3.4.11 \(\int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [311]

Optimal. Leaf size=295 \[ -\frac {b f \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}+\frac {b^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {b (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]

[Out]

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

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Rubi [A]
time = 0.51, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5692, 3403, 2296, 2221, 2317, 2438, 6874, 4269, 3556, 5559, 3855} \begin {gather*} -\frac {b f \text {ArcTan}(\sinh (c+d x))}{d^2 \left (a^2+b^2\right )}+\frac {b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {b^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {a f \log (\cosh (c+d x))}{d^2 \left (a^2+b^2\right )}+\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a (e+f x) \tanh (c+d x)}{d \left (a^2+b^2\right )}+\frac {b (e+f x) \text {sech}(c+d x)}{d \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*f*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^2)) + (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)])/((a^2 + b^2)^(3/2)*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*
d) - (a*f*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^2) + (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])
/((a^2 + b^2)^(3/2)*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2
) + (b*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d) + (a*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*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 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 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 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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5692

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]
time = 1.87, size = 284, normalized size = 0.96 \begin {gather*} \frac {-\frac {2 b f \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^2+b^2}-\frac {a f \log (\cosh (c+d x))}{a^2+b^2}+\frac {b^2 \left (-2 d e \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {d (e+f x) \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2+b^2}}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1927\) vs. \(2(275)=550\).
time = 3.11, size = 1928, normalized size = 6.54

method	result	size
risch	\(\text {Expression too large to display}\)	\(1928\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2*(f*x+e)*(-b*exp(d*x+c)+a)/d/(a^2+b^2)/(1+exp(2*d*x+2*c))+2/d^2/(a^2+b^2)^(3/2)*b^2*f/(2*a^2+2*b^2)*dilog((-
b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2+2/d^2/(a^2+b^2)^(3/2)*b^4*f/(2*a^2+2*b^2)*ln((-b*exp
(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/d^2/(a^2+b^2)^(3/2)*b^4*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+
(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/d/(a^2+b^2)^(3/2)*b^4*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^
(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-2/d/(a^2+b^2)^(3/2)*b^4*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(
a+(a^2+b^2)^(1/2)))*x-2/d/(a^2+b^2)^(3/2)*b^2*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2)
)*a^2+2/d^2/(a^2+b^2)^(3/2)*b^4*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d^2/(a^2
+b^2)^(1/2)*b^2*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d^2/(a^2+b^2)^(3/2)*b^2*
f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2-2/d^2/(a^2+b^2)^(3/2)*b^2*f/(2*a^2+2*b^2
)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a^2-1/d^2/(a^2+b^2)^2*a^3*f*ln(b*exp(2*d*x+2*c)+
2*a*exp(d*x+c)-b)+2/d^2/(a^2+b^2)*a*f*ln(exp(d*x+c))-2/d^2/(a^2+b^2)*a^3*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))+
2/d^2/(a^2+b^2)*a^3*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-4/d^2/(a^2+b^2)*b^3*f/(2*a^2+2*b^2)*
arctan(exp(d*x+c))+2/d^2/(a^2+b^2)^(5/2)*a^4*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d^2/(a^2+b^
2)^(3/2)*b^2*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a^2*c+2/d^2/(a^2+b^2)^(3
/2)*b^2*f*c/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2-1/2/d^2/(a^2+b^2)^2*a*b^2*f*ln
(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d^2/(a^2+b^2)^(1/2)*b^2*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)
/(a^2+b^2)^(1/2))+2/d^2/(a^2+b^2)^(5/2)*a^2*b^2*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/d^2/(a^2
+b^2)*a^2*b*f/(2*a^2+2*b^2)*arctan(exp(d*x+c))+2/d^2/(a^2+b^2)^(3/2)*b^4*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(
d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d/(a^2+b^2)^(3/2)*b^4*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2
)^(1/2))-2/d/(a^2+b^2)^(1/2)*b^2*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d^2/(a^2+
b^2)^(3/2)*b^4*f/(2*a^2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/d^2/(a^2+b^2)^(
3/2)*b^4*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-4/d^2/(a^2+b^2)^(1/2)*a^2
*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2/(a^2+b^2)*b^2*f/(2*a^2+2*b^2)*a*ln(b*
exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d^2/(a^2+b^2)*b^2*f/(2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))+2/d/(a^2+b^2)^(3/2
)*b^2*f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2*x-2/d/(a^2+b^2)^(3/2)*b^2
*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a^2*x+2/d^2/(a^2+b^2)^(3/2)*b^2*f/(2
*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2*c

