4 + a*b^3*f^2*sinh(d*x + c)^4 + 4*a^2*b^2*f^2*co
sh(d*x + c)^3 - 4*a^2*b^2*f^2*cosh(d*x + c) + a*b^3*f^2 + 2*(2*a^3*b - a*b^3)*f^2*cosh(d*x + c)^2 + 4*(a*b^3*f
^2*cosh(d*x + c) + a^2*b^2*f^2)*sinh(d*x + c)^3 + 2*(3*a*b^3*f^2*cosh(d*x + c)^2 + 6*a^2*b^2*f^2*cosh(d*x + c)
+ (2*a^3*b - a*b^3)*f^2)*sinh(d*x + c)^2 + 4*(a*b^3*f^2*cosh(d*x + c)^3 + 3*a^2*b^2*f^2*cosh(d*x + c)^2 - a^2
*b^2*f^2 + (2*a^3*b - a*b^3)*f^2*cosh(d*x + c))*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) - (a*b^3*d*f^2*x + a*b
^3*c*f^2 + (a*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x + c)^4 + (a*b^3*d*f^2*x + a*b^3*c*f^2)*sinh(d*x + c)^4 + 4*(
a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c)^3 + 4*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2 + (a*b^3*d*f^2*x + a*b^3
*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d*f^2*x + (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x + c)
^2 + 2*((2*a^3*b - a*b^3)*d*f^2*x + (2*a^3*b - a*b^3)*c*f^2 + 3*(a*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x + c)^2
+ 6*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cos
h(d*x + c) - 4*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2 - (a*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x + c)^3 - 3*(a^2*b^2*d
*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d*f^2*x + (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x + c))
*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) + (a*b^3*d*f^2*x + a*b^3*c*f^2 + (a*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x +
c)^4 + (a*b^3*d*f^2*x + a*b^3*c*f^2)*sinh(d*x + c)^4 + 4*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c)^3 + 4
*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2 + (a*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b
- a*b^3)*d*f^2*x + (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d*f^2*x + (2*a^3*b - a*b^3
)*c*f^2 + 3*(a*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x + c)^2 + 6*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c))
*sinh(d*x + c)^2 - 4*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c) - 4*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2 - (a
*b^3*d*f^2*x + a*b^3*c*f^2)*cosh(d*x + c)^3 - 3*(a^2*b^2*d*f^2*x + a^2*b^2*c*f^2)*cosh(d*x + c)^2 - ((2*a^3*b
- a*b^3)*d*f^2*x + (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x + c))*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) - 2*((a^3*b + a*
b^3)*d*f^2*x - 3*(a^3*b + a*b^3)*d*e*f + 4*(a^3*b + a*b^3)*c*f^2)*cosh(d*x + c) - ((a^2*b^2 + b^4)*f^2*cosh(d*
x + c)^4 + (a^2*b^2 + b^4)*f^2*sinh(d*x + c)^4 + 4*(a^3*b + a*b^3)*f^2*cosh(d*x + c)^3 + 2*(2*a^4 + a^2*b^2 -
b^4)*f^2*cosh(d*x + c)^2 - 4*(a^3*b + a*b^3)*f^2*cosh(d*x + c) + 4*((a^2*b^2 + b^4)*f^2*cosh(d*x + c) + (a^3*b
+ a*b^3)*f^2)*sinh(d*x + c)^3 + (a^2*b^2 + b^4)*f^2 + 2*(3*(a^2*b^2 + b^4)*f^2*cosh(d*x + c)^2 + 6*(a^3*b + a
*b^3)*f^2*cosh(d*x + c) + (2*a^4 + a^2*b^2 - b^4)*f^2)*sinh(d*x + c)^2 + 4*((a^2*b^2 + b^4)*f^2*cosh(d*x + c)^
3 + 3*(a^3*b + a*b^3)*f^2*cosh(d*x + c)^2 + (2*a^4 + a^2*b^2 - b^4)*f^2*cosh(d*x + c) - (a^3*b + a*b^3)*f^2)*s
inh(d*x + c) - (a*b^3*d*e*f - a*b^3*c*f^2 + (a*b^3*d*e*f - a*b^3*c*f^2)*cosh(d*x + c)^4 + (a*b^3*d*e*f - a*b^3
*c*f^2)*sinh(d*x + c)^4 + 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x + c)^3 + 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2
+ (a*b^3*d*e*f - a*b^3*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d*e*f - (2*a^3*b - a*b^3)
*c*f^2)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d*e*f - (2*a^3*b - a*b^3)*c*f^2 + 3*(a*b^3*d*e*f - a*b^3*c*f^2)
*cosh(d*x + c)^2 + 6*(a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(a^2*b^2*d*e*f - a^2*b
