Optimal. Leaf size=228 \[ \frac {3 e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {3 e^{a-\frac {b c}{d}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}} \]
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Rubi [A]
time = 0.29, antiderivative size = 228, 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 = {3393, 3389,
2211, 2235, 2236} \begin {gather*} \frac {3 \sqrt {\pi } e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {3 \sqrt {\pi } e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rubi steps
\begin {align*} \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx &=i \int \left (\frac {3 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\\ &=\frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{\sqrt {c+d x}} \, dx-\frac {3}{4} \int \frac {\sinh (a+b x)}{\sqrt {c+d x}} \, dx\\ &=\frac {1}{8} \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx-\frac {1}{8} \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx-\frac {3}{8} \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx+\frac {3}{8} \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx\\ &=-\frac {\text {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 )}{4 d}+\frac {\text {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 )}{4 d}+\frac {3 \text {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 )}{4 d}-\frac {3 \text {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 )}{4 d}\\ &=\frac {3 e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {3 e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 191, normalized size = 0.84 \begin {gather*} \frac {e^{-3 \left (a+\frac {b c}{d}\right )} \left (\sqrt {3} e^{6 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )-9 e^{4 a+\frac {2 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \left (-9 e^{2 a} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{24 b \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{3}\left (b x +a \right )}{\sqrt {d x +c}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 178, normalized size = 0.78 \begin {gather*} \frac {\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{\sqrt {\frac {b}{d}}} - \frac {9 \, \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 {9 \, \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}}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 252, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {3} \sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + \sqrt {3} \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 9 \, \sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) - \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - 9 \, \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) + \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right )}{24 \, b} \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 {\sinh ^{3}{\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {sinh}\left (a+b\,x\right )}^3}{\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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