Optimal. Leaf size=136 \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {4 a b \sinh (c+d x)}{d^3}-\frac {4 a b x \cosh (c+d x)}{d^2}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {24 b^2 \sinh (c+d x)}{d^5}-\frac {24 b^2 x \cosh (c+d x)}{d^4}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5277, 2637, 3296} \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {4 a b \sinh (c+d x)}{d^3}-\frac {4 a b x \cosh (c+d x)}{d^2}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {24 b^2 \sinh (c+d x)}{d^5}-\frac {24 b^2 x \cosh (c+d x)}{d^4}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5277
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^2 \cosh (c+d x) \, dx &=\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*}
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Mathematica [A] time = 0.20, size = 85, normalized size = 0.62 \[ \frac {\left (a^2 d^4+2 a b d^2 \left (d^2 x^2+2\right )+b^2 \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \sinh (c+d x)-4 b d x \left (a d^2+b \left (d^2 x^2+6\right )\right ) \cosh (c+d x)}{d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 98, normalized size = 0.72 \[ -\frac {4 \, {\left (b^{2} d^{3} x^{3} + {\left (a b d^{3} + 6 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) - {\left (b^{2} d^{4} x^{4} + a^{2} d^{4} + 4 \, a b d^{2} + 2 \, {\left (a b d^{4} + 6 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 180, normalized size = 1.32 \[ \frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} - 4 \, b^{2} d^{3} x^{3} + a^{2} d^{4} - 4 \, a b d^{3} x + 12 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} - 24 \, b^{2} d x + 24 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac {{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} + 4 \, b^{2} d^{3} x^{3} + a^{2} d^{4} + 4 \, a b d^{3} x + 12 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} + 24 \, b^{2} d x + 24 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 332, normalized size = 2.44 \[ \frac {\frac {b^{2} \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{4}}-\frac {4 b^{2} c \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4}}+\frac {6 b^{2} c^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{4}}+\frac {2 b a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{2}}-\frac {4 b^{2} c^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{4}}-\frac {4 b c a \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{2}}+\frac {b^{2} c^{4} \sinh \left (d x +c \right )}{d^{4}}+\frac {2 b \,c^{2} a \sinh \left (d x +c \right )}{d^{2}}+a^{2} \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 189, normalized size = 1.39 \[ \frac {a^{2} e^{\left (d x + c\right )}}{2 \, d} - \frac {a^{2} e^{\left (-d x - c\right )}}{2 \, d} + \frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{3}} - \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} a b e^{\left (-d x - c\right )}}{d^{3}} + \frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{2 \, d^{5}} - \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{2 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 114, normalized size = 0.84 \[ \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (a^2\,d^4+4\,a\,b\,d^2+24\,b^2\right )}{d^5}-\frac {4\,b^2\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {b^2\,x^4\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {4\,x\,\mathrm {cosh}\left (c+d\,x\right )\,\left (6\,b^2+a\,b\,d^2\right )}{d^4}+\frac {2\,x^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (6\,b^2+a\,b\,d^2\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 172, normalized size = 1.26 \[ \begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )}}{d} + \frac {2 a b x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {4 a b x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {4 a b \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b^{2} x^{4} \sinh {\left (c + d x \right )}}{d} - \frac {4 b^{2} x^{3} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {12 b^{2} x^{2} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {24 b^{2} x \cosh {\left (c + d x \right )}}{d^{4}} + \frac {24 b^{2} \sinh {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (a^{2} x + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{5}}{5}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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