\(\int \sinh ^2(e+\frac {f (a+b x)}{c+d x}) \, dx\) [299]

Optimal result

Integrand size = 19, antiderivative size = 129 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {(b c-a d) f \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2} \]

[Out]

-(-a*d+b*c)*f*cosh(2*e+2*b*f/d)*Shi(2*(-a*d+b*c)*f/d/(d*x+c))/d^2+(-a*d+b*c)*f*Chi(2*(-a*d+b*c)*f/d/(d*x+c))*s
inh(2*e+2*b*f/d)/d^2+(d*x+c)*sinh((b*f*x+d*e*x+a*f+c*e)/(d*x+c))^2/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5728, 5726, 3394, 12, 3384, 3379, 3382} \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {f (b c-a d) \sinh \left (2 \left (\frac {b f}{d}+e\right )\right ) \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2}-\frac {f (b c-a d) \cosh \left (2 \left (\frac {b f}{d}+e\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a f+b f x+c e+d e x}{c+d x}\right )}{d} \]

[In]

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

[Out]

((b*c - a*d)*f*CoshIntegral[(2*(b*c - a*d)*f)/(d*(c + d*x))]*Sinh[2*(e + (b*f)/d)])/d^2 + ((c + d*x)*Sinh[(c*e
+ a*f + d*e*x + b*f*x)/(c + d*x)]^2)/d - ((b*c - a*d)*f*Cosh[2*(e + (b*f)/d)]*SinhIntegral[(2*(b*c - a*d)*f)/
(d*(c + d*x))])/d^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3394

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

Rule 5726

Int[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> Dist[-d^(-1), Subst[Int[Sinh[b
*(e/d) - e*(b*c - a*d)*(x/d)]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*
c - a*d, 0]

Rule 5728

Int[Sinh[u_]^(n_.), x_Symbol] :> With[{lst = QuotientOfLinearsParts[u, x]}, Int[Sinh[(lst[[1]] + lst[[2]]*x)/(
lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \sinh ^2\left (\frac {c e+a f+(d e+b f) x}{c+d x}\right ) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh ^2\left (\frac {d e+b f}{d}-\frac {(-d (c e+a f)+c (d e+b f)) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(2 i (b c-a d) f) \text {Subst}\left (\int \frac {i \sinh \left (2 \left (e+\frac {b f}{d}\right )-\frac {2 (b c-a d) f x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac {((b c-a d) f) \text {Subst}\left (\int \frac {\sinh \left (2 \left (e+\frac {b f}{d}\right )-\frac {2 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {\left ((b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}+\frac {\left ((b c-a d) f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(b c-a d) f \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(572\) vs. \(2(129)=258\).

Time = 3.56 (sec) , antiderivative size = 572, normalized size of antiderivative = 4.43 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {2 (c e+a f+d e x+b f x)}{c+d x}}+c d e^{\frac {2 (c e+a f+d e x+b f x)}{c+d x}}+2 d^2 x \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \cosh \left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )+2 d^2 x \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \sinh \left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )-2 \left (d^2 x+(b c-a d) f \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \left (\cosh \left (2 \left (e+\frac {b f}{d}\right )\right )-\sinh \left (2 \left (e+\frac {b f}{d}\right )\right )\right )-(b c-a d) f \text {Chi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right ) \left (\cosh \left (2 \left (e+\frac {b f}{d}\right )\right )+\sinh \left (2 \left (e+\frac {b f}{d}\right )\right )\right )+b c f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )-a d f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )-b c f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )+a d f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )-b c f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )+a d f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )-b c f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )+a d f \sinh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )\right )}{4 d^2} \]

[In]

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

[Out]

((c*d)/E^((2*(c*e + a*f + d*e*x + b*f*x))/(c + d*x)) + c*d*E^((2*(c*e + a*f + d*e*x + b*f*x))/(c + d*x)) + 2*d
^2*x*Cosh[2*(e + (b*f)/d)]*Cosh[(2*(-(b*c*f) + a*d*f))/(d*(c + d*x))] + 2*d^2*x*Sinh[2*(e + (b*f)/d)]*Sinh[(2*
(-(b*c*f) + a*d*f))/(d*(c + d*x))] - 2*(d^2*x + (b*c - a*d)*f*CoshIntegral[(2*(b*c - a*d)*f)/(d*(c + d*x))]*(C
osh[2*(e + (b*f)/d)] - Sinh[2*(e + (b*f)/d)]) - (b*c - a*d)*f*CoshIntegral[(2*(-(b*c*f) + a*d*f))/(d*(c + d*x)
)]*(Cosh[2*(e + (b*f)/d)] + Sinh[2*(e + (b*f)/d)]) + b*c*f*Cosh[2*(e + (b*f)/d)]*SinhIntegral[(2*(b*c - a*d)*f
)/(d*(c + d*x))] - a*d*f*Cosh[2*(e + (b*f)/d)]*SinhIntegral[(2*(b*c - a*d)*f)/(d*(c + d*x))] - b*c*f*Sinh[2*(e
+ (b*f)/d)]*SinhIntegral[(2*(b*c - a*d)*f)/(d*(c + d*x))] + a*d*f*Sinh[2*(e + (b*f)/d)]*SinhIntegral[(2*(b*c
- a*d)*f)/(d*(c + d*x))] - b*c*f*Cosh[2*(e + (b*f)/d)]*SinhIntegral[(2*(-(b*c*f) + a*d*f))/(d*(c + d*x))] + a*
d*f*Cosh[2*(e + (b*f)/d)]*SinhIntegral[(2*(-(b*c*f) + a*d*f))/(d*(c + d*x))] - b*c*f*Sinh[2*(e + (b*f)/d)]*Sin
hIntegral[(2*(-(b*c*f) + a*d*f))/(d*(c + d*x))] + a*d*f*Sinh[2*(e + (b*f)/d)]*SinhIntegral[(2*(-(b*c*f) + a*d*
f))/(d*(c + d*x))]))/(4*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(471\) vs. \(2(131)=262\).

