\(\int (a+b x^2)^2 \cosh (c+d x) \, dx\) [51]

Optimal result

Integrand size = 16, antiderivative size = 136 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]

[Out]

-24*b^2*x*cosh(d*x+c)/d^4-4*a*b*x*cosh(d*x+c)/d^2-4*b^2*x^3*cosh(d*x+c)/d^2+24*b^2*sinh(d*x+c)/d^5+4*a*b*sinh(
d*x+c)/d^3+a^2*sinh(d*x+c)/d+12*b^2*x^2*sinh(d*x+c)/d^3+2*a*b*x^2*sinh(d*x+c)/d+b^2*x^4*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5385, 2717, 3377} \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {a^2 \sinh (c+d x)}{d}+\frac {4 a b \sinh (c+d x)}{d^3}-\frac {4 a b x \cosh (c+d x)}{d^2}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {24 b^2 \sinh (c+d x)}{d^5}-\frac {24 b^2 x \cosh (c+d x)}{d^4}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]

[In]

Int[(a + b*x^2)^2*Cosh[c + d*x],x]

[Out]

(-24*b^2*x*Cosh[c + d*x])/d^4 - (4*a*b*x*Cosh[c + d*x])/d^2 - (4*b^2*x^3*Cosh[c + d*x])/d^2 + (24*b^2*Sinh[c +
d*x])/d^5 + (4*a*b*Sinh[c + d*x])/d^3 + (a^2*Sinh[c + d*x])/d + (12*b^2*x^2*Sinh[c + d*x])/d^3 + (2*a*b*x^2*S
inh[c + d*x])/d + (b^2*x^4*Sinh[c + d*x])/d

Rule 2717

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5385

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, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cosh (c+d x)+2 a b x^2 \cosh (c+d x)+b^2 x^4 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \cosh (c+d x) \, dx+(2 a b) \int x^2 \cosh (c+d x) \, dx+b^2 \int x^4 \cosh (c+d x) \, dx \\ & = \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}-\frac {(4 a b) \int x \sinh (c+d x) \, dx}{d}-\frac {\left (4 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d} \\ & = -\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}+\frac {(4 a b) \int \cosh (c+d x) \, dx}{d^2}+\frac {\left (12 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}-\frac {\left (24 b^2\right ) \int x \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}+\frac {\left (24 b^2\right ) \int \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {-4 b d x \left (a d^2+b \left (6+d^2 x^2\right )\right ) \cosh (c+d x)+\left (a^2 d^4+2 a b d^2 \left (2+d^2 x^2\right )+b^2 \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^5} \]

[In]

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

[Out]

(-4*b*d*x*(a*d^2 + b*(6 + d^2*x^2))*Cosh[c + d*x] + (a^2*d^4 + 2*a*b*d^2*(2 + d^2*x^2) + b^2*(24 + 12*d^2*x^2
+ d^4*x^4))*Sinh[c + d*x])/d^5

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86

method	result	size
parallelrisch	\(\frac {4 d x b \left (\left (b \,x^{2}+a \right ) d^{2}+6 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \left (-\left (b \,x^{2}+a \right )^{2} d^{4}-4 b \left (3 b \,x^{2}+a \right ) d^{2}-24 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4 d x b \left (\left (b \,x^{2}+a \right ) d^{2}+6 b \right )}{d^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\)	\(117\)
risch	\(\frac {\left (b^{2} x^{4} d^{4}+2 a b \,d^{4} x^{2}-4 b^{2} d^{3} x^{3}+a^{2} d^{4}-4 a b \,d^{3} x +12 x^{2} d^{2} b^{2}+4 a \,d^{2} b -24 b^{2} d x +24 b^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{5}}-\frac {\left (b^{2} x^{4} d^{4}+2 a b \,d^{4} x^{2}+4 b^{2} d^{3} x^{3}+a^{2} d^{4}+4 a b \,d^{3} x +12 x^{2} d^{2} b^{2}+4 a \,d^{2} b +24 b^{2} d x +24 b^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{5}}\)	\(181\)
parts	\(\frac {b^{2} x^{4} \sinh \left (d x +c \right )}{d}+\frac {2 a b \,x^{2} \sinh \left (d x +c \right )}{d}+\frac {a^{2} \sinh \left (d x +c \right )}{d}-\frac {4 b \left (\frac {3 b \,c^{2} \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )}{d^{2}}-\frac {b \,c^{3} \cosh \left (d x +c \right )}{d^{2}}-\frac {3 b c \left (\left (d x +c \right )^{2} \cosh \left (d x +c \right )-2 \left (d x +c \right ) \sinh \left (d x +c \right )+2 \cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b \left (\left (d x +c \right )^{3} \cosh \left (d x +c \right )-3 \left (d x +c \right )^{2} \sinh \left (d x +c \right )+6 \left (d x +c \right ) \cosh \left (d x +c \right )-6 \sinh \left (d x +c \right )\right )}{d^{2}}+a \left (\left (d x +c \right ) \cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )-a c \cosh \left (d x +c \right )\right )}{d^{3}}\)	\(231\)
meijerg	\(-\frac {16 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {16 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}+\frac {8 i a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {8 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+\frac {a^{2} \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}\)	\(267\)
derivativedivides	\(\frac {\frac {b^{2} c^{4} \sinh \left (d x +c \right )}{d^{4}}-\frac {4 b^{2} c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\frac {6 b^{2} c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b \,c^{2} a \sinh \left (d x +c \right )}{d^{2}}-\frac {4 b^{2} c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}-\frac {4 b c a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+a^{2} \sinh \left (d x +c \right )}{d}\)	\(332\)
default	\(\frac {\frac {b^{2} c^{4} \sinh \left (d x +c \right )}{d^{4}}-\frac {4 b^{2} c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\frac {6 b^{2} c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b \,c^{2} a \sinh \left (d x +c \right )}{d^{2}}-\frac {4 b^{2} c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}-\frac {4 b c a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}+a^{2} \sinh \left (d x +c \right )}{d}\)	\(332\)

