Optimal. Leaf size=113 \[ \frac {a^2 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh (c+d x)}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^3 d}-\frac {a \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^4(c+d x)}{4 b d} \]
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Rubi [A] time = 0.16, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ \frac {\left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^3 d}-\frac {a \left (a^2+b^2\right ) \sinh (c+d x)}{b^4 d}+\frac {a^2 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^5 d}-\frac {a \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^4(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (-b^2-x^2\right )}{b^2 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (-b^2-x^2\right )}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^5 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a \left (a^2+b^2\right )-\left (a^2+b^2\right ) x+a x^2-x^3-\frac {a^2 \left (a^2+b^2\right )}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^5 d}\\ &=\frac {a^2 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh (c+d x)}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^3 d}-\frac {a \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^4(c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 98, normalized size = 0.87 \[ \frac {6 b^2 \left (a^2+b^2\right ) \sinh ^2(c+d x)-12 a b \left (a^2+b^2\right ) \sinh (c+d x)+12 a^2 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))-4 a b^3 \sinh ^3(c+d x)+3 b^4 \sinh ^4(c+d x)}{12 b^5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 1069, normalized size = 9.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 202, normalized size = 1.79 \[ \frac {\frac {3 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4} - 8 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 24 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 96 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 96 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{4}} + \frac {192 \, {\left (a^{4} + a^{2} b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{5}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 614, normalized size = 5.43 \[ \frac {\ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right ) a^{2}}{d \,b^{3}}+\frac {a}{d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{2}}{d \,b^{3}}+\frac {a}{d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{2}}{d \,b^{3}}+\frac {a}{2 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a^{2}}{2 d \,b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a}{2 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a^{2}}{2 d \,b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a}{3 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {a^{2}}{2 d \,b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a^{3}}{d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{5}}+\frac {a}{3 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {a^{2}}{2 d \,b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {a^{3}}{d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{5}}+\frac {a^{4} \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d \,b^{5}}+\frac {1}{4 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{4 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5}{8 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5}{8 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3}{8 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3}{8 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 234, normalized size = 2.07 \[ -\frac {{\left (8 \, a b^{2} e^{\left (-d x - c\right )} - 3 \, b^{3} - 12 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 24 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (a^{4} + a^{2} b^{2}\right )} {\left (d x + c\right )}}{b^{5} d} + \frac {8 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + 24 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-d x - c\right )} + 12 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (a^{4} + a^{2} b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 238, normalized size = 2.11 \[ \frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}-\frac {x\,\left (a^4+a^2\,b^2\right )}{b^5}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}+\frac {a\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b^2\,d}-\frac {a\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b^2\,d}+\frac {\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^4+a^2\,b^2\right )}{b^5\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^3+3\,a\,b^2\right )}{8\,b^4\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^3+3\,a\,b^2\right )}{8\,b^4\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (2\,a^2+b^2\right )}{16\,b^3\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+b^2\right )}{16\,b^3\,d} \]
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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