Optimal. Leaf size=85 \[ \frac {6 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {3 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5418, 5412,
3377, 2718} \begin {gather*} \frac {6 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {6 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5412
Rule 5418
Rubi steps
\begin {align*} \int \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int \sinh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {3 \text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {3 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d}\\ &=\frac {3 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {6 \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d}\\ &=\frac {6 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {3 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 65, normalized size = 0.76 \begin {gather*} \frac {3 \left (2+b^2 (c+d x)^{2/3}\right ) \cosh \left (a+b \sqrt [3]{c+d x}\right )-6 b \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 133, normalized size = 1.56
method | result | size |
derivativedivides | \(\frac {3 a^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3} d}\) | \(133\) |
default | \(\frac {3 a^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3} d}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 137, normalized size = 1.61 \begin {gather*} -\frac {b {\left (\frac {{\left ({\left (d x + c\right )} b^{3} e^{a} - 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 6 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{4}} - \frac {{\left ({\left (d x + c\right )} b^{3} + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 6\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{4}}\right )} - 2 \, {\left (d x + c\right )} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 58, normalized size = 0.68 \begin {gather*} -\frac {3 \, {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 2\right )} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 94, normalized size = 1.11 \begin {gather*} \begin {cases} x \sinh {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \sinh {\left (a + b \sqrt [3]{c} \right )} & \text {for}\: d = 0 \\\frac {3 \left (c + d x\right )^{\frac {2}{3}} \cosh {\left (a + b \sqrt [3]{c + d x} \right )}}{b d} - \frac {6 \sqrt [3]{c + d x} \sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} + \frac {6 \cosh {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 128, normalized size = 1.51 \begin {gather*} \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} - 2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 2\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{2 \, b^{3} d} + \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} + 2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 2\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{2 \, b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 75, normalized size = 0.88 \begin {gather*} \frac {6\,\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{b^3\,d}+\frac {3\,\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{2/3}}{b\,d}-\frac {6\,\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{1/3}}{b^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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