3.88 \(\int (a+b x^3)^2 \cosh (c+d x) \, dx\)

Optimal. Leaf size=186 \[ \frac {a^2 \sinh (c+d x)}{d}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]

[Out]

-12*a*b*cosh(d*x+c)/d^4-720*b^2*x*cosh(d*x+c)/d^6-6*a*b*x^2*cosh(d*x+c)/d^2-120*b^2*x^3*cosh(d*x+c)/d^4-6*b^2*
x^5*cosh(d*x+c)/d^2+720*b^2*sinh(d*x+c)/d^7+a^2*sinh(d*x+c)/d+12*a*b*x*sinh(d*x+c)/d^3+360*b^2*x^2*sinh(d*x+c)
/d^5+2*a*b*x^3*sinh(d*x+c)/d+30*b^2*x^4*sinh(d*x+c)/d^3+b^2*x^6*sinh(d*x+c)/d

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Rubi [A]  time = 0.29, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5277, 2637, 3296, 2638} \[ \frac {a^2 \sinh (c+d x)}{d}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-12*a*b*Cosh[c + d*x])/d^4 - (720*b^2*x*Cosh[c + d*x])/d^6 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cos
h[c + d*x])/d^4 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7 + (a^2*Sinh[c + d*x])/d + (12*a*
b*x*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (2*a*b*x^3*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c +
d*x])/d^3 + (b^2*x^6*Sinh[c + d*x])/d

Rule 2637

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

Rule 2638

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

Rule 3296

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 5277

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*} \int \left (a+b x^3\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^6 \cosh (c+d x) \, dx\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac {\left (6 b^2\right ) \int x^5 \sinh (c+d x) \, dx}{d}\\ &=-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac {\left (30 b^2\right ) \int x^4 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac {\left (120 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (360 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^4}\\ &=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {\left (720 b^2\right ) \int x \sinh (c+d x) \, dx}{d^5}\\ &=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (720 b^2\right ) \int \cosh (c+d x) \, dx}{d^6}\\ &=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {720 b^2 x \cosh (c+d x)}{d^6}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {a^2 \sinh (c+d x)}{d}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {b^2 x^6 \sinh (c+d x)}{d}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 111, normalized size = 0.60 \[ \frac {\left (a^2 d^6+2 a b d^4 x \left (d^2 x^2+6\right )+b^2 \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \sinh (c+d x)-6 b d \left (a d^2 \left (d^2 x^2+2\right )+b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \cosh (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2*Cosh[c + d*x],x]

[Out]

(-6*b*d*(a*d^2*(2 + d^2*x^2) + b*x*(120 + 20*d^2*x^2 + d^4*x^4))*Cosh[c + d*x] + (a^2*d^6 + 2*a*b*d^4*x*(6 + d
^2*x^2) + b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Sinh[c + d*x])/d^7

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fricas [A]  time = 0.65, size = 130, normalized size = 0.70 \[ -\frac {6 \, {\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} + 20 \, b^{2} d^{3} x^{3} + 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c),x, algorithm="fricas")

[Out]

-(6*(b^2*d^5*x^5 + a*b*d^5*x^2 + 20*b^2*d^3*x^3 + 2*a*b*d^3 + 120*b^2*d*x)*cosh(d*x + c) - (b^2*d^6*x^6 + 2*a*
b*d^6*x^3 + 30*b^2*d^4*x^4 + a^2*d^6 + 12*a*b*d^4*x + 360*b^2*d^2*x^2 + 720*b^2)*sinh(d*x + c))/d^7

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giac [A]  time = 0.14, size = 244, normalized size = 1.31 \[ \frac {{\left (b^{2} d^{6} x^{6} - 6 \, b^{2} d^{5} x^{5} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} - 6 \, a b d^{5} x^{2} + a^{2} d^{6} - 120 \, b^{2} d^{3} x^{3} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 12 \, a b d^{3} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac {{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} d^{5} x^{5} + 2 \, a b d^{6} x^{3} + 30 \, b^{2} d^{4} x^{4} + 6 \, a b d^{5} x^{2} + a^{2} d^{6} + 120 \, b^{2} d^{3} x^{3} + 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} + 12 \, a b d^{3} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c),x, algorithm="giac")

[Out]

1/2*(b^2*d^6*x^6 - 6*b^2*d^5*x^5 + 2*a*b*d^6*x^3 + 30*b^2*d^4*x^4 - 6*a*b*d^5*x^2 + a^2*d^6 - 120*b^2*d^3*x^3
+ 12*a*b*d^4*x + 360*b^2*d^2*x^2 - 12*a*b*d^3 - 720*b^2*d*x + 720*b^2)*e^(d*x + c)/d^7 - 1/2*(b^2*d^6*x^6 + 6*
b^2*d^5*x^5 + 2*a*b*d^6*x^3 + 30*b^2*d^4*x^4 + 6*a*b*d^5*x^2 + a^2*d^6 + 120*b^2*d^3*x^3 + 12*a*b*d^4*x + 360*
b^2*d^2*x^2 + 12*a*b*d^3 + 720*b^2*d*x + 720*b^2)*e^(-d*x - c)/d^7

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maple [B]  time = 0.04, size = 592, normalized size = 3.18 \[ \frac {\frac {b^{2} \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{6}}+\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{6}}+a^{2} \sinh \left (d x +c \right )-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d^{3}}-\frac {6 b^{2} c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{6}}+\frac {15 b^{2} c^{4} \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^{6}}-\frac {6 b^{2} c \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{6}}+\frac {15 b^{2} c^{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^{6}}-\frac {20 b^{2} c^{3} \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^{6}}+\frac {2 b a \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^{3}}-\frac {6 b c 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^{3}}+\frac {6 b \,c^{2} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*cosh(d*x+c),x)

