Optimal. Leaf size=129 \[ \frac {(b c-a d) f \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5728, 5726,
3394, 12, 3384, 3379, 3382} \begin {gather*} \frac {f (b c-a d) \sinh \left (2 \left (\frac {b f}{d}+e\right )\right ) \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2}-\frac {f (b c-a d) \cosh \left (2 \left (\frac {b f}{d}+e\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a f+b f x+c e+d e x}{c+d x}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 5726
Rule 5728
Rubi steps
\begin {align*} \int \sinh ^2\left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx &=\int \sinh ^2\left (\frac {c e+a f+(d e+b f) x}{c+d x}\right ) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {\sinh ^2\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 ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(2 i (b c-a d) f) \text {Subst}\left (\int \frac {i \sinh \left (2 \left (e+\frac {b f}{d}\right )-\frac {2 (b c-a d) f x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}+\frac {((b c-a d) f) \text {Subst}\left (\int \frac {\sinh \left (2 \left (e+\frac {b f}{d}\right )-\frac {2 (b c-a d) f x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^2\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 (2 \left (e+\frac {b f}{d}\right )\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 (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 (2 \left (e+\frac {b f}{d}\right )\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 (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 \text {Chi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right ) \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (b c-a d) f}{d (c+d x)}\right )}{d^2}\\ \end {align*}
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Mathematica [A]
time = 1.46, size = 136, normalized size = 1.05 \begin {gather*} \frac {d \left (-d x+(c+d x) \cosh \left (\frac {2 (c e+a f+d e x+b f x)}{c+d x}\right )\right )+2 (b c-a d) f \text {Chi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right ) \sinh \left (2 \left (e+\frac {b f}{d}\right )\right )+2 (b c-a d) f \cosh \left (2 \left (e+\frac {b f}{d}\right )\right ) \text {Shi}\left (\frac {2 (-b c f+a d f)}{d (c+d x)}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(467\) vs.
\(2(131)=262\).
time = 31.03, size = 468, normalized size = 3.63
method | result | size |
risch | \(-\frac {x}{2}+\frac {f \,{\mathrm e}^{-\frac {2 \left (b f x +d e x +f a +c e \right )}{d x +c}} a}{\frac {4 d f a}{d x +c}-\frac {4 f b c}{d x +c}}-\frac {f \,{\mathrm e}^{-\frac {2 \left (b f x +d e x +f a +c e \right )}{d x +c}} b c}{4 d \left (\frac {d f a}{d x +c}-\frac {f b c}{d x +c}\right )}-\frac {f \,{\mathrm e}^{-\frac {2 \left (b f +d e \right )}{d}} \expIntegral \left (1, \frac {2 \left (a d -b c \right ) f}{d \left (d x +c \right )}\right ) a}{2 d}+\frac {f \,{\mathrm e}^{-\frac {2 \left (b f +d e \right )}{d}} \expIntegral \left (1, \frac {2 \left (a d -b c \right ) f}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {f \,{\mathrm e}^{\frac {2 b f x +2 d e x +2 f a +2 c e}{d x +c}} a}{4 d \left (\frac {f a}{d x +c}-\frac {f b c}{d \left (d x +c \right )}\right )}-\frac {f \,{\mathrm e}^{\frac {2 b f x +2 d e x +2 f a +2 c e}{d x +c}} b c}{4 d^{2} \left (\frac {f a}{d x +c}-\frac {f b c}{d \left (d x +c \right )}\right )}+\frac {f \,{\mathrm e}^{\frac {2 b f +2 d e}{d}} \expIntegral \left (1, -\frac {2 \left (a d -b c \right ) f}{d \left (d x +c \right )}-\frac {2 \left (b f +d e \right )}{d}-\frac {2 \left (-b f -d e \right )}{d}\right ) a}{2 d}-\frac {f \,{\mathrm e}^{\frac {2 b f +2 d e}{d}} \expIntegral \left (1, -\frac {2 \left (a d -b c \right ) f}{d \left (d x +c \right )}-\frac {2 \left (b f +d e \right )}{d}-\frac {2 \left (-b f -d e \right )}{d}\right ) b c}{2 d^{2}}\) | \(468\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs.
\(2 (135) = 270\).
time = 0.51, size = 548, normalized size = 4.25 \begin {gather*} -\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2} + {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, {\left (b f + d \cosh \left (1\right ) + d \sinh \left (1\right )\right )}}{d}\right ) - d^{2} x - c d\right )} \sinh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2} - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2} - {\left (b c - a d\right )} f {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, {\left (b f + d \cosh \left (1\right ) + d \sinh \left (1\right )\right )}}{d}\right ) - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \cosh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2} - {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right ) \sinh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2} + {\left (b c - a d\right )} f {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, {\left (b f + d \cosh \left (1\right ) + d \sinh \left (1\right )\right )}}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b f x + a f + {\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}{d x + c}\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1596 vs.
\(2 (131) = 262\).
time = 20.37, size = 1596, normalized size = 12.37 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int {\mathrm {sinh}\left (e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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