[In]

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

[Out]

2*(2*d*x*b*((b*x^2+a)*d^2+6*b)*tanh(1/2*d*x+1/2*c)^2+(-(b*x^2+a)^2*d^4-4*b*(3*b*x^2+a)*d^2-24*b^2)*tanh(1/2*d*
x+1/2*c)+2*d*x*b*((b*x^2+a)*d^2+6*b))/d^5/(tanh(1/2*d*x+1/2*c)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.72 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=-\frac {4 \, {\left (b^{2} d^{3} x^{3} + {\left (a b d^{3} + 6 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{4} x^{4} + a^{2} d^{4} + 4 \, a b d^{2} + 2 \, {\left (a b d^{4} + 6 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{5}} \]

[In]

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

[Out]

-(4*(b^2*d^3*x^3 + (a*b*d^3 + 6*b^2*d)*x)*cosh(d*x + c) - (b^2*d^4*x^4 + a^2*d^4 + 4*a*b*d^2 + 2*(a*b*d^4 + 6*
b^2*d^2)*x^2 + 24*b^2)*sinh(d*x + c))/d^5

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )}}{d} + \frac {2 a b x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {4 a b x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {4 a b \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b^{2} x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {4 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {24 b^{2} x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {24 b^{2} \sinh {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (a^{2} x + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{5}}{5}\right ) \cosh {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*sinh(c + d*x)/d + 2*a*b*x**2*sinh(c + d*x)/d - 4*a*b*x*cosh(c + d*x)/d**2 + 4*a*b*sinh(c + d*x
)/d**3 + b**2*x**4*sinh(c + d*x)/d - 4*b**2*x**3*cosh(c + d*x)/d**2 + 12*b**2*x**2*sinh(c + d*x)/d**3 - 24*b**
2*x*cosh(c + d*x)/d**4 + 24*b**2*sinh(c + d*x)/d**5, Ne(d, 0)), ((a**2*x + 2*a*b*x**3/3 + b**2*x**5/5)*cosh(c)
, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.39 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {a^{2} e^{\left (d x + c\right )}}{2 \, d} - \frac {a^{2} e^{\left (-d x - c\right )}}{2 \, d} + \frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{3}} - \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} a b e^{\left (-d x - c\right )}}{d^{3}} + \frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{2 \, d^{5}} - \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]

[In]

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

[Out]

1/2*a^2*e^(d*x + c)/d - 1/2*a^2*e^(-d*x - c)/d + (d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*a*b*e^(d*x)/d^3 - (d^2*x^2
+ 2*d*x + 2)*a*b*e^(-d*x - c)/d^3 + 1/2*(d^4*x^4*e^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*b
^2*e^(d*x)/d^5 - 1/2*(d^4*x^4 + 4*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*b^2*e^(-d*x - c)/d^5

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.32 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} - 4 \, b^{2} d^{3} x^{3} + a^{2} d^{4} - 4 \, a b d^{3} x + 12 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} - 24 \, b^{2} d x + 24 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} + 4 \, b^{2} d^{3} x^{3} + a^{2} d^{4} + 4 \, a b d^{3} x + 12 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} + 24 \, b^{2} d x + 24 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]

[In]

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

[Out]

1/2*(b^2*d^4*x^4 + 2*a*b*d^4*x^2 - 4*b^2*d^3*x^3 + a^2*d^4 - 4*a*b*d^3*x + 12*b^2*d^2*x^2 + 4*a*b*d^2 - 24*b^2
*d*x + 24*b^2)*e^(d*x + c)/d^5 - 1/2*(b^2*d^4*x^4 + 2*a*b*d^4*x^2 + 4*b^2*d^3*x^3 + a^2*d^4 + 4*a*b*d^3*x + 12
*b^2*d^2*x^2 + 4*a*b*d^2 + 24*b^2*d*x + 24*b^2)*e^(-d*x - c)/d^5

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx=\frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+4\,a\,b\,d^2+24\,b^2\right )}{d^5}-\frac {4\,b^2\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {4\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (6\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (6\,b^2+a\,b\,d^2\right )}{d^3} \]

[In]

int(cosh(c + d*x)*(a + b*x^2)^2,x)

[Out]

(sinh(c + d*x)*(24*b^2 + a^2*d^4 + 4*a*b*d^2))/d^5 - (4*b^2*x^3*cosh(c + d*x))/d^2 + (b^2*x^4*sinh(c + d*x))/d
- (4*x*cosh(c + d*x)*(6*b^2 + a*b*d^2))/d^4 + (2*x^2*sinh(c + d*x)*(6*b^2 + a*b*d^2))/d^3