3.160 \(\int \frac {a+b \cosh (e+f x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac {a}{d (c+d x)}+\frac {b f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {b \cosh (e+f x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3317, 3297, 3303, 3298, 3301} \[ -\frac {a}{d (c+d x)}+\frac {b f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {b \cosh (e+f x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3297

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 3298

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 3301

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 3303

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 3317

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

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Mathematica [A]  time = 0.42, size = 71, normalized size = 0.82 \[ \frac {-\frac {d (a+b \cosh (e+f x))}{c+d x}+b f \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+b f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 162, normalized size = 1.86 \[ -\frac {2 \, b d \cosh \left (f x + e\right ) + 2 \, a d - {\left ({\left (b d f x + b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (b d f x + b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (b d f x + b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (b d f x + b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*d*cosh(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)

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giac [B]  time = 0.16, size = 683, normalized size = 7.85 \[ -\frac {1}{2} \, b {\left (\frac {{\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (\frac {c f - d e}{d}\right )} - c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (\frac {c f - d e}{d}\right )} + d f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (\frac {c f - d e}{d} + 1\right )} - d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c d^{4} f + d^{5} e\right )} f} - \frac {{\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (-\frac {c f - d e}{d}\right )} - c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (-\frac {c f - d e}{d}\right )} + d f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (-\frac {c f - d e}{d} + 1\right )} + d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c d^{4} f + d^{5} e\right )} f}\right )} - \frac {a}{{\left (d x + c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 149, normalized size = 1.71 \[ -\frac {a}{d \left (d x +c \right )}-\frac {f b \,{\mathrm e}^{-f x -e}}{2 d \left (d f x +c f \right )}+\frac {f b \,{\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {f b \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f b \,{\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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maxima [A]  time = 0.40, size = 87, normalized size = 1.00 \[ -\frac {1}{2} \, b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a}{d^{2} x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {cosh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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