Optimal. Leaf size=280 \[ \frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {30 b^2 \sinh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2}-\frac {5 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2+2}} \]
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Rubi [A] time = 0.11, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5880, 5878} \[ \frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {30 b^2 \sinh ^2\left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x}-\frac {5 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 5878
Rule 5880
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2} \, dx &=-\frac {5 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2}+\left (15 b^2\right ) \int \sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )} \, dx\\ &=-\frac {5 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}+\frac {30 b^2 \sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )} \sinh ^2\left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}\\ \end {align*}
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Mathematica [A] time = 3.61, size = 311, normalized size = 1.11 \[ \frac {x \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (4 \sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )} \left (\left (a^2+15 b^2\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-5 a b \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-b \cosh ^{-1}\left (d x^2+1\right ) \left (5 b \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-2 a \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )\right )+b^2 \cosh ^{-1}\left (d x^2+1\right )^2 \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )\right )+15 \sqrt {2 \pi } b^{5/2} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )-15 \sqrt {2 \pi } b^{5/2} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )\right )}{2 \sqrt {d x^2} \sqrt {\frac {d x^2}{d x^2+2}} \sqrt {d x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}+1\right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^{5/2} \,d x \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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