Optimal. Leaf size=261 \[ \frac{60 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac{6 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{360 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]
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Rubi [A] time = 0.318965, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5365, 5287, 3296, 2637, 2638} \[ \frac{60 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac{6 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{360 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]
Antiderivative was successfully verified.
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Rule 5365
Rule 5287
Rule 3296
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int x \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (-c+x) \cosh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{3 \operatorname{Subst}\left (\int x^2 \left (-c+x^3\right ) \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=\frac{3 \operatorname{Subst}\left (\int \left (-c x^2 \cosh (a+b x)+x^5 \cosh (a+b x)\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=\frac{3 \operatorname{Subst}\left (\int x^5 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}-\frac{(3 c) \operatorname{Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{15 \operatorname{Subst}\left (\int x^4 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}+\frac{(6 c) \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}\\ &=\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{60 \operatorname{Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{(6 c) \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}\\ &=\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{6 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{60 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{180 \operatorname{Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^2}\\ &=\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{6 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{60 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{360 \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^2}\\ &=\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{6 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{60 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{360 \operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^2}\\ &=-\frac{360 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac{6 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac{15 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{6 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac{3 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{60 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}\\ \end{align*}
Mathematica [A] time = 0.375067, size = 118, normalized size = 0.45 \[ \frac{3 b \left (b^4 d x (c+d x)^{2/3}+2 b^2 (9 c+10 d x)+120 \sqrt [3]{c+d x}\right ) \sinh \left (a+b \sqrt [3]{c+d x}\right )-3 \left (b^4 \sqrt [3]{c+d x} (3 c+5 d x)+60 b^2 (c+d x)^{2/3}+120\right ) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 659, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11862, size = 498, normalized size = 1.91 \begin{align*} \frac{2 \, d^{2} x^{2} \cosh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) -{\left (\frac{c^{2} e^{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}}{b} + \frac{c^{2} e^{\left (-{\left (d x + c\right )}^{\frac{1}{3}} b - a\right )}}{b} - \frac{2 \,{\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 )} c e^{\left ({\left (d x + c\right )}^{\frac{1}{3}} b\right )}}{b^{4}} - \frac{2 \,{\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 )} c e^{\left (-{\left (d x + c\right )}^{\frac{1}{3}} b - a\right )}}{b^{4}} + \frac{{\left ({\left (d x + c\right )}^{2} b^{6} e^{a} - 6 \,{\left (d x + c\right )}^{\frac{5}{3}} b^{5} e^{a} + 30 \,{\left (d x + c\right )}^{\frac{4}{3}} b^{4} e^{a} - 120 \,{\left (d x + c\right )} b^{3} e^{a} + 360 \,{\left (d x + c\right )}^{\frac{2}{3}} b^{2} e^{a} - 720 \,{\left (d x + c\right )}^{\frac{1}{3}} b e^{a} + 720 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac{1}{3}} b\right )}}{b^{7}} + \frac{{\left ({\left (d x + c\right )}^{2} b^{6} + 6 \,{\left (d x + c\right )}^{\frac{5}{3}} b^{5} + 30 \,{\left (d x + c\right )}^{\frac{4}{3}} b^{4} + 120 \,{\left (d x + c\right )} b^{3} + 360 \,{\left (d x + c\right )}^{\frac{2}{3}} b^{2} + 720 \,{\left (d x + c\right )}^{\frac{1}{3}} b + 720\right )} e^{\left (-{\left (d x + c\right )}^{\frac{1}{3}} b - a\right )}}{b^{7}}\right )} b}{4 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7847, size = 296, normalized size = 1.13 \begin{align*} -\frac{3 \,{\left ({\left (60 \,{\left (d x + c\right )}^{\frac{2}{3}} b^{2} +{\left (5 \, b^{4} d x + 3 \, b^{4} c\right )}{\left (d x + c\right )}^{\frac{1}{3}} + 120\right )} \cosh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) -{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{5} d x + 20 \, b^{3} d x + 18 \, b^{3} c + 120 \,{\left (d x + c\right )}^{\frac{1}{3}} b\right )} \sinh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh{\left (a + b \sqrt [3]{c + d x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.30918, size = 954, normalized size = 3.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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