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (277) = 554\).
time = 0.42, size = 1411, normalized size = 4.78 \begin {gather*} \frac {2 \, {\left (a^{3} + a b^{2}\right )} d f x \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} + a b^{2}\right )} d f x \sinh \left (d x + c\right )^{2} - 2 \, {\left (a^{3} + a b^{2}\right )} d \cosh \left (1\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} d \sinh \left (1\right ) + {\left (b^{3} f \cosh \left (d x + c\right )^{2} + 2 \, b^{3} f \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} f \sinh \left (d x + c\right )^{2} + b^{3} f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - {\left (b^{3} f \cosh \left (d x + c\right )^{2} + 2 \, b^{3} f \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} f \sinh \left (d x + c\right )^{2} + b^{3} f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right ) + {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right ) + {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (b^{3} c f - b^{3} d \cosh \left (1\right ) - b^{3} d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + {\left (b^{3} d f x + b^{3} c f + {\left (b^{3} d f x + b^{3} c f\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (b^{3} d f x + b^{3} c f\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (b^{3} d f x + b^{3} c f\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - {\left (b^{3} d f x + b^{3} c f + {\left (b^{3} d f x + b^{3} c f\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (b^{3} d f x + b^{3} c f\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (b^{3} d f x + b^{3} c f\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left ({\left (a^{2} b + b^{3}\right )} f \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b + b^{3}\right )} f \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} b + b^{3}\right )} f \sinh \left (d x + c\right )^{2} + {\left (a^{2} b + b^{3}\right )} f\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \, {\left ({\left (a^{2} b + b^{3}\right )} d f x + {\left (a^{2} b + b^{3}\right )} d \cosh \left (1\right ) + {\left (a^{2} b + b^{3}\right )} d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{3} + a b^{2}\right )} f \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} + a b^{2}\right )} f \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{3} + a b^{2}\right )} f \sinh \left (d x + c\right )^{2} + {\left (a^{3} + a b^{2}\right )} f\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (2 \, {\left (a^{3} + a b^{2}\right )} d f x \cosh \left (d x + c\right ) + {\left (a^{2} b + b^{3}\right )} d f x + {\left (a^{2} b + b^{3}\right )} d \cosh \left (1\right ) + {\left (a^{2} b + b^{3}\right )} d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} \sinh \left (d x + c\right )^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a^3 + a*b^2)*d*f*x*cosh(d*x + c)^2 + 2*(a^3 + a*b^2)*d*f*x*sinh(d*x + c)^2 - 2*(a^3 + a*b^2)*d*cosh(1) - 2
*(a^3 + a*b^2)*d*sinh(1) + (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c)*sinh(d*x + c) + b^3*f*sinh(d*x + c)^
2 + b^3*f)*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) - (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c)*sinh(d*x + c) + b^3*f*sin
h(d*x + c)^2 + b^3*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*si
nh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1) + (b^3*c*f - b^3*d*c
osh(1) - b^3*d*sinh(1))*cosh(d*x + c)^2 + 2*(b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*cosh(d*x + c)*sinh(d*x +
c) + (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*sinh(d*x + c)^2)*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) - (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1) + (b^3*c*f -
b^3*d*cosh(1) - b^3*d*sinh(1))*cosh(d*x + c)^2 + 2*(b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*cosh(d*x + c)*sin
h(d*x + c) + (b^3*c*f - b^3*d*cosh(1) - b^3*d*sinh(1))*sinh(d*x + c)^2)*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) + (b^3*d*f*x + b^3*c*f + (b^3*d*f*x + b^3*c*f)*co
sh(d*x + c)^2 + 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d*f*x + b^3*c*f)*sinh(d*x + c)^2)*s
qrt((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) - (b^3*d*f*x + b^3*c*f + (b^3*d*f*x + b^3*c*f)*cosh(d*x + c)^2 + 2*(b^3*d*f*x + b^3*c*f)*co
sh(d*x + c)*sinh(d*x + c) + (b^3*d*f*x + b^3*c*f)*sinh(d*x + c)^2)*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^2*b + b^3)*f*co
sh(d*x + c)^2 + 2*(a^2*b + b^3)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f*sinh(d*x + c)^2 + (a^2*b + b^3
)*f)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*
d*sinh(1))*cosh(d*x + c) - ((a^3 + a*b^2)*f*cosh(d*x + c)^2 + 2*(a^3 + a*b^2)*f*cosh(d*x + c)*sinh(d*x + c) +
(a^3 + a*b^2)*f*sinh(d*x + c)^2 + (a^3 + a*b^2)*f)*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(2
*(a^3 + a*b^2)*d*f*x*cosh(d*x + c) + (a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1))*
sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^2*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*cosh(d*x + c)*sinh
(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*d^2*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((f*x + e)*sech(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 {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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