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

Optimal. Leaf size=107 \[ \frac {\sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac {\cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \cosh ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5608, 3313, 12, 3303, 3298, 3301} \[ \frac {\sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac {\cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \cosh ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[(a + b*x)/(c + d*x)]^2,x]

[Out]

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

Rule 5608

Int[Cosh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> -Dist[d^(-1), Subst[Int[Cosh[(
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]

Rubi steps

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

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Mathematica [A]  time = 0.31, size = 111, normalized size = 1.04 \[ \frac {2 \sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (a d-b c)}{d (c+d x)}\right )+2 \cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (a d-b c)}{d (c+d x)}\right )+d \left ((c+d x) \cosh \left (\frac {2 (a+b x)}{c+d x}\right )+d x\right )}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.64, size = 366, normalized size = 3.42 \[ \frac {d^{2} x + {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (d^{2} x - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) + {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 15.29, size = 749, normalized size = 7.00 \[ \frac {{\left (2 \, b^{3} c^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - 4 \, a b^{2} c d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} + 2 \, a^{2} b d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} + \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, b^{3} c^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + 4 \, a b^{2} c d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, a^{2} b d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} - \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + b^{2} c^{2} d e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + 2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{4 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.42, size = 358, normalized size = 3.35 \[ \frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} a}{\frac {4 d a}{d x +c}-\frac {4 b c}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} c b}{4 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 b}{d}} \Ei \left (1, \frac {2 d a -2 c b}{d \left (d x +c \right )}\right ) a}{2 d}+\frac {{\mathrm e}^{-\frac {2 b}{d}} \Ei \left (1, \frac {2 d a -2 c b}{d \left (d x +c \right )}\right ) c b}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x a}{4 d a -4 c b}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x c b}{4 \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c a}{4 d a -4 c b}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c^{2} b}{4 d \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {2 b}{d}} \Ei \left (1, -\frac {2 \left (d a -c b \right )}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {2 b}{d}} \Ei \left (1, -\frac {2 \left (d a -c b \right )}{d \left (d x +c \right )}\right ) c b}{2 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x + \frac {1}{4} \, \int e^{\left (\frac {2 \, b c}{d^{2} x + c d} - \frac {2 \, a}{d x + c} - \frac {2 \, b}{d}\right )}\,{d x} + \frac {1}{4} \, \int e^{\left (-\frac {2 \, b c}{d^{2} x + c d} + \frac {2 \, a}{d x + c} + \frac {2 \, b}{d}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh((a + b*x)/(c + d*x))^2,x)

[Out]

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

[Out]

Timed out

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