Optimal. Leaf size=123 \[ \frac {a f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{2 d^3}+\frac {a f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {a f \sinh (e+f x)}{2 d^2 (c+d x)}-\frac {a \cosh (e+f x)}{2 d (c+d x)^2}-\frac {a}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.22, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3317, 3297, 3303, 3298, 3301} \[ \frac {a f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{2 d^3}+\frac {a f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {a f \sinh (e+f x)}{2 d^2 (c+d x)}-\frac {a \cosh (e+f x)}{2 d (c+d x)^2}-\frac {a}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 3317
Rubi steps
\begin {align*} \int \frac {a+a \cosh (e+f x)}{(c+d x)^3} \, dx &=\int \left (\frac {a}{(c+d x)^3}+\frac {a \cosh (e+f x)}{(c+d x)^3}\right ) \, dx\\ &=-\frac {a}{2 d (c+d x)^2}+a \int \frac {\cosh (e+f x)}{(c+d x)^3} \, dx\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a \cosh (e+f x)}{2 d (c+d x)^2}+\frac {(a f) \int \frac {\sinh (e+f x)}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a \cosh (e+f x)}{2 d (c+d x)^2}-\frac {a f \sinh (e+f x)}{2 d^2 (c+d x)}+\frac {\left (a f^2\right ) \int \frac {\cosh (e+f x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a \cosh (e+f x)}{2 d (c+d x)^2}-\frac {a f \sinh (e+f x)}{2 d^2 (c+d x)}+\frac {\left (a f^2 \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}+\frac {\left (a f^2 \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a \cosh (e+f x)}{2 d (c+d x)^2}+\frac {a f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{2 d^3}-\frac {a f \sinh (e+f x)}{2 d^2 (c+d x)}+\frac {a f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 90, normalized size = 0.73 \[ \frac {a \left (f^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \cosh \left (e-\frac {c f}{d}\right )+f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-\frac {d (f (c+d x) \sinh (e+f x)+d \cosh (e+f x)+d)}{(c+d x)^2}\right )}{2 d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 274, normalized size = 2.23 \[ -\frac {2 \, a d^{2} \cosh \left (f x + e\right ) + 2 \, a d^{2} - {\left ({\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + 2 \, {\left (a d^{2} f x + a c d f\right )} \sinh \left (f x + e\right ) + {\left ({\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 328, normalized size = 2.67 \[ \frac {a d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + a d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 2 \, a c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + 2 \, a c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + a c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + a c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} - a d^{2} f x e^{\left (f x + e\right )} + a d^{2} f x e^{\left (-f x - e\right )} - a c d f e^{\left (f x + e\right )} + a c d f e^{\left (-f x - e\right )} - a d^{2} e^{\left (f x + e\right )} - a d^{2} e^{\left (-f x - e\right )} - 2 \, a d^{2}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 296, normalized size = 2.41 \[ -\frac {a}{2 d \left (d x +c \right )^{2}}+\frac {f^{3} a \,{\mathrm e}^{-f x -e} x}{4 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} a \,{\mathrm e}^{-f x -e} c}{4 d^{2} \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a \,{\mathrm e}^{-f x -e}}{4 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a \,{\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{4 d^{3}}-\frac {f^{2} a \,{\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} a \,{\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} a \,{\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{4 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 98, normalized size = 0.80 \[ -\frac {1}{2} \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
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 \[ \int \frac {a+a\,\mathrm {cosh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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