Optimal. Leaf size=246 \[ -\frac {3 \sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {\sqrt {3 \pi } \sqrt {b} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}} \]
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Rubi [A] time = 0.42, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3313, 3308, 2180, 2204, 2205} \[ -\frac {3 \sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {\sqrt {3 \pi } \sqrt {b} e^{\frac {3 b c}{d}-3 a} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {3 \pi } \sqrt {b} e^{3 a-\frac {3 b c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 3313
Rubi steps
\begin {align*} \int \frac {\cosh ^3(a+b x)}{(c+d x)^{3/2}} \, dx &=-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {(6 i b) \int \left (-\frac {i \sinh (a+b x)}{4 \sqrt {c+d x}}-\frac {i \sinh (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{d}\\ &=-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {(3 b) \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx}{2 d}+\frac {(3 b) \int \frac {\sinh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{2 d}\\ &=-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}+\frac {(3 b) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 d}-\frac {(3 b) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{4 d}+\frac {(3 b) \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{4 d}-\frac {(3 b) \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{4 d}\\ &=-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {(3 b) \operatorname {Subst}\left (\int e^{i \left (3 i a-\frac {3 i b c}{d}\right )-\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}-\frac {(3 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 )}{2 d^2}+\frac {(3 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 )}{2 d^2}+\frac {(3 b) \operatorname {Subst}\left (\int e^{-i \left (3 i a-\frac {3 i b c}{d}\right )+\frac {3 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{2 d^2}\\ &=-\frac {2 \cosh ^3(a+b x)}{d \sqrt {c+d x}}-\frac {3 \sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {\sqrt {b} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {3 \sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}\\ \end {align*}
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Mathematica [B] time = 2.89, size = 717, normalized size = 2.91 \[ \frac {e^{-\frac {3 b (c+d x)}{d}} \left (\sqrt {3 \pi } \sqrt {b} \sqrt {c+d x} e^{\frac {3 b (c+d x)}{d}} \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+\sqrt {3 \pi } \sqrt {b} \sqrt {c+d x} e^{\frac {3 b (c+d x)}{d}} \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )-\sqrt {d} e^{\frac {6 b (c+d x)}{d}} \sinh \left (3 a-\frac {3 b c}{d}\right )+3 \sqrt {d} e^{\frac {2 b (c+d x)}{d}} \sinh \left (a-\frac {b c}{d}\right )-3 \sqrt {d} e^{\frac {4 b (c+d x)}{d}} \sinh \left (a-\frac {b c}{d}\right )-\sqrt {d} e^{\frac {6 b (c+d x)}{d}} \cosh \left (3 a-\frac {3 b c}{d}\right )-3 \sqrt {d} e^{\frac {2 b (c+d x)}{d}} \cosh \left (a-\frac {b c}{d}\right )-3 \sqrt {d} e^{\frac {4 b (c+d x)}{d}} \cosh \left (a-\frac {b c}{d}\right )+\sqrt {3} \sqrt {d} e^{\frac {3 b (c+d x)}{d}} \sqrt {-\frac {b (c+d x)}{d}} \cosh \left (3 a-\frac {3 b c}{d}\right ) \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )+\sqrt {3} \sqrt {d} e^{\frac {3 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \cosh \left (3 a-\frac {3 b c}{d}\right ) \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )+3 \sqrt {d} e^{\frac {3 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \cosh \left (a-\frac {b c}{d}\right ) \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )-3 \sqrt {d} e^{\frac {3 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \sinh \left (a-\frac {b c}{d}\right ) \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )+3 \sqrt {d} e^{\frac {3 b (c+d x)}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right ) \left (\sinh \left (a-\frac {b c}{d}\right )+\cosh \left (a-\frac {b c}{d}\right )\right )+\sqrt {d} \sinh \left (3 a-\frac {3 b c}{d}\right )-\sqrt {d} \cosh \left (3 a-\frac {3 b c}{d}\right )\right )}{4 d^{3/2} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 1344, normalized size = 5.46 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}\left (b x +a \right )}{\left (d x +c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 196, normalized size = 0.80 \[ -\frac {\frac {\sqrt {3} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {\sqrt {3} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cosh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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