Optimal. Leaf size=617 \[ -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {b \cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
[Out]
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Rubi [A]
time = 0.66, antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3294, 1252,
213, 1192, 1180, 211, 214} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {b \cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 213
Rule 214
Rule 1180
Rule 1192
Rule 1252
Rule 3294
Rubi steps
\begin {align*} \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {1}{a^3 \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^3}+\frac {b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )^2}+\frac {b-b x^2}{a^3 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-4 a b^2+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a^3 (a-b) b d}+\frac {\text {Subst}\left (\int \frac {-12 a b^2+10 a b^2 x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a^2 (a-b) b d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {b \cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {4 a (13 a-b) b^3-4 a b^3 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^3 (a-b)^2 b^2 d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right ) d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {b \cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (\left (5 \sqrt {a}-2 \sqrt {b}\right ) b\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^2 d}+\frac {\left (\left (5 \sqrt {a}+2 \sqrt {b}\right ) b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^2 d}\\ &=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {b \cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.85, size = 1189, normalized size = 1.93 \begin {gather*} \frac {\frac {32 a b \cosh (c+d x) (-41 a+23 b+(13 a-7 b) \cosh (2 (c+d x)))}{(a-b)^2 (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {512 a^2 b (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{(a-b) (-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+256 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-45 a^2 c+71 a b c-32 b^2 c-45 a^2 d x+71 a b d x-32 b^2 d x-90 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+142 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-64 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+199 a^2 c \text {$\#$1}^2-253 a b c \text {$\#$1}^2+96 b^2 c \text {$\#$1}^2+199 a^2 d x \text {$\#$1}^2-253 a b d x \text {$\#$1}^2+96 b^2 d x \text {$\#$1}^2+398 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-506 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+192 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-199 a^2 c \text {$\#$1}^4+253 a b c \text {$\#$1}^4-96 b^2 c \text {$\#$1}^4-199 a^2 d x \text {$\#$1}^4+253 a b d x \text {$\#$1}^4-96 b^2 d x \text {$\#$1}^4-398 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+506 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-192 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+45 a^2 c \text {$\#$1}^6-71 a b c \text {$\#$1}^6+32 b^2 c \text {$\#$1}^6+45 a^2 d x \text {$\#$1}^6-71 a b d x \text {$\#$1}^6+32 b^2 d x \text {$\#$1}^6+90 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-142 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6+64 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{(a-b)^2}}{256 a^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 4.05, size = 647, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\frac {8 b \left (\frac {-\frac {a^{2} \left (8 a -5 b \right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 a^{2}+86 a b -64 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}+\frac {a \left (104 a^{2}-327 a b +208 b^{2}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 \left (105 a^{3}-358 a^{2} b +576 a \,b^{2}-256 b^{3}\right ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (400 a^{2}-1161 a b +560 b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (257 a^{2}-370 a b +128 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (80 a -53 b \right ) a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right )}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {a \left (\frac {\left (-45 \sqrt {a b}\, a^{2}+71 a b \sqrt {a b}-32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {\sqrt {a b}\, a -a b}}-\frac {\left (45 \sqrt {a b}\, a^{2}-71 a b \sqrt {a b}+32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {-\sqrt {a b}\, a -a b}}\right )}{64 a^{2}-128 a b +64 b^{2}}\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(647\) |
default | \(\frac {\frac {8 b \left (\frac {-\frac {a^{2} \left (8 a -5 b \right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 a^{2}+86 a b -64 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}+\frac {a \left (104 a^{2}-327 a b +208 b^{2}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 \left (105 a^{3}-358 a^{2} b +576 a \,b^{2}-256 b^{3}\right ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (400 a^{2}-1161 a b +560 b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (257 a^{2}-370 a b +128 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (80 a -53 b \right ) a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right )}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {a \left (\frac {\left (-45 \sqrt {a b}\, a^{2}+71 a b \sqrt {a b}-32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {\sqrt {a b}\, a -a b}}-\frac {\left (45 \sqrt {a b}\, a^{2}-71 a b \sqrt {a b}+32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {-\sqrt {a b}\, a -a b}}\right )}{64 a^{2}-128 a b +64 b^{2}}\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(647\) |
risch | \(\text {Expression too large to display}\) | \(1496\) |
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. 28586 vs.
\(2 (481) = 962\).
time = 1.43, size = 28586, normalized size = 46.33 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1783 vs.
\(2 (481) = 962\).
time = 0.64, size = 1783, normalized size = 2.89 \begin {gather*} \frac {\frac {{\left ({\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}^{2} {\left (180 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} - 59 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b - 227 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{2} + 160 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{3}\right )} {\left | b \right |} - {\left (244 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{8} b - 507 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b^{2} + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{3} + 695 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{4} - 597 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{5} + 160 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{6}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |} {\left | b \right |} + 2 \, {\left (32 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{12} b - 108 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{11} b^{2} + 87 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{10} b^{3} + 92 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{9} b^{4} - 198 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{8} b^{5} + 120 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b^{6} - 25 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} + \sqrt {{\left (a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right )} {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} + {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )}^{2}}}{a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}}}}\right )}{{\left (4 \, a^{12} b^{2} - 15 \, a^{11} b^{3} + 15 \, a^{10} b^{4} + 10 \, a^{9} b^{5} - 30 \, a^{8} b^{6} + 21 \, a^{7} b^{7} - 5 \, a^{6} b^{8}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |}} - \frac {{\left ({\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}^{2} {\left (180 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} - 59 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b - 227 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{2} + 160 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{3}\right )} {\left | b \right |} + {\left (244 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{8} b - 507 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b^{2} + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{3} + 695 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{4} - 597 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{5} + 160 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{6}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |} {\left | b \right |} + 2 \, {\left (32 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{12} b - 108 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{11} b^{2} + 87 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{10} b^{3} + 92 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{9} b^{4} - 198 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{8} b^{5} + 120 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b^{6} - 25 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} - \sqrt {{\left (a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right )} {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} + {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )}^{2}}}{a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}}}}\right )}{{\left (4 \, a^{12} b^{2} - 15 \, a^{11} b^{3} + 15 \, a^{10} b^{4} + 10 \, a^{9} b^{5} - 30 \, a^{8} b^{6} + 21 \, a^{7} b^{7} - 5 \, a^{6} b^{8}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |}} - \frac {4 \, {\left (13 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 7 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 212 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 116 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 272 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 1248 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 592 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 2240 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 3200 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 960 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )}^{2} {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )}} - \frac {32 \, \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a^{3}} + \frac {32 \, \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a^{3}}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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