Optimal. Leaf size=55 \[ \text {Ci}(b x) \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x) \]
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Rubi [A]
time = 0.11, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6852, 3384,
3380, 3383} \begin {gather*} \sin (a) \text {CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \text {Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 6852
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (a+b x)}{x} \, dx\\ &=\left (\cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (b x)}{x} \, dx+\left (\csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\cos (b x)}{x} \, dx\\ &=\text {Ci}(b x) \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 0.65 \begin {gather*} \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} (\text {Ci}(b x) \sin (a)+\cos (a) \text {Si}(b x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 228, normalized size = 4.15
method | result | size |
risch | \(-\frac {\expIntegral \left (1, -i b x \right ) \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i \left (b x +2 a \right )}}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \pi \,\mathrm {csgn}\left (b x \right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \sinIntegral \left (b x \right )}{{\mathrm e}^{2 i \left (b x +a \right )}-1}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \expIntegral \left (1, -i b x \right )}{2 \,{\mathrm e}^{2 i \left (b x +a \right )}-2}\) | \(228\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.58, size = 42, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, {\left ({\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \cos \left (a\right ) + {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \sin \left (a\right )\right )} c^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 80, normalized size = 1.45 \begin {gather*} -\frac {4^{\frac {1}{3}} {\left (2 \cdot 4^{\frac {2}{3}} \cos \left (a\right ) \operatorname {Si}\left (b x\right ) + {\left (4^{\frac {2}{3}} \operatorname {Ci}\left (b x\right ) + 4^{\frac {2}{3}} \operatorname {Ci}\left (-b x\right )\right )} \sin \left (a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}} \sin \left (b x + a\right )}{8 \, {\left (\cos \left (b x + a\right )^{2} - 1\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 \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{x}\, 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 = -0.02 \begin {gather*} \int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{1/3}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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