Optimal. Leaf size=85 \[ \frac {6 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {6 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.08, 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 = {5311, 5305, 3296, 2637} \[ \frac {6 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {6 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5305
Rule 5311
Rubi steps
\begin {align*} \int \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \cosh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {3 \operatorname {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {3 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d}\\ &=-\frac {6 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {6 \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d}\\ &=-\frac {6 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {6 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {3 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 65, normalized size = 0.76 \[ \frac {3 \left (b^2 (c+d x)^{2/3}+2\right ) \sinh \left (a+b \sqrt [3]{c+d x}\right )-6 b \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 58, normalized size = 0.68 \[ -\frac {3 \, {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 2\right )} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 128, normalized size = 1.51 \[ \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} - 2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 2\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{2 \, b^{3} d} - \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} + 2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 2\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{2 \, b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 133, normalized size = 1.56 \[ \frac {3 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2}-6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\cosh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+3 a^{2} \sinh \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{d \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 136, normalized size = 1.60 \[ -\frac {b {\left (\frac {{\left ({\left (d x + c\right )} b^{3} e^{a} - 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 6 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{4}} + \frac {{\left ({\left (d x + c\right )} b^{3} + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 6\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{4}}\right )} - 2 \, {\left (d x + c\right )} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 75, normalized size = 0.88 \[ \frac {6\,\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{b^3\,d}-\frac {6\,\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{1/3}}{b^2\,d}+\frac {3\,\mathrm {sinh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{2/3}}{b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 94, normalized size = 1.11 \[ \begin {cases} x \cosh {\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \cosh {\left (a + b \sqrt [3]{c} \right )} & \text {for}\: d = 0 \\\frac {3 \left (c + d x\right )^{\frac {2}{3}} \sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{b d} - \frac {6 \sqrt [3]{c + d x} \cosh {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} + \frac {6 \sinh {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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