Optimal. Leaf size=71 \[ \frac {\sqrt {b} (a+b) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{a^2 d}+\frac {\cosh ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4218, 470, 327,
211} \begin {gather*} \frac {\sqrt {b} (a+b) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{a^2 d}+\frac {\cosh ^3(c+d x)}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 327
Rule 470
Rule 4218
Rubi steps
\begin {align*} \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^2 \left (1-x^2\right )}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x)}{3 a d}-\frac {(a+b) \text {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=-\frac {(a+b) \cosh (c+d x)}{a^2 d}+\frac {\cosh ^3(c+d x)}{3 a d}+\frac {(b (a+b)) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{a^2 d}\\ &=\frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{a^2 d}+\frac {\cosh ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.49, size = 372, normalized size = 5.24 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \left (3 \left (a^2+8 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+3 \left (a^2+8 a b+8 b^2\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )-3 a^2 \left (\text {ArcTan}\left (\frac {\sqrt {a}-i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\text {ArcTan}\left (\frac {\sqrt {a}+i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )-6 \sqrt {a} \sqrt {b} (3 a+4 b) \cosh (c+d x)+2 a^{3/2} \sqrt {b} \cosh (3 (c+d x))\right )}{48 a^{5/2} \sqrt {b} d \left (b+a \cosh ^2(c+d x)\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs.
\(2(61)=122\).
time = 2.06, size = 170, normalized size = 2.39
method | result | size |
derivativedivides | \(\frac {\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 b +a}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a +b \right ) b \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-2 b -a}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(170\) |
default | \(\frac {\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 b +a}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a +b \right ) b \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-2 b -a}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(170\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 a d}-\frac {3 \,{\mathrm e}^{d x +c}}{8 a d}-\frac {{\mathrm e}^{d x +c} b}{2 a^{2} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a d}-\frac {{\mathrm e}^{-d x -c} b}{2 a^{2} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 a d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{2} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{2 a^{3} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{2 a^{3} d}\) | \(274\) |
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. 565 vs.
\(2 (61) = 122\).
time = 0.38, size = 1246, normalized size = 17.55 \begin {gather*} \left [\frac {a \cosh \left (d x + c\right )^{6} + 6 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + a \sinh \left (d x + c\right )^{6} - 3 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, a \cosh \left (d x + c\right )^{2} - 3 \, a - 4 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, a \cosh \left (d x + c\right )^{3} - 3 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a \cosh \left (d x + c\right )^{4} - 6 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - 4 \, b\right )} \sinh \left (d x + c\right )^{2} + 12 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + a \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a}} + a}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right ) + 6 \, {\left (a \cosh \left (d x + c\right )^{5} - 2 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}{24 \, {\left (a^{2} d \cosh \left (d x + c\right )^{3} + 3 \, a^{2} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} d \sinh \left (d x + c\right )^{3}\right )}}, \frac {a \cosh \left (d x + c\right )^{6} + 6 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + a \sinh \left (d x + c\right )^{6} - 3 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, a \cosh \left (d x + c\right )^{2} - 3 \, a - 4 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, a \cosh \left (d x + c\right )^{3} - 3 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a \cosh \left (d x + c\right )^{4} - 6 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - 4 \, b\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + {\left (a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 4 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + 24 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + 6 \, {\left (a \cosh \left (d x + c\right )^{5} - 2 \, {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}{24 \, {\left (a^{2} d \cosh \left (d x + c\right )^{3} + 3 \, a^{2} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} d \sinh \left (d x + c\right )^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \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 = 1.84, size = 473, normalized size = 6.66 \begin {gather*} \frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,a\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a^4\,b\,d\,\sqrt {a^2\,b+2\,a\,b^2+b^3}+2\,a^2\,b^3\,d\,\sqrt {a^2\,b+2\,a\,b^2+b^3}+4\,a^3\,b^2\,d\,\sqrt {a^2\,b+2\,a\,b^2+b^3}\right )}{a^{11}\,d^2\,\left (a+b\right )}+\frac {2\,\left (b^4\,\sqrt {a^5\,d^2}+3\,a^2\,b^2\,\sqrt {a^5\,d^2}+3\,a\,b^3\,\sqrt {a^5\,d^2}+a^3\,b\,\sqrt {a^5\,d^2}\right )}{a^8\,d\,\sqrt {b\,{\left (a+b\right )}^2}\,\sqrt {a^5\,d^2}}\right )\,\sqrt {a^5\,d^2}}{4\,a^2\,b+8\,a\,b^2+4\,b^3}+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (b^4\,\sqrt {a^5\,d^2}+3\,a^2\,b^2\,\sqrt {a^5\,d^2}+3\,a\,b^3\,\sqrt {a^5\,d^2}+a^3\,b\,\sqrt {a^5\,d^2}\right )}{a^2\,d\,\sqrt {b\,{\left (a+b\right )}^2}\,\left (4\,a^2\,b+8\,a\,b^2+4\,b^3\right )}\right )-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a+b\right )\,\sqrt {a^5\,d^2}}{2\,a^2\,d\,\sqrt {b\,{\left (a+b\right )}^2}}\right )\right )\,\sqrt {a^2\,b+2\,a\,b^2+b^3}}{2\,\sqrt {a^5\,d^2}}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,a\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+4\,b\right )}{8\,a^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a+4\,b\right )}{8\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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