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

Optimal. Leaf size=81 \[ \frac {b \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {\cosh ^2(a+b x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3313, 12, 3303, 3298, 3301} \[ \frac {b \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {\cosh ^2(a+b x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^2/(c + d*x)^2,x]

[Out]

-(Cosh[a + b*x]^2/(d*(c + d*x))) + (b*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/d^2 + (b*Cosh[2*a
- (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*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 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 3313

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]

Rubi steps

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

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Mathematica [A]  time = 0.41, size = 75, normalized size = 0.93 \[ \frac {b \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b (c+d x)}{d}\right )+b \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )-\frac {d \cosh ^2(a+b x)}{c+d x}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 164, normalized size = 2.02 \[ -\frac {d \cosh \left (b x + a\right )^{2} + d \sinh \left (b x + a\right )^{2} - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.19, size = 574, normalized size = 7.09 \[ -\frac {{\left (2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} + 2 \, b^{3} c {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} - 2 \, a b^{2} d {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} - 2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} - 2 \, b^{3} c {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} + 2 \, a b^{2} d {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} + b^{2} d e^{\left (\frac {2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} + b^{2} d e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} + 2 \, b^{2} d\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.24, size = 152, normalized size = 1.88 \[ -\frac {1}{2 d \left (d x +c \right )}-\frac {b \,{\mathrm e}^{-2 b x -2 a}}{4 \left (b d x +c b \right ) d}+\frac {b \,{\mathrm e}^{-\frac {2 \left (d a -c b \right )}{d}} \Ei \left (1, 2 b x +2 a -\frac {2 \left (d a -c b \right )}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{2 b x +2 a}}{4 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {b \,{\mathrm e}^{\frac {2 d a -2 c b}{d}} \Ei \left (1, -2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right )}{2 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 88, normalized size = 1.09 \[ -\frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )} d} - \frac {1}{2 \, {\left (d^{2} x + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^2/(c + d*x)^2,x)

[Out]

int(cosh(a + b*x)^2/(c + d*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x)**2/(c + d*x)**2, x)

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