3.3.98 \(\int \sinh (e+\frac {f (a+b x)}{c+d x}) \, dx\) [298]

Optimal. Leaf size=121 \[ \frac {(b c-a d) f \cosh \left (e+\frac {b f}{d}\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5728, 5726, 3378, 3384, 3379, 3382} \begin {gather*} \frac {f (b c-a d) \cosh \left (\frac {b f}{d}+e\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}-\frac {f (b c-a d) \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a f+b f x+c e+d e x}{c+d x}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3378

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(449\) vs. \(2(121)=242\).
time = 1.03, size = 449, normalized size = 3.71 \begin {gather*} \frac {(b c-a d) f \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right ) \left (\cosh \left (e+\frac {b f}{d}\right )-\sinh \left (e+\frac {b f}{d}\right )\right )+(b c-a d) f \text {Chi}\left (\frac {-b c f+a d f}{d (c+d x)}\right ) \left (\cosh \left (e+\frac {b f}{d}\right )+\sinh \left (e+\frac {b f}{d}\right )\right )+2 c d \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )+2 d^2 x \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )+b c f \cosh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-a d f \cosh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-b c f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+a d f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+b c f \cosh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {-b c f+a d f}{d (c+d x)}\right )-a d f \cosh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {-b c f+a d f}{d (c+d x)}\right )+b c f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {-b c f+a d f}{d (c+d x)}\right )-a d f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {-b c f+a d f}{d (c+d x)}\right )}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(121)=242\).
time = 1.44, size = 459, normalized size = 3.79

