Optimal. Leaf size=107 \[ \frac {\sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac {\cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5607, 3313, 12, 3303, 3298, 3301} \[ \frac {\sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac {\cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 3313
Rule 5607
Rubi steps
\begin {align*} \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh ^2\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 ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(2 i (b c-a d)) \operatorname {Subst}\left (\int \frac {i \sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {\left ((b c-a d) \cosh \left (\frac {2 b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 (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 {2 b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 112, normalized size = 1.05 \[ \frac {2 \sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (a d-b c)}{d (c+d x)}\right )+2 \cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (a d-b c)}{d (c+d x)}\right )+d \left ((c+d x) \cosh \left (\frac {2 (a+b x)}{c+d x}\right )-d x\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.33, size = 370, normalized size = 3.46 \[ -\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (d^{2} x - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 17.26, size = 749, normalized size = 7.00 \[ \frac {{\left (2 \, b^{3} c^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - 4 \, a b^{2} c d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} + 2 \, a^{2} b d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} + \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, b^{3} c^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + 4 \, a b^{2} c d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, a^{2} b d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} - \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + b^{2} c^{2} d e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{4 \, {\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 = 0.21, size = 358, normalized size = 3.35 \[ -\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} a}{\frac {4 d a}{d x +c}-\frac {4 b c}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} c b}{4 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 b}{d}} \Ei \left (1, \frac {2 d a -2 c b}{d \left (d x +c \right )}\right ) a}{2 d}+\frac {{\mathrm e}^{-\frac {2 b}{d}} \Ei \left (1, \frac {2 d a -2 c b}{d \left (d x +c \right )}\right ) c b}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x a}{4 d a -4 c b}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x c b}{4 \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c a}{4 d a -4 c b}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c^{2} b}{4 d \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {2 b}{d}} \Ei \left (1, -\frac {2 \left (d a -c b \right )}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {2 b}{d}} \Ei \left (1, -\frac {2 \left (d a -c b \right )}{d \left (d x +c \right )}\right ) c b}{2 d^{2}} \]
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 \[ -\frac {1}{2} \, x + \frac {1}{4} \, \int e^{\left (\frac {2 \, b c}{d^{2} x + c d} - \frac {2 \, a}{d x + c} - \frac {2 \, b}{d}\right )}\,{d x} + \frac {1}{4} \, \int e^{\left (-\frac {2 \, b c}{d^{2} x + c d} + \frac {2 \, a}{d x + c} + \frac {2 \, b}{d}\right )}\,{d x} \]
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 \[ \int {\mathrm {sinh}\left (\frac {a+b\,x}{c+d\,x}\right )}^2 \,d x \]
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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