Optimal. Leaf size=79 \[ \frac {a^2 \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} b^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d} \]
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Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 398, 211}
\begin {gather*} \frac {a^2 \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{5/2} d \sqrt {a-b}}-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 398
Rule 3265
Rubi steps
\begin {align*} \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {a+b}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a-b+b x^2\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{b^2 d}\\ &=\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} b^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 134, normalized size = 1.70 \begin {gather*} \frac {\frac {12 a^2 \left (\text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{\sqrt {a-b}}-3 \sqrt {b} (4 a+3 b) \cosh (c+d x)+b^{3/2} \cosh (3 (c+d x))}{12 b^{5/2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs.
\(2(69)=138\).
time = 1.09, size = 179, normalized size = 2.27
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b^{2} \sqrt {a b -b^{2}}}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(179\) |
default | \(\frac {\frac {a^{2} \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b^{2} \sqrt {a b -b^{2}}}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(179\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{d x +c}}{2 b^{2} d}-\frac {3 \,{\mathrm e}^{d x +c}}{8 b d}-\frac {{\mathrm e}^{-d x -c} a}{2 b^{2} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 b d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}-\frac {a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d \,b^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d \,b^{2}}\) | \(212\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 780 vs.
\(2 (69) = 138\).
time = 0.46, size = 1668, normalized size = 21.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 348, normalized size = 4.41 \begin {gather*} \frac {\left (2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^5\,d^2\,\left (a-b\right )}}{2\,b^2\,d\,\left (a-b\right )\,\sqrt {a^4}}\right )+2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^2}{b^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^4}}-\frac {4\,\left (2\,a^3\,b^3\,d\,\sqrt {a^4}-2\,a^4\,b^2\,d\,\sqrt {a^4}\right )}{a^5\,b^6\,\left (a-b\right )\,\sqrt {a\,b^5\,d^2-b^6\,d^2}\,\sqrt {b^5\,d^2\,\left (a-b\right )}}\right )+\frac {2\,a^2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{b^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^4}}\right )\,\left (\frac {b^7\,\sqrt {a\,b^5\,d^2-b^6\,d^2}}{4}-\frac {a\,b^6\,\sqrt {a\,b^5\,d^2-b^6\,d^2}}{4}\right )\right )\right )\,\sqrt {a^4}}{2\,\sqrt {a\,b^5\,d^2-b^6\,d^2}}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a+3\,b\right )}{8\,b^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a+3\,b\right )}{8\,b^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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