Optimal. Leaf size=95 \[ \frac {2 a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {x \left (2 a^2+b^2\right )}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \]
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Rubi [A] time = 0.18, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2865, 2735, 2660, 618, 204} \[ \frac {2 a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {x \left (2 a^2+b^2\right )}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2735
Rule 2865
Rubi steps
\begin {align*} \int \frac {\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}+\frac {i \int \frac {i a b-i \left (2 a^2+b^2\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{2 b^2}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{b^3}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}+\frac {\left (2 i a \left (a^2+b^2\right )\right ) \operatorname {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^3 d}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}-\frac {\left (4 i a \left (a^2+b^2\right )\right ) \operatorname {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^3 d}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}+\frac {2 a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 109, normalized size = 1.15 \[ \frac {8 a \sqrt {-a^2-b^2} \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+4 a^2 c+4 a^2 d x-4 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))+2 b^2 c+2 b^2 d x}{4 b^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 446, normalized size = 4.69 \[ \frac {b^{2} \cosh \left (d x + c\right )^{4} + b^{2} \sinh \left (d x + c\right )^{4} + 4 \, {\left (2 \, a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) - b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 155, normalized size = 1.63 \[ \frac {\frac {4 \, {\left (2 \, a^{2} + b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )}}{b^{2}} - \frac {{\left (4 \, a b e^{\left (d x + c\right )} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{b^{3}} - \frac {8 \, {\left (a^{3} + a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{3}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 260, normalized size = 2.74 \[ \frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a}{d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{2}}{d \,b^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d b}-\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a}{d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{2}}{d \,b^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d b}-\frac {2 a \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 160, normalized size = 1.68 \[ -\frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, b^{2} d} - \frac {\sqrt {a^{2} + b^{2}} a \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{3} d} + \frac {{\left (2 \, a^{2} + b^{2}\right )} {\left (d x + c\right )}}{2 \, b^{3} d} - \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 212, normalized size = 2.23 \[ \frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}+\frac {x\,\left (2\,a^2+b^2\right )}{2\,b^3}-\frac {a\,{\mathrm {e}}^{-c-d\,x}}{2\,b^2\,d}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,b^2\,d}-\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^4}-\frac {2\,a\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^4}\right )\,\sqrt {a^2+b^2}}{b^3\,d}+\frac {a\,\ln \left (\frac {2\,a\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^4}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^4}\right )\,\sqrt {a^2+b^2}}{b^3\,d} \]
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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