\(\int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx\) [161]

Optimal result

Integrand size = 18, antiderivative size = 87 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}+\frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {b \sinh (e+f x)}{d (c+d x)}+\frac {b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378, 3384, 3379, 3382} \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}+\frac {b f \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {b \sinh (e+f x)}{d (c+d x)} \]

[In]

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

[Out]

-(a/(d*(c + d*x))) + (b*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 - (b*Sinh[e + f*x])/(d*(c + d*x))
+ (b*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*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 3398

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-\frac {d (a+b \sinh (e+f x))}{c+d x}+b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.71

method	result	size
risch	\(-\frac {a}{d \left (d x +c \right )}+\frac {f b \,{\mathrm e}^{-f x -e}}{2 d \left (d f x +c f \right )}-\frac {f b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {f b \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f b \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}}\)	\(149\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx=-\frac {2 \, b d \sinh \left (f x + e\right ) + 2 \, a d - {\left ({\left (b d f x + b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (b d f x + b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (b d f x + b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (b d f x + b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} \, b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a}{d^{2} x + c d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (90) = 180\).

Time = 0.31 (sec) , antiderivative size = 630, normalized size of antiderivative = 7.24 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} \, b {\left (\frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} + \frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f}\right )} - \frac {a}{{\left (d x + c\right )} d} \]

[In]

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

[Out]

1/2*b*(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) -
d*e + c*f)/d)*e^((d*e - c*f)/d) - d*e*f^2*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^
((d*e - c*f)/d) + c*f^3*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^((d*e - c*f)/d) -
d*f^2*e^((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d))*d^2/(((d*x + c)*d^4*(d*e/(d*x + c) - c*f/(d*x + c)
+ f) - d^5*e + c*d^4*f)*f) + ((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-((d*x + c)*(d*e/(d*x + c)
- c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-(d*e - c*f)/d) - d*e*f^2*Ei(-((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c
) + f) - d*e + c*f)/d)*e^(-(d*e - c*f)/d) + c*f^3*Ei(-((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c
*f)/d)*e^(-(d*e - c*f)/d) + d*f^2*e^(-(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d))*d^2/(((d*x + c)*d^4*(d
*e/(d*x + c) - c*f/(d*x + c) + f) - d^5*e + c*d^4*f)*f)) - a/((d*x + c)*d)

Mupad [F(-1)]

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

[In]

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

[Out]

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