^2*c*f^2)*cosh(d*x + c) - 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2 - (a*b^3*d*e*f - a*b^3*c*f^2)*cosh(d*x + c)^3 - 3*(
a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d*e*f - (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x
+ c))*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) - ((a^2*b^2 + b^4)*f^2*cosh(d*x + c)^4 + (a^2*b^2 + b^4)*f^2*sinh(d*x + c)^4 + 4*(a^3*b + a*b^3)*f^
2*cosh(d*x + c)^3 + 2*(2*a^4 + a^2*b^2 - b^4)*f^2*cosh(d*x + c)^2 - 4*(a^3*b + a*b^3)*f^2*cosh(d*x + c) + 4*((
a^2*b^2 + b^4)*f^2*cosh(d*x + c) + (a^3*b + a*b^3)*f^2)*sinh(d*x + c)^3 + (a^2*b^2 + b^4)*f^2 + 2*(3*(a^2*b^2
+ b^4)*f^2*cosh(d*x + c)^2 + 6*(a^3*b + a*b^3)*f^2*cosh(d*x + c) + (2*a^4 + a^2*b^2 - b^4)*f^2)*sinh(d*x + c)^
2 + 4*((a^2*b^2 + b^4)*f^2*cosh(d*x + c)^3 + 3*(a^3*b + a*b^3)*f^2*cosh(d*x + c)^2 + (2*a^4 + a^2*b^2 - b^4)*f
^2*cosh(d*x + c) - (a^3*b + a*b^3)*f^2)*sinh(d*x + c) + (a*b^3*d*e*f - a*b^3*c*f^2 + (a*b^3*d*e*f - a*b^3*c*f^
2)*cosh(d*x + c)^4 + (a*b^3*d*e*f - a*b^3*c*f^2)*sinh(d*x + c)^4 + 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x
+ c)^3 + 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2 + (a*b^3*d*e*f - a*b^3*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2
*a^3*b - a*b^3)*d*e*f - (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d*e*f - (2*a^3*b - a*b
^3)*c*f^2 + 3*(a*b^3*d*e*f - a*b^3*c*f^2)*cosh(d*x + c)^2 + 6*(a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x + c))*s
inh(d*x + c)^2 - 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x + c) - 4*(a^2*b^2*d*e*f - a^2*b^2*c*f^2 - (a*b^3*d
*e*f - a*b^3*c*f^2)*cosh(d*x + c)^3 - 3*(a^2*b^2*d*e*f - a^2*b^2*c*f^2)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d
*e*f - (2*a^3*b - a*b^3)*c*f^2)*cosh(d*x + c))*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) - 2*((a^3*b + a*b^3)*d*f^2*x - 3*(a^3*b + a*b^3)*d*e*f + 4
*(a^3*b + a*b^3)*c*f^2 - 4*((a^2*b^2 + b^4)*d*f^2*x + (a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)^3 - 3*(3*(a^3*b + a
*b^3)*d*f^2*x - (a^3*b + a*b^3)*d*e*f + 4*(a^3*b + a*b^3)*c*f^2)*cosh(d*x + c)^2 - 2*((a^4 + 2*a^2*b^2 + b^4)*
d^2*f^2*x^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2 - (2*a^4 + a^2*b^2 - b^4)*d*e*f + 2*(2*a^4 + a^2*b^2 - b^4)*c*f^
2 + (2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f + (2*a^4 + a^2*b^2 - b^4)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4
*b^3 + 2*a^2*b^5 + b^7)*d^3*cosh(d*x + c)^4 + (a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*sinh(d*x + c)^4 + 4*(a^5*b^2 + 2
*a^3*b^4 + a*b^6)*d^3*cosh(d*x + c)^3 + 2*(2*a^6*b + 3*a^4*b^3 - b^7)*d^3*cosh(d*x + c)^2 - 4*(a^5*b^2 + 2*a^3
*b^4 + a*b^6)*d^3*cosh(d*x + c) + (a^4*b^3 + 2*a^2*b^5 + b^7)*d^3 + 4*((a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*cosh(d*
x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^3)*sinh(d*x + c)^3 + 2*(3*(a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*cosh(d*x +
c)^2 + 6*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^3*cosh(d*x + c) + (2*a^6*b + 3*a^4*b^3 - b^7)*d^3)*sinh(d*x + c)^2 +
4*((a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*cosh(d*x + c)^3 + 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^3*cosh(d*x + c)^2 + (2*
a^6*b + 3*a^4*b^3 - b^7)*d^3*cosh(d*x + c) - (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^3)*sinh(d*x + c))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**2*cosh(d*x+c)/(a+b*sinh(d*x+c))**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*cosh(d*x+c)/(a+b*sinh(d*x+c))^3,x, algorithm="giac")
[Out]
integrate((f*x + e)^2*cosh(d*x + c)/(b*sinh(d*x + c) + a)^3, x)