[Out]

1/d*(1/d^6*b^2*((d*x+c)^6*sinh(d*x+c)-6*(d*x+c)^5*cosh(d*x+c)+30*(d*x+c)^4*sinh(d*x+c)-120*(d*x+c)^3*cosh(d*x+
c)+360*(d*x+c)^2*sinh(d*x+c)-720*(d*x+c)*cosh(d*x+c)+720*sinh(d*x+c))+1/d^6*b^2*c^6*sinh(d*x+c)+a^2*sinh(d*x+c
)-2/d^3*b*c^3*a*sinh(d*x+c)-6/d^6*b^2*c^5*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+15/d^6*b^2*c^4*((d*x+c)^2*sinh(d*x
+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-6/d^6*b^2*c*((d*x+c)^5*sinh(d*x+c)-5*(d*x+c)^4*cosh(d*x+c)+20*(d*x+c)
^3*sinh(d*x+c)-60*(d*x+c)^2*cosh(d*x+c)+120*(d*x+c)*sinh(d*x+c)-120*cosh(d*x+c))+15/d^6*b^2*c^2*((d*x+c)^4*sin
h(d*x+c)-4*(d*x+c)^3*cosh(d*x+c)+12*(d*x+c)^2*sinh(d*x+c)-24*(d*x+c)*cosh(d*x+c)+24*sinh(d*x+c))-20/d^6*b^2*c^
3*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))+2/d^3*b*a*((d*x+c)^3*sin
h(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))-6/d^3*b*c*a*((d*x+c)^2*sinh(d*x+c)-2*(d*
x+c)*cosh(d*x+c)+2*sinh(d*x+c))+6/d^3*b*c^2*a*((d*x+c)*sinh(d*x+c)-cosh(d*x+c)))

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maxima [A]  time = 0.35, size = 243, normalized size = 1.31 \[ \frac {a^{2} e^{\left (d x + c\right )}}{2 \, d} - \frac {a^{2} e^{\left (-d x - c\right )}}{2 \, d} + \frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{4}} - \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a b e^{\left (-d x - c\right )}}{d^{4}} + \frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{2 \, d^{7}} - \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{2} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+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^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*a*b*e^(d*x)
/d^4 - (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*a*b*e^(-d*x - c)/d^4 + 1/2*(d^6*x^6*e^c - 6*d^5*x^5*e^c + 30*d^4*x^4*
e^c - 120*d^3*x^3*e^c + 360*d^2*x^2*e^c - 720*d*x*e^c + 720*e^c)*b^2*e^(d*x)/d^7 - 1/2*(d^6*x^6 + 6*d^5*x^5 +
30*d^4*x^4 + 120*d^3*x^3 + 360*d^2*x^2 + 720*d*x + 720)*b^2*e^(-d*x - c)/d^7

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mupad [B]  time = 0.24, size = 182, normalized size = 0.98 \[ \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^6+720\,b^2\right )}{d^7}-\frac {6\,b^2\,x^5\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}-\frac {120\,b^2\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}+\frac {b^2\,x^6\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {30\,b^2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}+\frac {360\,b^2\,x^2\,\mathrm {sinh}\left (c+d\,x\right )}{d^5}-\frac {12\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}-\frac {720\,b^2\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^6}-\frac {6\,a\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,a\,b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d}+\frac {12\,a\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)*(a + b*x^3)^2,x)

[Out]

(sinh(c + d*x)*(720*b^2 + a^2*d^6))/d^7 - (6*b^2*x^5*cosh(c + d*x))/d^2 - (120*b^2*x^3*cosh(c + d*x))/d^4 + (b
^2*x^6*sinh(c + d*x))/d + (30*b^2*x^4*sinh(c + d*x))/d^3 + (360*b^2*x^2*sinh(c + d*x))/d^5 - (12*a*b*cosh(c +
d*x))/d^4 - (720*b^2*x*cosh(c + d*x))/d^6 - (6*a*b*x^2*cosh(c + d*x))/d^2 + (2*a*b*x^3*sinh(c + d*x))/d + (12*
a*b*x*sinh(c + d*x))/d^3

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sympy [A]  time = 5.86, size = 226, normalized size = 1.22 \[ \begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )}}{d} + \frac {2 a b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {6 a b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b^{2} x^{6} \sinh {\left (c + d x \right )}}{d} - \frac {6 b^{2} x^{5} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {360 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {720 b^{2} x \cosh {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \sinh {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (a^{2} x + \frac {a b x^{4}}{2} + \frac {b^{2} x^{7}}{7}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*cosh(d*x+c),x)

[Out]

Piecewise((a**2*sinh(c + d*x)/d + 2*a*b*x**3*sinh(c + d*x)/d - 6*a*b*x**2*cosh(c + d*x)/d**2 + 12*a*b*x*sinh(c
+ d*x)/d**3 - 12*a*b*cosh(c + d*x)/d**4 + b**2*x**6*sinh(c + d*x)/d - 6*b**2*x**5*cosh(c + d*x)/d**2 + 30*b**
2*x**4*sinh(c + d*x)/d**3 - 120*b**2*x**3*cosh(c + d*x)/d**4 + 360*b**2*x**2*sinh(c + d*x)/d**5 - 720*b**2*x*c
osh(c + d*x)/d**6 + 720*b**2*sinh(c + d*x)/d**7, Ne(d, 0)), ((a**2*x + a*b*x**4/2 + b**2*x**7/7)*cosh(c), True
))

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