Optimal. Leaf size=184 \[ -\frac {a^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {2 a b x^3 \sinh (c+d x)}{d}-\frac {120 b^2 \cosh (c+d x)}{d^6}+\frac {120 b^2 x \sinh (c+d x)}{d^5}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {b^2 x^5 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.28, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5287, 3296, 2638} \[ -\frac {a^2 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}+\frac {12 a b x \sinh (c+d x)}{d^3}-\frac {12 a b \cosh (c+d x)}{d^4}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac {120 b^2 x \sinh (c+d x)}{d^5}-\frac {120 b^2 \cosh (c+d x)}{d^6}+\frac {b^2 x^5 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5287
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 x \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^5 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^5 \cosh (c+d x) \, dx\\ &=\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}-\frac {a^2 \int \sinh (c+d x) \, dx}{d}-\frac {(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac {\left (5 b^2\right ) \int x^4 \sinh (c+d x) \, dx}{d}\\ &=-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}+\frac {(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac {\left (20 b^2\right ) \int x^3 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}-\frac {(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac {\left (60 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}+\frac {\left (120 b^2\right ) \int x \cosh (c+d x) \, dx}{d^4}\\ &=-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {120 b^2 x \sinh (c+d x)}{d^5}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}-\frac {\left (120 b^2\right ) \int \sinh (c+d x) \, dx}{d^5}\\ &=-\frac {120 b^2 \cosh (c+d x)}{d^6}-\frac {12 a b \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {6 a b x^2 \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {120 b^2 x \sinh (c+d x)}{d^5}+\frac {12 a b x \sinh (c+d x)}{d^3}+\frac {a^2 x \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {2 a b x^3 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 113, normalized size = 0.61 \[ \frac {d x \left (a^2 d^4+2 a b d^2 \left (d^2 x^2+6\right )+b^2 \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)-\left (a^2 d^4+6 a b d^2 \left (d^2 x^2+2\right )+5 b^2 \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 126, normalized size = 0.68 \[ -\frac {{\left (5 \, b^{2} d^{4} x^{4} + a^{2} d^{4} + 12 \, a b d^{2} + 6 \, {\left (a b d^{4} + 10 \, b^{2} d^{2}\right )} x^{2} + 120 \, b^{2}\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{5} x^{5} + 2 \, {\left (a b d^{5} + 10 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} + 12 \, a b d^{3} + 120 \, b^{2} d\right )} x\right )} \sinh \left (d x + c\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 239, normalized size = 1.30 \[ \frac {{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} - 5 \, b^{2} d^{4} x^{4} + a^{2} d^{5} x - 6 \, a b d^{4} x^{2} + 20 \, b^{2} d^{3} x^{3} - a^{2} d^{4} + 12 \, a b d^{3} x - 60 \, b^{2} d^{2} x^{2} - 12 \, a b d^{2} + 120 \, b^{2} d x - 120 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} + 5 \, b^{2} d^{4} x^{4} + a^{2} d^{5} x + 6 \, a b d^{4} x^{2} + 20 \, b^{2} d^{3} x^{3} + a^{2} d^{4} + 12 \, a b d^{3} x + 60 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} + 120 \, b^{2} d x + 120 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 513, normalized size = 2.79 \[ \frac {\frac {b^{2} \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^{4}}-\frac {5 b^{2} c \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 {10 b^{2} c^{2} \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 {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^{2}}-\frac {10 b^{2} c^{3} \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 {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^{2}}+\frac {5 b^{2} c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}+\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^{2}}+a^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-\frac {b^{2} c^{5} \sinh \left (d x +c \right )}{d^{4}}-\frac {2 b \,c^{3} a \sinh \left (d x +c \right )}{d^{2}}-c \,a^{2} \sinh \left (d x +c \right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 353, normalized size = 1.92 \[ \frac {{\left (b x^{2} + a\right )}^{3} \cosh \left (d x + c\right )}{6 \, b} - \frac {{\left (\frac {a^{3} e^{\left (d x + c\right )}}{d} + \frac {a^{3} e^{\left (-d x - c\right )}}{d} + \frac {3 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a^{2} b e^{\left (d x\right )}}{d^{3}} + \frac {3 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a^{2} b e^{\left (-d x - c\right )}}{d^{3}} + \frac {3 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a b^{2} e^{\left (d x\right )}}{d^{5}} + \frac {3 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a b^{2} e^{\left (-d x - c\right )}}{d^{5}} + \frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{3} e^{\left (d x\right )}}{d^{7}} + \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{3} e^{\left (-d x - c\right )}}{d^{7}}\right )} d}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 148, normalized size = 0.80 \[ \frac {b^2\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {5\,b^2\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^4+12\,a\,b\,d^2+120\,b^2\right )}{d^6}+\frac {x\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+12\,a\,b\,d^2+120\,b^2\right )}{d^5}-\frac {6\,x^2\,\mathrm {cosh}\left (c+d\,x\right )\,\left (10\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (10\,b^2+a\,b\,d^2\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.58, size = 226, normalized size = 1.23 \[ \begin {cases} \frac {a^{2} x \sinh {\left (c + d x \right )}}{d} - \frac {a^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a b x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {6 a b x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b^{2} x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b^{2} x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b^{2} x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b^{2} x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b^{2} x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b^{2} \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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