Optimal. Leaf size=275 \[ -\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}+\frac {3 \sqrt {d} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {3 \sqrt {d} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}} \]
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Rubi [A]
time = 0.38, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3393, 3377,
3388, 2211, 2235, 2236} \begin {gather*} \frac {3 \sqrt {\pi } \sqrt {d} e^{\frac {b c}{d}-a} \text {Erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} e^{\frac {3 b c}{d}-3 a} \text {Erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {d} e^{a-\frac {b c}{d}} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} e^{3 a-\frac {3 b c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}-\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3377
Rule 3388
Rule 3393
Rubi steps
\begin {align*} \int \sqrt {c+d x} \sinh ^3(a+b x) \, dx &=i \int \left (\frac {3}{4} i \sqrt {c+d x} \sinh (a+b x)-\frac {1}{4} i \sqrt {c+d x} \sinh (3 a+3 b x)\right ) \, dx\\ &=\frac {1}{4} \int \sqrt {c+d x} \sinh (3 a+3 b x) \, dx-\frac {3}{4} \int \sqrt {c+d x} \sinh (a+b x) \, dx\\ &=-\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}-\frac {d \int \frac {\cosh (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{24 b}+\frac {(3 d) \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{8 b}\\ &=-\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}-\frac {d \int \frac {e^{-i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{48 b}-\frac {d \int \frac {e^{i (3 i a+3 i b x)}}{\sqrt {c+d x}} \, dx}{48 b}+\frac {(3 d) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{16 b}+\frac {(3 d) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{16 b}\\ &=-\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}-\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 )}{24 b}-\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 )}{24 b}+\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 )}{8 b}+\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 )}{8 b}\\ &=-\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}+\frac {3 \sqrt {d} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} 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 )}{48 b^{3/2}}+\frac {3 \sqrt {d} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} 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 )}{48 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 209, normalized size = 0.76 \begin {gather*} \frac {e^{-3 \left (a+\frac {b c}{d}\right )} \sqrt {c+d x} \left (\sqrt {3} e^{6 a} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {3 b (c+d x)}{d}\right )-27 e^{4 a+\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \left (-27 e^{2 a} \Gamma \left (\frac {3}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{72 b \sqrt {-\frac {b^2 (c+d x)^2}{d^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (\sinh ^{3}\left (b x +a \right )\right ) \sqrt {d x +c}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 333, normalized size = 1.21 \begin {gather*} -\frac {\frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} + \frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - \frac {27 \, \sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} - \frac {27 \, \sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - \frac {6 \, \sqrt {d x + c} d e^{\left (3 \, a + \frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b} + \frac {54 \, \sqrt {d x + c} d e^{\left (a + \frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b} + \frac {54 \, \sqrt {d x + c} d e^{\left (-a - \frac {{\left (d x + c\right )} b}{d} + \frac {b c}{d}\right )}}{b} - \frac {6 \, \sqrt {d x + c} d e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d} + \frac {3 \, b c}{d}\right )}}{b}}{144 \, d} \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. 1216 vs.
\(2 (201) = 402\).
time = 0.38, size = 1216, normalized size = 4.42 \begin {gather*} -\frac {\sqrt {3} \sqrt {\pi } {\left (d \cosh \left (b x + a\right )^{3} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - d \cosh \left (b x + a\right )^{3} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - d \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - d \cosh \left (b x + a\right ) \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (d \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - d \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - \sqrt {3} \sqrt {\pi } {\left (d \cosh \left (b x + a\right )^{3} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + d \cosh \left (b x + a\right )^{3} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + d \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + d \cosh \left (b x + a\right ) \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (d \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + d \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 27 \, \sqrt {\pi } {\left (d \cosh \left (b x + a\right )^{3} \cosh \left (-\frac {b c - a d}{d}\right ) - d \cosh \left (b x + a\right )^{3} \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d \cosh \left (-\frac {b c - a d}{d}\right ) - d \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - d \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (d \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {b c - a d}{d}\right ) - d \cosh \left (b x + a\right )^{2} \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 ) + 27 \, \sqrt {\pi } {\left (d \cosh \left (b x + a\right )^{3} \cosh \left (-\frac {b c - a d}{d}\right ) + d \cosh \left (b x + a\right )^{3} \sinh \left (-\frac {b c - a d}{d}\right ) + {\left (d \cosh \left (-\frac {b c - a d}{d}\right ) + d \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (d \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + d \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (d \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {b c - a d}{d}\right ) + d \cosh \left (b x + a\right )^{2} \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 ) - 6 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - 9 \, b \cosh \left (b x + a\right )^{4} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} - 3 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 9 \, b \cosh \left (b x + a\right )^{2} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{4} - 18 \, b \cosh \left (b x + a\right )^{2} - 3 \, b\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (b \cosh \left (b x + a\right )^{5} - 6 \, b \cosh \left (b x + a\right )^{3} - 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )} \sqrt {d x + c}}{144 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3}\right )}} \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 \sqrt {c + d x} \sinh ^{3}{\left (a + b x \right )}\, 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 {\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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