Optimal. Leaf size=84 \[ \frac {\sinh (c+d x) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{a d}+\frac {\sinh ^3(c+d x) \, _2F_1\left (1,\frac {3}{n};\frac {n+3}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{3 a d} \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3223, 1893, 245, 364} \[ \frac {\sinh ^3(c+d x) \, _2F_1\left (1,\frac {3}{n};\frac {n+3}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{3 a d}+\frac {\sinh (c+d x) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 245
Rule 364
Rule 1893
Rule 3223
Rubi steps
\begin {align*} \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^n(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{a+b x^n} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a+b x^n}+\frac {x^2}{a+b x^n}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^n} \, dx,x,\sinh (c+d x)\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {x^2}{a+b x^n} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right ) \sinh (c+d x)}{a d}+\frac {\, _2F_1\left (1,\frac {3}{n};\frac {3+n}{n};-\frac {b \sinh ^n(c+d x)}{a}\right ) \sinh ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.98 \[ \frac {\frac {\sinh (c+d x) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{a}+\frac {\sinh ^3(c+d x) \, _2F_1\left (1,\frac {3}{n};1+\frac {3}{n};-\frac {b \sinh ^n(c+d x)}{a}\right )}{3 a}}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{n} + a}, x\right ) \]
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 \[ \int \frac {\cosh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}\left (d x +c \right )}{a +b \left (\sinh ^{n}\left (d x +c \right )\right )}\, dx \]
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 \[ \int \frac {\cosh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3}{a+b\,{\mathrm {sinh}\left (c+d\,x\right )}^n} \,d x \]
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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