Optimal. Leaf size=101 \[ \frac {\cosh \left (\frac {b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}-\frac {\sinh \left (\frac {b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5607, 3297, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}-\frac {\sinh \left (\frac {b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5607
Rubi steps
\begin {align*} \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}+\frac {\left ((b c-a d) \cosh \left (\frac {b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}-\frac {\left ((b c-a d) \sinh \left (\frac {b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}\\ \end {align*}
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Mathematica [B] time = 0.71, size = 373, normalized size = 3.69 \[ \frac {(b c-a d) \left (\cosh \left (\frac {b}{d}\right )-\sinh \left (\frac {b}{d}\right )\right ) \text {Chi}\left (\frac {b c-a d}{x d^2+c d}\right )+(b c-a d) \left (\sinh \left (\frac {b}{d}\right )+\cosh \left (\frac {b}{d}\right )\right ) \text {Chi}\left (\frac {a d-b c}{d (c+d x)}\right )+a d \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{x d^2+c d}\right )-b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{x d^2+c d}\right )-a d \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{x d^2+c d}\right )+b c \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{x d^2+c d}\right )+2 d^2 x \sinh \left (\frac {a+b x}{c+d x}\right )-a d \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {a d-b c}{d (c+d x)}\right )+b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {a d-b c}{d (c+d x)}\right )-a d \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {a d-b c}{d (c+d x)}\right )+b c \cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {a d-b c}{d (c+d x)}\right )+2 c d \sinh \left (\frac {a+b x}{c+d x}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 171, normalized size = 1.69 \[ \frac {{\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.23, size = 764, normalized size = 7.56 \[ \frac {{\left (b^{3} c^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - 2 \, a b^{2} c d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - \frac {{\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + a^{2} b d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} + \frac {2 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {b x + a}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {b x + a}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {b x + a}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} + \frac {{\left (b^{3} c^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - 2 \, a b^{2} c d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - \frac {{\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} + a^{2} b d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - b^{2} c^{2} d e^{\left (-\frac {b x + a}{d x + c}\right )} + 2 \, a b c d^{2} e^{\left (-\frac {b x + a}{d x + c}\right )} - a^{2} d^{3} e^{\left (-\frac {b x + a}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 347, normalized size = 3.44 \[ -\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} a}{2 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} c b}{2 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b}{d}} \Ei \left (1, \frac {d a -c b}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{-\frac {b}{d}} \Ei \left (1, \frac {d a -c b}{d \left (d x +c \right )}\right ) c b}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {b x +a}{d x +c}} x a}{2 d a -2 c b}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} x c b}{2 \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c a}{2 d a -2 c b}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c^{2} b}{2 d \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {b}{d}} \Ei \left (1, -\frac {d a -c b}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {b}{d}} \Ei \left (1, -\frac {d a -c b}{d \left (d x +c \right )}\right ) c b}{2 d^{2}} \]
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 \[ \int \sinh \left (\frac {b x + a}{d x + c}\right )\,{d x} \]
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 \[ \int \mathrm {sinh}\left (\frac {a+b\,x}{c+d\,x}\right ) \,d x \]
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 \[ \int \sinh {\left (\frac {a + b x}{c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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