Optimal. Leaf size=274 \[ -\frac {4 b \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {b} (7 a+b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 \sqrt {a} (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a+5 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3275, 482, 541,
539, 429, 422} \begin {gather*} -\frac {\sqrt {b} (7 a+b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 \sqrt {a} f (a-b)^3 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac {(3 a+5 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 a f (a-b)^3 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\tanh (e+f x)}{f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {4 b \sinh (e+f x) \cosh (e+f x)}{3 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 482
Rule 539
Rule 541
Rule 3275
Rubi steps
\begin {align*} \int \frac {\tanh ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^{3/2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {a-3 b x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{(-a+b) f}\\ &=-\frac {4 b \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {a (3 a+b)-4 a b x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a (a-b) (-a+b) f}\\ &=-\frac {4 b \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (b (7 a+b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^2 (-a+b) f}-\frac {\left ((3 a+5 b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^2 (-a+b) f}\\ &=-\frac {4 b \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {b} (7 a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 \sqrt {a} (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a+5 b) F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.14, size = 215, normalized size = 0.78 \begin {gather*} \frac {-2 i a^2 (7 a+b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 i a^2 (a-b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-\frac {\left (24 a^3-4 a^2 b+5 a b^2-b^3+4 a (11 a-3 b) b \cosh (2 (e+f x))+b^2 (7 a+b) \cosh (4 (e+f x))\right ) \tanh (e+f x)}{\sqrt {2}}}{6 a (a-b)^3 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \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. \(798\) vs.
\(2(342)=684\).
time = 2.19, size = 799, normalized size = 2.92
method | result | size |
default | \(\frac {-7 \sqrt {-\frac {b}{a}}\, a \,b^{2} \left (\sinh ^{5}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{3} \left (\sinh ^{5}\left (f x +e \right )\right )+3 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b \left (\sinh ^{2}\left (f x +e \right )\right )-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2} \left (\sinh ^{2}\left (f x +e \right )\right )-\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3} \left (\sinh ^{2}\left (f x +e \right )\right )+7 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2} \left (\sinh ^{2}\left (f x +e \right )\right )+\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3} \left (\sinh ^{2}\left (f x +e \right )\right )-11 \sqrt {-\frac {b}{a}}\, a^{2} b \left (\sinh ^{3}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{3} \left (\sinh ^{3}\left (f x +e \right )\right )+3 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+7 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b +\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-3 \sqrt {-\frac {b}{a}}\, a^{3} \sinh \left (f x +e \right )-5 \sinh \left (f x +e \right ) b \,a^{2} \sqrt {-\frac {b}{a}}}{3 \sqrt {-\frac {b}{a}}\, \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a -b \right )^{3} a \cosh \left (f x +e \right ) f}\) | \(799\) |
risch | \(\text {Expression too large to display}\) | \(74496\) |
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. 8226 vs.
\(2 (280) = 560\).
time = 0.29, size = 8226, normalized size = 30.02 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, 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: RuntimeError} \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 {tanh}\left (e+f\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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