method	result	size
risch	\(-\frac {f \,{\mathrm e}^{-\frac {b f x +d e x +f a +c e}{d x +c}} a}{2 \left (\frac {d f a}{d x +c}-\frac {f b c}{d x +c}\right )}+\frac {f \,{\mathrm e}^{-\frac {b f x +d e x +f a +c e}{d x +c}} b c}{2 d \left (\frac {d f a}{d x +c}-\frac {f b c}{d x +c}\right )}+\frac {f \,{\mathrm e}^{-\frac {b f +d e}{d}} \expIntegral \left (1, \frac {\left (a d -b c \right ) f}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {f \,{\mathrm e}^{-\frac {b f +d e}{d}} \expIntegral \left (1, \frac {\left (a d -b c \right ) f}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {f \,{\mathrm e}^{\frac {b f x +d e x +f a +c e}{d x +c}} a}{2 d \left (\frac {f a}{d x +c}-\frac {f b c}{d \left (d x +c \right )}\right )}-\frac {f \,{\mathrm e}^{\frac {b f x +d e x +f a +c e}{d x +c}} b c}{2 d^{2} \left (\frac {f a}{d x +c}-\frac {f b c}{d \left (d x +c \right )}\right )}+\frac {f \,{\mathrm e}^{\frac {b f +d e}{d}} \expIntegral \left (1, -\frac {\left (a d -b c \right ) f}{d \left (d x +c \right )}-\frac {b f +d e}{d}-\frac {-b f -d e}{d}\right ) a}{2 d}-\frac {f \,{\mathrm e}^{\frac {b f +d e}{d}} \expIntegral \left (1, -\frac {\left (a d -b c \right ) f}{d \left (d x +c \right )}-\frac {b f +d e}{d}-\frac {-b f -d e}{d}\right ) b c}{2 d^{2}}\)	\(459\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*f*exp(-(b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(d*f/(d*x+c)*a-f/(d*x+c)*b*c)*a+1/2/d*f*exp(-(b*f*x+d*e*x+a*f+c*e)/
(d*x+c))/(d*f/(d*x+c)*a-f/(d*x+c)*b*c)*b*c+1/2/d*f*exp(-(b*f+d*e)/d)*Ei(1,(a*d-b*c)*f/d/(d*x+c))*a-1/2/d^2*f*e
xp(-(b*f+d*e)/d)*Ei(1,(a*d-b*c)*f/d/(d*x+c))*b*c+1/2/d*f*exp((b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(f/(d*x+c)*a-f/d/(
d*x+c)*b*c)*a-1/2/d^2*f*exp((b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(f/(d*x+c)*a-f/d/(d*x+c)*b*c)*b*c+1/2/d*f*exp((b*f+
d*e)/d)*Ei(1,-(a*d-b*c)*f/d/(d*x+c)-(b*f+d*e)/d-(-b*f-d*e)/d)*a-1/2/d^2*f*exp((b*f+d*e)/d)*Ei(1,-(a*d-b*c)*f/d
/(d*x+c)-(b*f+d*e)/d-(-b*f-d*e)/d)*b*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(sinh(e+f*(b*x+a)/(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.49, size = 220, normalized size = 1.82 \begin {gather*} \frac {{\left ({\left (b c - a d\right )} f {\rm Ei}\left (\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b f + d \cosh \left (1\right ) + d \sinh \left (1\right )}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right ) - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b f + d \cosh \left (1\right ) + d \sinh \left (1\right )}{d}\right )}{2 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1624 vs. \(2 (121) = 242\).
time = 4.57, size = 1624, normalized size = 13.42 \begin {gather*} \frac {{\left (b^{2} c^{2} d e f^{2} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )} - 2 \, a b c d^{2} e f^{2} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )} + a^{2} d^{3} e f^{2} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )} + b^{3} c^{2} f^{3} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )} - 2 \, a b^{2} c d f^{3} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )} + a^{2} b d^{2} f^{3} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )} - \frac {{\left (d e x + b f x + c e + a f\right )} b^{2} c^{2} d f^{2} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )}}{d x + c} + \frac {2 \, {\left (d e x + b f x + c e + a f\right )} a b c d^{2} f^{2} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )}}{d x + c} - \frac {{\left (d e x + b f x + c e + a f\right )} a^{2} d^{3} f^{2} {\rm Ei}\left (-\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (\frac {d e + b f}{d}\right )}}{d x + c} + b^{2} c^{2} d f^{2} e^{\left (\frac {d e x + b f x + c e + a f}{d x + c}\right )} - 2 \, a b c d^{2} f^{2} e^{\left (\frac {d e x + b f x + c e + a f}{d x + c}\right )} + a^{2} d^{3} f^{2} e^{\left (\frac {d e x + b f x + c e + a f}{d x + c}\right )}\right )} {\left (\frac {{\left (d e + b f\right )} c}{{\left (b c f - a d f\right )}^{2}} - \frac {{\left (c e + a f\right )} d}{{\left (b c f - a d f\right )}^{2}}\right )}}{2 \, {\left (d^{3} e + b d^{2} f - \frac {{\left (d e x + b f x + c e + a f\right )} d^{3}}{d x + c}\right )}} + \frac {{\left (b^{2} c^{2} d e f^{2} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )} - 2 \, a b c d^{2} e f^{2} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )} + a^{2} d^{3} e f^{2} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )} + b^{3} c^{2} f^{3} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )} - 2 \, a b^{2} c d f^{3} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )} + a^{2} b d^{2} f^{3} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )} - \frac {{\left (d e x + b f x + c e + a f\right )} b^{2} c^{2} d f^{2} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )}}{d x + c} + \frac {2 \, {\left (d e x + b f x + c e + a f\right )} a b c d^{2} f^{2} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )}}{d x + c} - \frac {{\left (d e x + b f x + c e + a f\right )} a^{2} d^{3} f^{2} {\rm Ei}\left (\frac {d e + b f - \frac {{\left (d e x + b f x + c e + a f\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {d e + b f}{d}\right )}}{d x + c} - b^{2} c^{2} d f^{2} e^{\left (-\frac {d e x + b f x + c e + a f}{d x + c}\right )} + 2 \, a b c d^{2} f^{2} e^{\left (-\frac {d e x + b f x + c e + a f}{d x + c}\right )} - a^{2} d^{3} f^{2} e^{\left (-\frac {d e x + b f x + c e + a f}{d x + c}\right )}\right )} {\left (\frac {{\left (d e + b f\right )} c}{{\left (b c f - a d f\right )}^{2}} - \frac {{\left (c e + a f\right )} d}{{\left (b c f - a d f\right )}^{2}}\right )}}{2 \, {\left (d^{3} e + b d^{2} f - \frac {{\left (d e x + b f x + c e + a f\right )} d^{3}}{d x + c}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {sinh}\left (e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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