Optimal. Leaf size=264 \[ -\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^6}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^6}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}+\frac {3 a^2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sinh (c+d x)}{b^3 d} \]
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Rubi [A] time = 0.62, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ -\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^6}+\frac {3 a^2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^6}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sinh (c+d x)}{b^3 d} \]
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^3 \cosh (c+d x)}{(a+b x)^3} \, dx &=\int \left (\frac {\cosh (c+d x)}{b^3}-\frac {a^3 \cosh (c+d x)}{b^3 (a+b x)^3}+\frac {3 a^2 \cosh (c+d x)}{b^3 (a+b x)^2}-\frac {3 a \cosh (c+d x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {\int \cosh (c+d x) \, dx}{b^3}-\frac {(3 a) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^3}+\frac {\left (3 a^2\right ) \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b^3}-\frac {a^3 \int \frac {\cosh (c+d x)}{(a+b x)^3} \, dx}{b^3}\\ &=\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}+\frac {\sinh (c+d x)}{b^3 d}+\frac {\left (3 a^2 d\right ) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^4}-\frac {\left (a^3 d\right ) \int \frac {\sinh (c+d x)}{(a+b x)^2} \, dx}{2 b^4}-\frac {\left (3 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^3}-\frac {\left (3 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^3}\\ &=\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\sinh (c+d x)}{b^3 d}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {\left (a^3 d^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{2 b^5}+\frac {\left (3 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^4}+\frac {\left (3 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^4}\\ &=\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {3 a^2 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}+\frac {\sinh (c+d x)}{b^3 d}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {\left (a^3 d^2 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^5}-\frac {\left (a^3 d^2 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^5}\\ &=\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^6}+\frac {3 a^2 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}+\frac {\sinh (c+d x)}{b^3 d}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^6}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 236, normalized size = 0.89 \[ -\frac {a d (a+b x)^2 \left (\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (a^2 d^2+6 b^2\right ) \cosh \left (c-\frac {a d}{b}\right )-6 a b d \sinh \left (c-\frac {a d}{b}\right )\right )+\text {Shi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (a^2 d^2+6 b^2\right ) \sinh \left (c-\frac {a d}{b}\right )-6 a b d \cosh \left (c-\frac {a d}{b}\right )\right )\right )+b \cosh (d x) \left (a^2 b d \cosh (c) (5 a+6 b x)-\sinh (c) (a+b x) \left (a^3 d^2+2 a b^2+2 b^3 x\right )\right )-b \sinh (d x) \left (\cosh (c) (a+b x) \left (a^3 d^2+2 a b^2+2 b^3 x\right )-a^2 b d \sinh (c) (5 a+6 b x)\right )}{2 b^6 d (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 566, normalized size = 2.14 \[ -\frac {2 \, {\left (6 \, a^{2} b^{3} d x + 5 \, a^{3} b^{2} d\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a^{4} b d^{2} + 2 \, b^{5} x^{2} + 2 \, a^{2} b^{3} + {\left (a^{3} b^{2} d^{2} + 4 \, a b^{4}\right )} x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 879, normalized size = 3.33 \[ -\frac {a^{3} b^{2} d^{3} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{3} b^{2} d^{3} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{4} b d^{3} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 6 \, a^{2} b^{3} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{4} b d^{3} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{5} d^{3} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 12 \, a^{3} b^{2} d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a b^{4} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{5} d^{3} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 12 \, a^{3} b^{2} d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 6 \, a b^{4} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{3} b^{2} d^{2} x e^{\left (-d x - c\right )} - 6 \, a^{4} b d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 12 \, a^{2} b^{3} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a^{4} b d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 12 \, a^{2} b^{3} d x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{4} b d^{2} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d x e^{\left (d x + c\right )} - 2 \, b^{5} x^{2} e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (-d x - c\right )} + 6 \, a^{2} b^{3} d x e^{\left (-d x - c\right )} + 2 \, b^{5} x^{2} e^{\left (-d x - c\right )} + 6 \, a^{3} b^{2} d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a^{3} b^{2} d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 5 \, a^{3} b^{2} d e^{\left (d x + c\right )} - 4 \, a b^{4} x e^{\left (d x + c\right )} + 5 \, a^{3} b^{2} d e^{\left (-d x - c\right )} + 4 \, a b^{4} x e^{\left (-d x - c\right )} - 2 \, a^{2} b^{3} e^{\left (d x + c\right )} + 2 \, a^{2} b^{3} e^{\left (-d x - c\right )}}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 571, normalized size = 2.16 \[ \frac {d^{2} {\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a^{3}}{4 b^{6}}+\frac {3 d \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a^{2}}{2 b^{5}}+\frac {3 \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a}{2 b^{4}}-\frac {{\mathrm e}^{-d x -c}}{2 d \,b^{3}}-\frac {5 d^{2} {\mathrm e}^{-d x -c} a^{3}}{4 b^{4} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{3} {\mathrm e}^{-d x -c} a^{4}}{4 b^{5} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{3} {\mathrm e}^{-d x -c} a^{3} x}{4 b^{4} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {3 d^{2} {\mathrm e}^{-d x -c} a^{2} x}{2 b^{3} \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {3 d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a^{2}}{2 b^{5}}-\frac {3 d \,{\mathrm e}^{d x +c} a^{2}}{2 b^{5} \left (\frac {a d}{b}+d x \right )}+\frac {{\mathrm e}^{d x +c}}{2 d \,b^{3}}+\frac {3 \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a}{2 b^{4}}+\frac {d^{2} {\mathrm e}^{d x +c} a^{3}}{4 b^{6} \left (\frac {a d}{b}+d x \right )}+\frac {d^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a^{3}}{4 b^{6}}+\frac {d^{2} {\mathrm e}^{d x +c} a^{3}}{4 b^{6} \left (\frac {a d}{b}+d x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{2} \, a^{2} d \int \frac {x e^{\left (d x + c\right )}}{b^{5} d^{2} x^{4} + 4 \, a b^{4} d^{2} x^{3} + 6 \, a^{2} b^{3} d^{2} x^{2} + 4 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}}\,{d x} - \frac {3}{2} \, a^{2} d \int \frac {x}{b^{5} d^{2} x^{4} e^{\left (d x + c\right )} + 4 \, a b^{4} d^{2} x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (d x + c\right )}}\,{d x} - 3 \, a b \int \frac {x e^{\left (d x + c\right )}}{b^{5} d^{2} x^{4} + 4 \, a b^{4} d^{2} x^{3} + 6 \, a^{2} b^{3} d^{2} x^{2} + 4 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}}\,{d x} - 3 \, a b \int \frac {x}{b^{5} d^{2} x^{4} e^{\left (d x + c\right )} + 4 \, a b^{4} d^{2} x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (d x + c\right )}}\,{d x} + \frac {{\left (b d x^{3} e^{\left (2 \, c\right )} - 3 \, a x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (b d x^{3} + 3 \, a x\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{4} d^{2} x^{3} e^{c} + 3 \, a b^{3} d^{2} x^{2} e^{c} + 3 \, a^{2} b^{2} d^{2} x e^{c} + a^{3} b d^{2} e^{c}\right )}} - \frac {3 \, a^{2} e^{\left (-c + \frac {a d}{b}\right )} E_{4}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{2 \, {\left (b x + a\right )}^{3} b^{2} d^{2}} - \frac {3 \, a^{2} e^{\left (c - \frac {a d}{b}\right )} E_{4}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{2 \, {\left (b x + a\right )}^{3} b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{3} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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