Optimal. Leaf size=208 \[ -\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A]
time = 0.15, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3273, 91, 79,
53, 65, 214} \begin {gather*} -\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 91
Rule 214
Rule 3273
Rubi steps
\begin {align*} \int \frac {\coth ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {(1+x)^2}{x^3 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (8 a-7 b)+2 a x}{x^2 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^3 f}\\ &=\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^4 f}\\ &=\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{8 a^4 b f}\\ &=-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.30, size = 117, normalized size = 0.56 \begin {gather*} -\frac {\text {csch}^2(e+f x) \left (3 a \text {csch}^2(e+f x) \left (8 a-7 b+2 a \text {csch}^2(e+f x)\right )+\left (-8 a^2+40 a b-35 b^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1+\frac {b \sinh ^2(e+f x)}{a}\right )\right )}{24 a^3 f \left (b+a \text {csch}^2(e+f x)\right ) \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.03, size = 73, normalized size = 0.35
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\cosh ^{4}\left (f x +e \right )}{\left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right ) \sinh \left (f x +e \right )^{5} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(73\) |
risch | \(\text {Expression too large to display}\) | \(2628058\) |
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. 7450 vs.
\(2 (184) = 368\).
time = 1.03, size = 15102, normalized size = 72.61 \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 [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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