Time = 9.18 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.66

method	result	size
risch	\(-\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b f x +d e x +a f +c e \right )}{d x +c}} a f}{\frac {4 d f a}{d x +c}-\frac {4 b c f}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b f x +d e x +a f +c e \right )}{d x +c}} b c f}{4 d \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 \left (f b +d e \right )}{d}} \operatorname {Ei}_{1}\left (\frac {2 a d f -2 b c f}{\left (d x +c \right ) d}\right ) a f}{2 d}+\frac {{\mathrm e}^{-\frac {2 \left (f b +d e \right )}{d}} \operatorname {Ei}_{1}\left (\frac {2 a d f -2 b c f}{\left (d x +c \right ) d}\right ) b c f}{2 d^{2}}+\frac {{\mathrm e}^{\frac {2 b f x +2 d e x +2 a f +2 c e}{d x +c}} a f}{4 d \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}-\frac {{\mathrm e}^{\frac {2 b f x +2 d e x +2 a f +2 c e}{d x +c}} b c f}{4 d^{2} \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}+\frac {{\mathrm e}^{\frac {2 f b +2 d e}{d}} \operatorname {Ei}_{1}\left (-\frac {2 \left (a d f -b c f \right )}{d \left (d x +c \right )}-\frac {2 \left (f b +d e \right )}{d}-\frac {2 \left (-f b -d e \right )}{d}\right ) a f}{2 d}-\frac {{\mathrm e}^{\frac {2 f b +2 d e}{d}} \operatorname {Ei}_{1}\left (-\frac {2 \left (a d f -b c f \right )}{d \left (d x +c \right )}-\frac {2 \left (f b +d e \right )}{d}-\frac {2 \left (-f b -d e \right )}{d}\right ) b c f}{2 d^{2}}\)	\(472\)

[In]

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

[Out]

-1/2*x+1/4*exp(-2*(b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(d*f/(d*x+c)*a-1/(d*x+c)*b*c*f)*a*f-1/4/d*exp(-2*(b*f*x+d*e*x
+a*f+c*e)/(d*x+c))/(d*f/(d*x+c)*a-1/(d*x+c)*b*c*f)*b*c*f-1/2/d*exp(-2*(b*f+d*e)/d)*Ei(1,2/d*(a*d*f-b*c*f)/(d*x
+c))*a*f+1/2/d^2*exp(-2*(b*f+d*e)/d)*Ei(1,2/d*(a*d*f-b*c*f)/(d*x+c))*b*c*f+1/4/d*exp(2*(b*f*x+d*e*x+a*f+c*e)/(
d*x+c))/(f/(d*x+c)*a-1/(d*x+c)/d*b*c*f)*a*f-1/4/d^2*exp(2*(b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(f/(d*x+c)*a-1/(d*x+c
)/d*b*c*f)*b*c*f+1/2/d*exp(2*(b*f+d*e)/d)*Ei(1,-2/d*(a*d*f-b*c*f)/(d*x+c)-2*(b*f+d*e)/d-2*(-b*f-d*e)/d)*a*f-1/
2/d^2*exp(2*(b*f+d*e)/d)*Ei(1,-2/d*(a*d*f-b*c*f)/(d*x+c)-2*(b*f+d*e)/d-2*(-b*f-d*e)/d)*b*c*f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (131) = 262\).

Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.70 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=-\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} + {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, {\left (d e + b f\right )}}{d}\right ) - d^{2} x - c d\right )} \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - {\left (b c - a d\right )} f {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, {\left (d e + b f\right )}}{d}\right ) - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} + {\left (b c - a d\right )} f {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, {\left (d e + b f\right )}}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right )^{2}\right )}} \]

[In]

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

[Out]

-1/2*(d^2*x - (d^2*x + c*d)*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 + ((b*c - a*d)*f*Ei(-2*(b*c - a*d)*f
/(d^2*x + c*d))*cosh(2*(d*e + b*f)/d) - d^2*x - c*d)*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 - ((b*c - a
*d)*f*Ei(-2*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 - (b*c - a*d)*f*Ei(2*(b
*c - a*d)*f/(d^2*x + c*d)))*cosh(2*(d*e + b*f)/d) - ((b*c - a*d)*f*Ei(-2*(b*c - a*d)*f/(d^2*x + c*d))*cosh((c*
e + a*f + (d*e + b*f)*x)/(d*x + c))^2 - (b*c - a*d)*f*Ei(-2*(b*c - a*d)*f/(d^2*x + c*d))*sinh((c*e + a*f + (d*
e + b*f)*x)/(d*x + c))^2 + (b*c - a*d)*f*Ei(2*(b*c - a*d)*f/(d^2*x + c*d)))*sinh(2*(d*e + b*f)/d))/(d^2*cosh((
c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2 - d^2*sinh((c*e + a*f + (d*e + b*f)*x)/(d*x + c))^2)

Sympy [F(-1)]

Timed out. \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\int { \sinh \left (e + \frac {{\left (b x + a\right )} f}{d x + c}\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1596 vs. \(2 (131) = 262\).

Time = 22.53 (sec) , antiderivative size = 1596, normalized size of antiderivative = 12.37 \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(2*b^2*c^2*d*e*f^2*Ei(-2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d) - 4*
a*b*c*d^2*e*f^2*Ei(-2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d) + 2*a^2*d^3
*e*f^2*Ei(-2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d) + 2*b^3*c^2*f^3*Ei(-
2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d) - 4*a*b^2*c*d*f^3*Ei(-2*(d*e +
b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d) + 2*a^2*b*d^2*f^3*Ei(-2*(d*e + b*f - (d*
e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d) - 2*b^2*c^2*d*e*f^2*Ei(2*(d*e + b*f - (d*e*x + b*
f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-2*(d*e + b*f)/d) + 4*a*b*c*d^2*e*f^2*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c
*e + a*f)*d/(d*x + c))/d)*e^(-2*(d*e + b*f)/d) - 2*a^2*d^3*e*f^2*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)
*d/(d*x + c))/d)*e^(-2*(d*e + b*f)/d) - 2*b^3*c^2*f^3*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c
))/d)*e^(-2*(d*e + b*f)/d) + 4*a*b^2*c*d*f^3*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(
-2*(d*e + b*f)/d) - 2*a^2*b*d^2*f^3*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-2*(d*e +
b*f)/d) - 2*(d*e*x + b*f*x + c*e + a*f)*b^2*c^2*d*f^2*Ei(-2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x +
c))/d)*e^(2*(d*e + b*f)/d)/(d*x + c) + 4*(d*e*x + b*f*x + c*e + a*f)*a*b*c*d^2*f^2*Ei(-2*(d*e + b*f - (d*e*x
+ b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d)/(d*x + c) - 2*(d*e*x + b*f*x + c*e + a*f)*a^2*d^3*f^2
*Ei(-2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(2*(d*e + b*f)/d)/(d*x + c) + 2*(d*e*x + b*f
*x + c*e + a*f)*b^2*c^2*d*f^2*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-2*(d*e + b*f)/
d)/(d*x + c) - 4*(d*e*x + b*f*x + c*e + a*f)*a*b*c*d^2*f^2*Ei(2*(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*
x + c))/d)*e^(-2*(d*e + b*f)/d)/(d*x + c) + 2*(d*e*x + b*f*x + c*e + a*f)*a^2*d^3*f^2*Ei(2*(d*e + b*f - (d*e*x
+ b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-2*(d*e + b*f)/d)/(d*x + c) + b^2*c^2*d*f^2*e^(2*(d*e*x + b*f*x + c*e
+ a*f)/(d*x + c)) - 2*a*b*c*d^2*f^2*e^(2*(d*e*x + b*f*x + c*e + a*f)/(d*x + c)) + a^2*d^3*f^2*e^(2*(d*e*x + b
*f*x + c*e + a*f)/(d*x + c)) + b^2*c^2*d*f^2*e^(-2*(d*e*x + b*f*x + c*e + a*f)/(d*x + c)) - 2*a*b*c*d^2*f^2*e^
(-2*(d*e*x + b*f*x + c*e + a*f)/(d*x + c)) + a^2*d^3*f^2*e^(-2*(d*e*x + b*f*x + c*e + a*f)/(d*x + c)) - 2*b^2*
c^2*d*f^2 + 4*a*b*c*d^2*f^2 - 2*a^2*d^3*f^2)*((d*e + b*f)*c/(b*c*f - a*d*f)^2 - (c*e + a*f)*d/(b*c*f - a*d*f)^
2)/(d^3*e + b*d^2*f - (d*e*x + b*f*x + c*e + a*f)*d^3/(d*x + c))

Mupad [F(-1)]

Timed out. \[ \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\int {\mathrm {sinh}\left (e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}\right )}^2 \,d x \]

[In]

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

[Out]

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