Optimal. Leaf size=147 \[ \frac {a^2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac {2 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {2 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\sinh (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.37, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6742, 2637, 3297, 3303, 3298, 3301} \[ \frac {a^2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac {2 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {2 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\sinh (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2 \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac {\cosh (c+d x)}{b^2}+\frac {a^2 \cosh (c+d x)}{b^2 (a+b x)^2}-\frac {2 a \cosh (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {\int \cosh (c+d x) \, dx}{b^2}-\frac {(2 a) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^2}+\frac {a^2 \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b^2}\\ &=-\frac {a^2 \cosh (c+d x)}{b^3 (a+b x)}+\frac {\sinh (c+d x)}{b^2 d}+\frac {\left (a^2 d\right ) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^3}-\frac {\left (2 a \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}-\frac {\left (2 a \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=-\frac {a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac {2 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\sinh (c+d x)}{b^2 d}-\frac {2 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\left (a^2 d \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^3}+\frac {\left (a^2 d \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^3}\\ &=-\frac {a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac {2 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a^2 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^4}+\frac {\sinh (c+d x)}{b^2 d}+\frac {a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {2 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 115, normalized size = 0.78 \[ \frac {b \left (\frac {b \sinh (c+d x)}{d}-\frac {a^2 \cosh (c+d x)}{a+b x}\right )+a \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac {a d}{b}\right )-2 b \cosh \left (c-\frac {a d}{b}\right )\right )+a \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac {a d}{b}\right )-2 b \sinh \left (c-\frac {a d}{b}\right )\right )}{b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 274, normalized size = 1.86 \[ -\frac {2 \, a^{2} b d \cosh \left (d x + c\right ) - {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (b^{3} x + a b^{2}\right )} \sinh \left (d x + c\right ) + {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{5} d x + a b^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 1308, normalized size = 8.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 254, normalized size = 1.73 \[ -\frac {{\mathrm e}^{-d x -c}}{2 d \,b^{2}}-\frac {d \,{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} \left (b d x +d a \right )}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a^{2}}{2 b^{4}}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a}{b^{3}}+\frac {{\mathrm e}^{d x +c}}{2 d \,b^{2}}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a}{b^{3}}-\frac {d \,{\mathrm e}^{d x +c} a^{2}}{2 b^{4} \left (\frac {a d}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a^{2}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 236, normalized size = 1.61 \[ \frac {1}{2} \, {\left (a^{2} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{4}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{4}}\right )} + \frac {2 \, a {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{2} d} - \frac {\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}}{b^{2}} + \frac {4 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{3} d}\right )} d - {\left (\frac {a^{2}}{b^{4} x + a b^{3}} - \frac {x}{b^{2}} + \frac {2 \, a \log \left (b x + a\right )}{b^{3}}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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