Optimal. Leaf size=234 \[ \frac {2 a^2 \sinh (c+d x)}{d^3}-\frac {2 a^2 x \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {48 a b \sinh (c+d x)}{d^5}-\frac {48 a b x \cosh (c+d x)}{d^4}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.39, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5287, 3296, 2637} \[ \frac {2 a^2 \sinh (c+d x)}{d^3}-\frac {2 a^2 x \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}-\frac {48 a b x \cosh (c+d x)}{d^4}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac {720 b^2 \sinh (c+d x)}{d^7}-\frac {720 b^2 x \cosh (c+d x)}{d^6}+\frac {b^2 x^6 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5287
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 x^2 \cosh (c+d x)+2 a b x^4 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x^2 \cosh (c+d x) \, dx+(2 a b) \int x^4 \cosh (c+d x) \, dx+b^2 \int x^6 \cosh (c+d x) \, dx\\ &=\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {\left (2 a^2\right ) \int x \sinh (c+d x) \, dx}{d}-\frac {(8 a b) \int x^3 \sinh (c+d x) \, dx}{d}-\frac {\left (6 b^2\right ) \int x^5 \sinh (c+d x) \, dx}{d}\\ &=-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {\left (2 a^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac {(24 a b) \int x^2 \cosh (c+d x) \, dx}{d^2}+\frac {\left (30 b^2\right ) \int x^4 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}-\frac {(48 a b) \int x \sinh (c+d x) \, dx}{d^3}-\frac {\left (120 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}+\frac {(48 a b) \int \cosh (c+d x) \, dx}{d^4}+\frac {\left (360 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^4}\\ &=-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\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 {720 b^2 x \cosh (c+d x)}{d^6}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\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 {720 b^2 x \cosh (c+d x)}{d^6}-\frac {48 a b x \cosh (c+d x)}{d^4}-\frac {2 a^2 x \cosh (c+d x)}{d^2}-\frac {120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac {8 a b x^3 \cosh (c+d x)}{d^2}-\frac {6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac {720 b^2 \sinh (c+d x)}{d^7}+\frac {48 a b \sinh (c+d x)}{d^5}+\frac {2 a^2 \sinh (c+d x)}{d^3}+\frac {360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac {24 a b x^2 \sinh (c+d x)}{d^3}+\frac {a^2 x^2 \sinh (c+d x)}{d}+\frac {30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac {2 a b x^4 \sinh (c+d x)}{d}+\frac {b^2 x^6 \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 138, normalized size = 0.59 \[ \frac {\left (a^2 d^4 \left (d^2 x^2+2\right )+2 a b d^2 \left (d^4 x^4+12 d^2 x^2+24\right )+b^2 \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \sinh (c+d x)-2 d x \left (a^2 d^4+4 a b d^2 \left (d^2 x^2+6\right )+3 b^2 \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \cosh (c+d x)}{d^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 155, normalized size = 0.66 \[ -\frac {2 \, {\left (3 \, b^{2} d^{5} x^{5} + 4 \, {\left (a b d^{5} + 15 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} + 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{6} x^{6} + 2 \, a^{2} d^{4} + 2 \, {\left (a b d^{6} + 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} + {\left (a^{2} d^{6} + 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 304, normalized size = 1.30 \[ \frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} - 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} - 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} - 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} - 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} - 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} + 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} + 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} + 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} + 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 738, normalized size = 3.15 \[ \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^{4}}+\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^{4}}+\frac {2 b a \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^{2}}-\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^{4}}-\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^{4}}+\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^{4}}+\frac {b^{2} c^{6} \sinh \left (d x +c \right )}{d^{4}}-2 a^{2} c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a^{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 )+\frac {2 b \,c^{4} a \sinh \left (d x +c \right )}{d^{2}}-\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^{4}}+a^{2} c^{2} \sinh \left (d x +c \right )-\frac {8 b \,c^{3} a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}-\frac {8 b a 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^{2}}+\frac {12 b \,c^{2} 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}}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 383, normalized size = 1.64 \[ -\frac {1}{210} \, d {\left (\frac {35 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac {35 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac {42 \, {\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{6}} + \frac {42 \, {\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} a b e^{\left (-d x - c\right )}}{d^{6}} + \frac {15 \, {\left (d^{7} x^{7} e^{c} - 7 \, d^{6} x^{6} e^{c} + 42 \, d^{5} x^{5} e^{c} - 210 \, d^{4} x^{4} e^{c} + 840 \, d^{3} x^{3} e^{c} - 2520 \, d^{2} x^{2} e^{c} + 5040 \, d x e^{c} - 5040 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{8}} + \frac {15 \, {\left (d^{7} x^{7} + 7 \, d^{6} x^{6} + 42 \, d^{5} x^{5} + 210 \, d^{4} x^{4} + 840 \, d^{3} x^{3} + 2520 \, d^{2} x^{2} + 5040 \, d x + 5040\right )} b^{2} e^{\left (-d x - c\right )}}{d^{8}}\right )} + \frac {1}{105} \, {\left (15 \, b^{2} x^{7} + 42 \, a b x^{5} + 35 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 182, normalized size = 0.78 \[ \frac {2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^7}-\frac {6\,b^2\,x^5\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^6\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {2\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^6}+\frac {x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+24\,a\,b\,d^2+360\,b^2\right )}{d^5}-\frac {8\,x^3\,\mathrm {cosh}\left (c+d\,x\right )\,\left (15\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )\,\left (15\,b^2+a\,b\,d^2\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.02, size = 286, normalized size = 1.22 \[ \begin {cases} \frac {a^{2} x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a^{2} x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a^{2} \sinh {\left (c + d x \right )}}{d^{3}} + \frac {2 a b x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {8 a b x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {48 a b \sinh {\left (c + d x \right )}}{d^{5}} + \frac {b^{2} x^{6} \sinh {\left (c + d x \right )}}{d} - \frac {6 b^{2} x^{5} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {360 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {720 b^{2} x \cosh {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \sinh {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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