Optimal. Leaf size=112 \[ -\frac {a f \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5572, 2743, 12,
2739, 632, 210} \begin {gather*} -\frac {a f \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac {f \cosh (c+d x)}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2743
Rule 5572
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx &=-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}+\frac {f \int \frac {1}{(a+b \sinh (c+d x))^2} \, dx}{2 b d}\\ &=-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac {f \int \frac {a}{a+b \sinh (c+d x)} \, dx}{2 b \left (a^2+b^2\right ) d}\\ &=-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac {(a f) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{2 b \left (a^2+b^2\right ) d}\\ &=-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {(i a f) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b \left (a^2+b^2\right ) d^2}\\ &=-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac {(2 i a f) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b \left (a^2+b^2\right ) d^2}\\ &=-\frac {a f \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac {f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.91, size = 112, normalized size = 1.00 \begin {gather*} -\frac {\frac {f \cosh (c+d x)}{\left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac {\frac {2 a f \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\frac {d (e+f x)}{(a+b \sinh (c+d x))^2}}{b}}{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. \(307\) vs.
\(2(106)=212\).
time = 6.91, size = 308, normalized size = 2.75
method | result | size |
risch | \(-\frac {2 a^{2} d f x \,{\mathrm e}^{2 d x +2 c}+2 b^{2} d f x \,{\mathrm e}^{2 d x +2 c}+2 a^{2} d e \,{\mathrm e}^{2 d x +2 c}-a b f \,{\mathrm e}^{3 d x +3 c}+2 b^{2} d e \,{\mathrm e}^{2 d x +2 c}-2 a^{2} f \,{\mathrm e}^{2 d x +2 c}+b^{2} f \,{\mathrm e}^{2 d x +2 c}+3 f a \,{\mathrm e}^{d x +c} b -f \,b^{2}}{b \,d^{2} \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )^{2} \left (a^{2}+b^{2}\right )}+\frac {f a \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}-\frac {f a \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} b}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs.
\(2 (106) = 212\).
time = 0.59, size = 413, normalized size = 3.69 \begin {gather*} \frac {1}{2} \, f {\left (\frac {2 \, {\left (a b e^{\left (3 \, d x + 3 \, c\right )} - 3 \, a b e^{\left (d x + c\right )} + b^{2} + {\left (2 \, a^{2} e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )} - 2 \, {\left (a^{2} d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}\right )}}{a^{2} b^{3} d^{2} + b^{5} d^{2} + {\left (a^{2} b^{3} d^{2} e^{\left (4 \, c\right )} + b^{5} d^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{3} b^{2} d^{2} e^{\left (3 \, c\right )} + a b^{4} d^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 2 \, {\left (2 \, a^{4} b d^{2} e^{\left (2 \, c\right )} + a^{2} b^{3} d^{2} e^{\left (2 \, c\right )} - b^{5} d^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 4 \, {\left (a^{3} b^{2} d^{2} e^{c} + a b^{4} d^{2} e^{c}\right )} e^{\left (d x\right )}} + \frac {a \log \left (\frac {b e^{\left (d x + 2 \, c\right )} + a e^{c} - \sqrt {a^{2} + b^{2}} e^{c}}{b e^{\left (d x + 2 \, c\right )} + a e^{c} + \sqrt {a^{2} + b^{2}} e^{c}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a^{2} + b^{2}} d^{2}}\right )} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c + 1\right )}}{{\left (4 \, a b^{2} e^{\left (-d x - c\right )} - 4 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + b^{3} + 2 \, {\left (2 \, a^{2} b - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} \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. 1293 vs.
\(2 (106) = 212\).
time = 0.38, size = 1293, normalized size = 11.54 \begin {gather*} \frac {2 \, {\left (a^{3} b + a b^{3}\right )} f \cosh \left (d x + c\right )^{3} + 2 \, {\left (a^{3} b + a b^{3}\right )} f \sinh \left (d x + c\right )^{3} - 6 \, {\left (a^{3} b + a b^{3}\right )} f \cosh \left (d x + c\right ) - 2 \, {\left (2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d f x + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (1\right ) + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \sinh \left (1\right ) - {\left (2 \, a^{4} + a^{2} b^{2} - b^{4}\right )} f\right )} \cosh \left (d x + c\right )^{2} - 2 \, {\left (2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d f x + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (1\right ) - 3 \, {\left (a^{3} b + a b^{3}\right )} f \cosh \left (d x + c\right ) + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \sinh \left (1\right ) - {\left (2 \, a^{4} + a^{2} b^{2} - b^{4}\right )} f\right )} \sinh \left (d x + c\right )^{2} + {\left (a b^{2} f \cosh \left (d x + c\right )^{4} + a b^{2} f \sinh \left (d x + c\right )^{4} + 4 \, a^{2} b f \cosh \left (d x + c\right )^{3} - 4 \, a^{2} b f \cosh \left (d x + c\right ) + a b^{2} f + 2 \, {\left (2 \, a^{3} - a b^{2}\right )} f \cosh \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} f \cosh \left (d x + c\right ) + a^{2} b f\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (3 \, a b^{2} f \cosh \left (d x + c\right )^{2} + 6 \, a^{2} b f \cosh \left (d x + c\right ) + {\left (2 \, a^{3} - a b^{2}\right )} f\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} f \cosh \left (d x + c\right )^{3} + 3 \, a^{2} b f \cosh \left (d x + c\right )^{2} - a^{2} b f + {\left (2 \, a^{3} - a b^{2}\right )} f \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + 2 \, {\left (a^{2} b^{2} + b^{4}\right )} f + 2 \, {\left (3 \, {\left (a^{3} b + a b^{3}\right )} f \cosh \left (d x + c\right )^{2} - 3 \, {\left (a^{3} b + a b^{3}\right )} f - 2 \, {\left (2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d f x + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (1\right ) + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \sinh \left (1\right ) - {\left (2 \, a^{4} + a^{2} b^{2} - b^{4}\right )} f\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d^{2} \cosh \left (d x + c\right )^{4} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d^{2} \sinh \left (d x + c\right )^{4} + 4 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d^{2} \cosh \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{6} b + 3 \, a^{4} b^{3} - b^{7}\right )} d^{2} \cosh \left (d x + c\right )^{2} - 4 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d^{2} \cosh \left (d x + c\right ) + 4 \, {\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d^{2} \cosh \left (d x + c\right ) + {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d^{2}\right )} \sinh \left (d x + c\right )^{3} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d^{2} + 2 \, {\left (3 \, {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d^{2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d^{2} \cosh \left (d x + c\right ) + {\left (2 \, a^{6} b + 3 \, a^{4} b^{3} - b^{7}\right )} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d^{2} \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{6} b + 3 \, a^{4} b^{3} - b^{7}\right )} d^{2} \cosh \left (d x + c\right ) - {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d^{2}\right )} \sinh \left (d x + c\right )\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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