Optimal. Leaf size=259 \[ -\frac {2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^{10} d}+\frac {2 \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}}{b^9 d}-\frac {a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac {2 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)}{5 b^5 d}-\frac {a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac {2 a^2 \sinh ^{\frac {7}{2}}(c+d x)}{7 b^3 d}-\frac {a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac {2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)}{3 b^7 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {2 \sinh ^{\frac {9}{2}}(c+d x)}{9 b d} \]
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Rubi [A] time = 0.30, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3223, 1890, 1620} \[ \frac {2 a^2 \sinh ^{\frac {7}{2}}(c+d x)}{7 b^3 d}-\frac {a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac {2 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)}{5 b^5 d}-\frac {a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac {2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)}{3 b^7 d}-\frac {a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac {2 \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}}{b^9 d}-\frac {2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^{10} d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {2 \sinh ^{\frac {9}{2}}(c+d x)}{9 b d} \]
Antiderivative was successfully verified.
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Rule 1620
Rule 1890
Rule 3223
Rubi steps
\begin {align*} \int \frac {\cosh ^5(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{a+b \sqrt {x}} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (1+x^4\right )^2}{a+b x} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {\left (a^4+b^4\right )^2}{b^9}-\frac {a^3 \left (a^4+2 b^4\right ) x}{b^8}+\frac {a^2 \left (a^4+2 b^4\right ) x^2}{b^7}-\frac {a \left (a^4+2 b^4\right ) x^3}{b^6}+\frac {\left (a^4+2 b^4\right ) x^4}{b^5}-\frac {a^3 x^5}{b^4}+\frac {a^2 x^6}{b^3}-\frac {a x^7}{b^2}+\frac {x^8}{b}-\frac {a \left (a^4+b^4\right )^2}{b^9 (a+b x)}\right ) \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=-\frac {2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^{10} d}+\frac {2 \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}}{b^9 d}-\frac {a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac {2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)}{3 b^7 d}-\frac {a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac {2 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)}{5 b^5 d}-\frac {a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac {2 a^2 \sinh ^{\frac {7}{2}}(c+d x)}{7 b^3 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {2 \sinh ^{\frac {9}{2}}(c+d x)}{9 b d}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 220, normalized size = 0.85 \[ \frac {-630 a b^4 \left (a^4+2 b^4\right ) \sinh ^2(c+d x)+2520 b \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}-2520 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )+504 b^5 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)-420 a^3 b^6 \sinh ^3(c+d x)+360 a^2 b^7 \sinh ^{\frac {7}{2}}(c+d x)-1260 a^3 b^2 \left (a^4+2 b^4\right ) \sinh (c+d x)+840 a^2 b^3 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)-315 a b^8 \sinh ^4(c+d x)+280 b^9 \sinh ^{\frac {9}{2}}(c+d x)}{1260 b^{10} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.91, size = 2595, normalized size = 10.02 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (d x + c\right )^{5}}{b \sqrt {\sinh \left (d x + c\right )} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 780, normalized size = 3.01 \[ -\frac {7 a}{8 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {7 a}{8 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {9 a}{8 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {9 a}{8 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a^{9} \ln \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}{d \,b^{10}}-\frac {2 a^{5} \ln \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}{d \,b^{6}}-\frac {a \ln \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}{d \,b^{2}}+\frac {a^{7}}{d \,b^{8} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a^{5}}{2 d \,b^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{3}}{3 d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {a^{5}}{2 d \,b^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a^{3}}{2 d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {a^{3}}{3 d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a^{5}}{2 d \,b^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a^{3}}{2 d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a^{9} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{10}}+\frac {a^{9} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{10}}+\frac {2 a^{5} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{6}}+\frac {2 a^{5} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{6}}-\frac {a}{4 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {a}{4 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {a^{7}}{d \,b^{8} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a^{5}}{2 d \,b^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{2}}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{2}}-\frac {a}{2 d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 a^{3}}{d \,b^{4} \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 )^{3}}+\frac {2 a^{3}}{d \,b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\mathit {`\,int/indef0`\,}\left (-\frac {\left (\cosh ^{4}\left (d x +c \right )\right ) b \left (\sqrt {\sinh }\left (d x +c \right )\right )}{-b^{2} \sinh \left (d x +c \right )+a^{2}}, \sinh \left (d x +c \right )\right )}{d} \]
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 \[ \int \frac {\cosh \left (d x + c\right )^{5}}{b \sqrt {\sinh \left (d x + c\right )} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^5}{a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]
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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