Optimal. Leaf size=55 \[ \sin (a) \text {Ci}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \text {Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, 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 = {6720, 3303, 3299, 3302} \[ \sin (a) \text {CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \text {Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3299
Rule 3302
Rule 3303
Rule 6720
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 36, normalized size = 0.65 \[ \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} (\sin (a) \text {Ci}(b x)+\cos (a) \text {Si}(b x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 80, normalized size = 1.45 \[ -\frac {4^{\frac {1}{3}} {\left (2 \cdot 4^{\frac {2}{3}} \cos \relax (a) \operatorname {Si}\left (b x\right ) + {\left (4^{\frac {2}{3}} \operatorname {Ci}\left (b x\right ) + 4^{\frac {2}{3}} \operatorname {Ci}\left (-b x\right )\right )} \sin \relax (a)\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}} \sin \left (b x + a\right )}{8 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )^{3}\right )^{\frac {1}{3}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.19, size = 228, normalized size = 4.15 \[ -\frac {\Ei \left (1, -i b x \right ) \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i \left (b x +2 a \right )}}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \pi \,\mathrm {csgn}\left (b x \right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \Si \left (b x \right )}{{\mathrm e}^{2 i \left (b x +a \right )}-1}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \Ei \left (1, -i b x \right )}{2 \,{\mathrm e}^{2 i \left (b x +a \right )}-2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.21, size = 42, normalized size = 0.76 \[ \frac {1}{4} \, {\left ({\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \cos \relax (a) + {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \sin \relax (a)\right )} c^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{1/3}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________