Optimal. Leaf size=288 \[ \frac {6 (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {360 f \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 f (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {360 f \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {6 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 f (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3512, 3377,
2718, 2717} \begin {gather*} \frac {360 f \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}-\frac {360 f \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {180 f (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {6 (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {60 f (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {6 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {3 (c+d x)^{2/3} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
Rule 3377
Rule 3512
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \text {Subst}\left (\int \left (\frac {(d e-c f) x^2 \sin (a+b x)}{d}+\frac {f x^5 \sin (a+b x)}{d}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {(3 f) \text {Subst}\left (\int x^5 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}+\frac {(3 (d e-c f)) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {(15 f) \text {Subst}\left (\int x^4 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}+\frac {(6 (d e-c f)) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}\\ &=-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {(60 f) \text {Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {(6 (d e-c f)) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}\\ &=\frac {6 (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 f (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {(180 f) \text {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^2}\\ &=\frac {6 (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 f (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 f (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {(360 f) \text {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^2}\\ &=\frac {6 (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {360 f \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 f (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {6 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 f (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {(360 f) \text {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^2}\\ &=\frac {6 (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {360 f \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 (d e-c f) (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {60 f (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 f (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {360 f \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {6 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 f (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 f (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 147, normalized size = 0.51 \begin {gather*} \frac {-3 b \left (120 f \sqrt [3]{c+d x}+b^4 d (c+d x)^{2/3} (e+f x)-2 b^2 (9 c f+d (e+10 f x))\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )+3 \left (2 b^4 d e \sqrt [3]{c+d x}+f \left (120-60 b^2 (c+d x)^{2/3}+b^4 \sqrt [3]{c+d x} (3 c+5 d x)\right )\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(800\) vs.
\(2(258)=516\).
time = 0.02, size = 801, normalized size = 2.78 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 684 vs.
\(2 (261) = 522\).
time = 0.34, size = 684, normalized size = 2.38 \begin {gather*} \frac {3 \, {\left (\frac {a^{2} c f \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{d} - a^{2} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e - \frac {2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a c f}{d} + \frac {a^{5} f \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{3} d} + 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a e - \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{4} f}{b^{3} d} + \frac {{\left ({\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} c f}{d} - {\left ({\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} e + \frac {10 \, {\left ({\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{3} f}{b^{3} d} - \frac {10 \, {\left ({\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 6 \, a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{2} f}{b^{3} d} + \frac {5 \, {\left ({\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 12 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 24\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 4 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 6 \, a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a f}{b^{3} d} - \frac {{\left ({\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 120 \, a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 5 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 12 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 24\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} f}{b^{3} d}\right )}}{b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 145, normalized size = 0.50 \begin {gather*} \frac {3 \, {\left ({\left (20 \, b^{3} d f x + 18 \, b^{3} c f + 2 \, b^{3} d e - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b f - {\left (b^{5} d f x + b^{5} d e\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (60 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} f - {\left (5 \, b^{4} d f x + 3 \, b^{4} c f + 2 \, b^{4} d e\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 120 \, f\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \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 \left (e + f x\right ) \sin {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.12, size = 454, normalized size = 1.58 \begin {gather*} \frac {3 \, {\left ({\left (\frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b} - \frac {{\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\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{2}}\right )} e + \frac {f {\left (\frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} b^{3} c - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} a - 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{3} - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{4} + a^{5} - 2 \, b^{3} c + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{2} - 20 \, a^{3} - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}} - \frac {{\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} b^{3} c - 2 \, a b^{3} c - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a - 30 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{2} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{3} - 5 \, a^{4} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 120 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + 60 \, a^{2} - 120\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}}\right )}}{d}\right )}}{b d} \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 \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,\left (e+f\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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