Optimal. Leaf size=56 \[ \frac {\cosh (c+d x)}{b d}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{3/2} d \sqrt {a-b}} \]
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Rubi [A] time = 0.09, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3186, 388, 205} \[ \frac {\cosh (c+d x)}{b d}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{3/2} d \sqrt {a-b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sinh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x)}{b d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{b d}\\ &=-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} b^{3/2} d}+\frac {\cosh (c+d x)}{b d}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 107, normalized size = 1.91 \[ \frac {\sqrt {b} \cosh (c+d x)-\frac {a \left (\tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{\sqrt {a-b}}}{b^{3/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 746, normalized size = 13.32 \[ \left [\frac {{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} - \sqrt {-a b + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right )\right )} \sqrt {-a b + b^{2}} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) + a b - b^{2}}{2 \, {\left ({\left (a b^{2} - b^{3}\right )} d \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d \sinh \left (d x + c\right )\right )}}, \frac {{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, \sqrt {a b - b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \arctan \left (-\frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a b - b^{2}}}\right ) + 2 \, \sqrt {a b - b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \arctan \left (-\frac {\sqrt {a b - b^{2}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, {\left (a - b\right )}}\right ) + a b - b^{2}}{2 \, {\left ({\left (a b^{2} - b^{3}\right )} d \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d \sinh \left (d x + c\right )\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 98, normalized size = 1.75 \[ -\frac {a \arctan \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{d b \sqrt {a b -b^{2}}}+\frac {1}{d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )} e^{\left (-d x - c\right )}}{2 \, b d} - \frac {1}{8} \, \int \frac {16 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - a e^{\left (d x + c\right )}\right )}}{b^{2} e^{\left (4 \, d x + 4 \, c\right )} + b^{2} + 2 \, {\left (2 \, a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 293, normalized size = 5.23 \[ \frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^3\,d^2\,\left (a-b\right )}}{2\,b\,d\,\left (a-b\right )\,{\left (a^2\right )}^{3/2}}\right )+2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^3}{b^5\,d\,{\left (a-b\right )}^2\,{\left (a^2\right )}^{3/2}}-\frac {4\,\left (2\,b^2\,d\,{\left (a^2\right )}^{3/2}-2\,a\,b\,d\,{\left (a^2\right )}^{3/2}\right )}{a^3\,b^4\,\left (a-b\right )\,\sqrt {a\,b^3\,d^2-b^4\,d^2}\,\sqrt {b^3\,d^2\,\left (a-b\right )}}\right )+\frac {2\,a^3\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{b^5\,d\,{\left (a-b\right )}^2\,{\left (a^2\right )}^{3/2}}\right )\,\left (\frac {b^5\,\sqrt {a\,b^3\,d^2-b^4\,d^2}}{4}-\frac {a\,b^4\,\sqrt {a\,b^3\,d^2-b^4\,d^2}}{4}\right )\right )\right )\,\sqrt {a^2}}{2\,\sqrt {a\,b^3\,d^2-b^4\,d^2}}+\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d} \]
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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