Optimal. Leaf size=121 \[ \frac {f (b c-a d) \cosh \left (\frac {b f}{d}+e\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}-\frac {f (b c-a d) \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a f+b f x+c e+d e x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.26, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5609, 5607, 3297, 3303, 3298, 3301} \[ \frac {f (b c-a d) \cosh \left (\frac {b f}{d}+e\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}-\frac {f (b c-a d) \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a f+b f x+c e+d e x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5607
Rule 5609
Rubi steps
\begin {align*} \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx &=\int \sinh \left (\frac {c e+a f+(d e+b f) x}{c+d x}\right ) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh \left (\frac {d e+b f}{d}-\frac {(-d (c e+a f)+c (d e+b f)) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac {((b c-a d) f) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {d e+b f}{d}-\frac {(b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac {\left ((b c-a d) f \cosh \left (e+\frac {b f}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {(b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}-\frac {\left ((b c-a d) f \sinh \left (e+\frac {b f}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {(b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(b c-a d) f \cosh \left (e+\frac {b f}{d}\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}\\ \end {align*}
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Mathematica [B] time = 1.48, size = 449, normalized size = 3.71 \[ \frac {f (b c-a d) \left (\cosh \left (\frac {b f}{d}+e\right )-\sinh \left (\frac {b f}{d}+e\right )\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+f (b c-a d) \left (\sinh \left (\frac {b f}{d}+e\right )+\cosh \left (\frac {b f}{d}+e\right )\right ) \text {Chi}\left (\frac {a d f-b c f}{d (c+d x)}\right )+2 d^2 x \sinh \left (\frac {a f+b f x+c e+d e x}{c+d x}\right )+a d f \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-a d f \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {a d f-b c f}{d (c+d x)}\right )-b c f \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+b c f \sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {a d f-b c f}{d (c+d x)}\right )-a d f \cosh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-a d f \cosh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {a d f-b c f}{d (c+d x)}\right )+b c f \cosh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+b c f \cosh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {a d f-b c f}{d (c+d x)}\right )+2 c d \sinh \left (\frac {a f+b f x+c e+d e x}{c+d x}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.61, size = 202, normalized size = 1.67 \[ \frac {{\left ({\left (b c - a d\right )} f {\rm Ei}\left (\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {d e + b f}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right ) - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {d e + b f}{d}\right )}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 34.35, size = 1736, normalized size = 14.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 459, normalized size = 3.79 \[ -\frac {f \,{\mathrm e}^{-\frac {b f x +d e x +a f +c e}{d x +c}} a}{2 \left (\frac {d a f}{d x +c}-\frac {f c b}{d x +c}\right )}+\frac {f \,{\mathrm e}^{-\frac {b f x +d e x +a f +c e}{d x +c}} c b}{2 d \left (\frac {d a f}{d x +c}-\frac {f c b}{d x +c}\right )}+\frac {f \,{\mathrm e}^{-\frac {b f +d e}{d}} \Ei \left (1, \frac {\left (d a -c b \right ) f}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {f \,{\mathrm e}^{-\frac {b f +d e}{d}} \Ei \left (1, \frac {\left (d a -c b \right ) f}{d \left (d x +c \right )}\right ) c b}{2 d^{2}}+\frac {f \,{\mathrm e}^{\frac {b f x +d e x +a f +c e}{d x +c}} a}{2 d \left (\frac {a f}{d x +c}-\frac {f c b}{d \left (d x +c \right )}\right )}-\frac {f \,{\mathrm e}^{\frac {b f x +d e x +a f +c e}{d x +c}} c b}{2 d^{2} \left (\frac {a f}{d x +c}-\frac {f c b}{d \left (d x +c \right )}\right )}+\frac {f \,{\mathrm e}^{\frac {b f +d e}{d}} \Ei \left (1, -\frac {\left (d a -c b \right ) f}{d \left (d x +c \right )}-\frac {b f +d e}{d}-\frac {-b f -d e}{d}\right ) a}{2 d}-\frac {f \,{\mathrm e}^{\frac {b f +d e}{d}} \Ei \left (1, -\frac {\left (d a -c b \right ) f}{d \left (d x +c \right )}-\frac {b f +d e}{d}-\frac {-b f -d e}{d}\right ) c b}{2 d^{2}} \]
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 \[ \int \sinh \left (e + \frac {{\left (b x + a\right )} f}{d x + c}\right )\,{d x} \]
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 \[ \int \mathrm {sinh}\left (e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}\right ) \,d x \]
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 \[ \int \sinh {\left (e + \frac {f \left (a + b x\right )}{c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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