\(\int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx\) [59]

Optimal result

Integrand size = 19, antiderivative size = 226 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}}+\frac {\sinh (c+d x)}{b d}-\frac {\sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5401, 2717, 5389, 3384, 3379, 3382} \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {\sqrt {-a} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sinh (c+d x)}{b d} \]

[In]

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

[Out]

(Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Cos
h[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2)) + Sinh[c + d*x]/(b*d) - (Sqr
t[-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Sinh[c
- (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2))

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 5389

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (a
+ b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 5401

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{b}-\frac {a \cosh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \cosh (c+d x) \, dx}{b}-\frac {a \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{b} \\ & = \frac {\sinh (c+d x)}{b d}-\frac {a \int \left (\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{b} \\ & = \frac {\sinh (c+d x)}{b d}-\frac {\sqrt {-a} \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b}-\frac {\sqrt {-a} \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b} \\ & = \frac {\sinh (c+d x)}{b d}-\frac {\left (\sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b}-\frac {\left (\sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b}-\frac {\left (\sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b}+\frac {\left (\sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b} \\ & = \frac {\sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}}+\frac {\sinh (c+d x)}{b d}-\frac {\sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {i \sqrt {a} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{4 b^{3/2}}-\frac {i \sqrt {a} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{4 b^{3/2}}+\frac {\cosh (d x) \sinh (c)}{b d}+\frac {\cosh (c) \sinh (d x)}{b d} \]

[In]

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

[Out]

((I/4)*Sqrt[a]*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sq
rt[b] + x)] - ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]))/b^(3/2) - ((I/4)*Sqrt[a]*E^(-c - (I*Sqrt[a]*d)/Sqrt
[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] - ExpIntegralEi[(I*Sqrt[a]*d
)/Sqrt[b] - d*x]))/b^(3/2) + (Cosh[d*x]*Sinh[c])/(b*d) + (Cosh[c]*Sinh[d*x])/(b*d)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.15

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a}{4 b \sqrt {-a b}}-\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a}{4 b \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a}{4 b \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a}{4 b \sqrt {-a b}}-\frac {{\mathrm e}^{-d x -c}}{2 d b}+\frac {{\mathrm e}^{d x +c}}{2 d b}\)	\(259\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (170) = 340\).

Time = 0.26 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.19 \[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\frac {{\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (\sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - \sqrt {-\frac {a d^{2}}{b}} {\left (\cosh \left (d x + c\right )^{2} - \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + 4 \, \sinh \left (d x + c\right )}{4 \, {\left (b d \cosh \left (d x + c\right )^{2} - b d \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

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

[Out]

1/4*((sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x - sqrt(-a*d^2/b)) + sqrt(-a*d^2/b)*(cosh(d*x +
c)^2 - sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqrt(-a*d^2/b)) - (sqrt(-a*d^2/b)*(cosh(d*x + c)^
2 - sinh(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) + sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(-d*x - s
qrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b)) + (sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x - sqrt(
-a*d^2/b)) - sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^
2/b)) + (sqrt(-a*d^2/b)*(cosh(d*x + c)^2 - sinh(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) - sqrt(-a*d^2/b)*(cosh(d*
x + c)^2 - sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c + sqrt(-a*d^2/b)) + 4*sinh(d*x + c))/(b*d*cosh(
d*x + c)^2 - b*d*sinh(d*x + c)^2)

Sympy [F]

\[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\int \frac {x^{2} \cosh {\left (c + d x \right )}}{a + b x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{b x^{2} + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{b x^{2} + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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