Optimal. Leaf size=119 \[ -\frac {\sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}} \]
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Rubi [A] time = 0.18, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3297, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3297
Rule 3308
Rubi steps
\begin {align*} \int \frac {\cosh (a+b x)}{(c+d x)^{3/2}} \, dx &=-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}}+\frac {(2 b) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}}+\frac {b \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{d}-\frac {b \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}}-\frac {(2 b) \operatorname {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d^2}+\frac {(2 b) \operatorname {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 \cosh (a+b x)}{d \sqrt {c+d x}}-\frac {\sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 118, normalized size = 0.99 \[ \frac {e^{-a} \left (e^{2 a-\frac {b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{-b x} \left (e^{2 (a+b x)}+1\right )+e^{\frac {b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},b \left (\frac {c}{d}+x\right )\right )\right )}{d \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 338, normalized size = 2.84 \[ -\frac {\sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + \sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (d x + c\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + \sqrt {d x + c} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )}}{{\left (d^{2} x + c d\right )} \cosh \left (b x + a\right ) + {\left (d^{2} x + c d\right )} \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x +a \right )}{\left (d x +c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 104, normalized size = 0.87 \[ \frac {\frac {{\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{\sqrt {\frac {b}{d}}}\right )} b}{d} - \frac {2 \, \cosh \left (b x + a\right )}{\sqrt {d x + c}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cosh {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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