Optimal. Leaf size=313 \[ -\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
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Rubi [A]
time = 0.39, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1219,
1192, 1180, 211, 214} \begin {gather*} -\frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{32 a b d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b 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.
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Rule 211
Rule 214
Rule 1180
Rule 1192
Rule 1219
Rule 3294
Rubi steps
\begin {align*} \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {2 a (a-7 b)+10 a b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a (a-b) b d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 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 \left (3 a^2-17 a b+2 b^2\right )-8 a b^2 (2 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 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^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {b} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 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^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} d}\\ &=-\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b 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 = 1.36, size = 1019, normalized size = 3.26 \begin {gather*} -\frac {\frac {32 \cosh (c+d x) \left (a^2-9 a b-b^2+b (2 a+b) \cosh (2 (c+d x))\right )}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}-\frac {512 (a-b) \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {2 a b c+b^2 c+2 a b d x+b^2 d x+4 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 a^2 c \text {$\#$1}^2-32 a b c \text {$\#$1}^2+5 b^2 c \text {$\#$1}^2+6 a^2 d x \text {$\#$1}^2-32 a b d x \text {$\#$1}^2+5 b^2 d x \text {$\#$1}^2+12 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-64 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+10 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-6 a^2 c \text {$\#$1}^4+32 a b c \text {$\#$1}^4-5 b^2 c \text {$\#$1}^4-6 a^2 d x \text {$\#$1}^4+32 a b d x \text {$\#$1}^4-5 b^2 d x \text {$\#$1}^4-12 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+64 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-10 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-2 a b c \text {$\#$1}^6-b^2 c \text {$\#$1}^6-2 a b d x \text {$\#$1}^6-b^2 d x \text {$\#$1}^6-4 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-2 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a}}{128 (a-b)^2 b 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. \(649\) vs.
\(2(261)=522\).
time = 10.68, size = 650, normalized size = 2.08
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (3 a^{2}-13 a b +4 b^{2}\right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (7 a^{2}-33 a b +8 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (63 a^{3}-225 a^{2} b +68 a \,b^{2}+64 b^{3}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right ) a}+\frac {3 \left (35 a^{3}-61 a^{2} b +32 a \,b^{2}+128 b^{3}\right ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (105 a^{3}+9 a^{2} b -452 a \,b^{2}-64 b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (21 a^{2}+29 a b -40 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (21 a^{2}+37 a b -4 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \left (a +b \right )}{16 b \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 {\frac {\left (4 \sqrt {a b}\, a +2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 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 )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\left (-4 \sqrt {a b}\, a -2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 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 )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(650\) |
default | \(\frac {\frac {-\frac {\left (3 a^{2}-13 a b +4 b^{2}\right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (7 a^{2}-33 a b +8 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (63 a^{3}-225 a^{2} b +68 a \,b^{2}+64 b^{3}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right ) a}+\frac {3 \left (35 a^{3}-61 a^{2} b +32 a \,b^{2}+128 b^{3}\right ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (105 a^{3}+9 a^{2} b -452 a \,b^{2}-64 b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (21 a^{2}+29 a b -40 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (21 a^{2}+37 a b -4 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \left (a +b \right )}{16 b \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 {\frac {\left (4 \sqrt {a b}\, a +2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 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 )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\left (-4 \sqrt {a b}\, a -2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 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 )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(650\) |
risch | \(\text {Expression too large to display}\) | \(1427\) |
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. 22506 vs.
\(2 (262) = 524\).
time = 0.83, size = 22506, normalized size = 71.90 \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. 1797 vs.
\(2 (262) = 524\).
time = 0.93, size = 1797, normalized size = 5.74 \begin {gather*} \frac {\frac {{\left (2 \, {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2} {\left (8 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 14 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | b \right |} - {\left (12 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b - 77 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{2} + 41 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{3} + 111 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{4} - 97 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{5} + 10 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{6}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |} {\left | b \right |} - {\left (12 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{8} b^{2} - 85 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b^{3} + 171 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{4} - 54 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{5} - 214 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{6} + 279 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{7} - 129 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{8} + 20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{9}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + \sqrt {{\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} + {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}^{2}}}{a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}}}}\right )}{{\left (4 \, a^{8} b^{4} - 15 \, a^{7} b^{5} + 15 \, a^{6} b^{6} + 10 \, a^{5} b^{7} - 30 \, a^{4} b^{8} + 21 \, a^{3} b^{9} - 5 \, a^{2} b^{10}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |}} + \frac {{\left (2 \, {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2} {\left (8 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 14 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | b \right |} - {\left (12 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b - 77 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{2} + 41 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{3} + 111 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{4} - 97 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{5} + 10 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{6}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |} {\left | b \right |} - {\left (12 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{8} b^{2} - 85 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b^{3} + 171 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{4} - 54 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{5} - 214 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{6} + 279 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{7} - 129 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{8} + 20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{9}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - \sqrt {{\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} + {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}^{2}}}{a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}}}}\right )}{{\left (4 \, a^{8} b^{4} - 15 \, a^{7} b^{5} + 15 \, a^{6} b^{6} + 10 \, a^{5} b^{7} - 30 \, a^{4} b^{8} + 21 \, a^{3} b^{9} - 5 \, a^{2} b^{10}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |}} + \frac {8 \, {\left (2 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 2 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 38 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 12 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 80 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 224 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 48 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 96 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 384 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 416 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 64 \, 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^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}}}{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 {{\mathrm {sinh}\left (c+d\,x\right )}^5}{{\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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