Optimal. Leaf size=144 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} d} \]
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Rubi [A] time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3223, 200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 3223
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} d}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} d}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt [3]{a} d}-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b} d}\\ &=-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 116, normalized size = 0.81 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{6 a^{2/3} \sqrt [3]{b} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 401, normalized size = 2.78 \[ \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (-\frac {3 \, \left (a^{2} b\right )^{\frac {1}{3}} a \sin \left (d x + c\right ) + a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b \cos \left (d x + c\right )^{2} - 2 \, a b - \left (a^{2} b\right )^{\frac {2}{3}} \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} + 2 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \sin \left (d x + c\right )}{{\left (b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (-a b \cos \left (d x + c\right )^{2} + a b - \left (a^{2} b\right )^{\frac {2}{3}} \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b d}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} \sin \left (d x + c\right ) - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (-a b \cos \left (d x + c\right )^{2} + a b - \left (a^{2} b\right )^{\frac {2}{3}} \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 137, normalized size = 0.95 \[ -\frac {\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 120, normalized size = 0.83 \[ \frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 d b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 121, normalized size = 0.84 \[ \frac {\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 123, normalized size = 0.85 \[ \frac {\ln \left (b^{1/3}\,\sin \left (c+d\,x\right )+a^{1/3}\right )}{3\,a^{2/3}\,b^{1/3}\,d}+\frac {\ln \left (3\,b^2\,\sin \left (c+d\,x\right )+\frac {3\,a^{1/3}\,b^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,d}-\frac {\ln \left (3\,b^2\,\sin \left (c+d\,x\right )-\frac {3\,a^{1/3}\,b^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{1/3}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.85, size = 250, normalized size = 1.74 \[ \begin {cases} \frac {\tilde {\infty } x \cos {\relax (c )}}{\sin ^{3}{\relax (c )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {1}{2 b d \sin ^{2}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{a + b \sin ^{3}{\relax (c )}} & \text {for}\: d = 0 \\- \frac {\sqrt [3]{-1} \sqrt [3]{\frac {1}{b}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sin {\left (c + d x \right )} \right )}}{3 a^{\frac {2}{3}} d} + \frac {\sqrt [3]{-1} \sqrt [3]{\frac {1}{b}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} \sin {\left (c + d x \right )} + 4 \sin ^{2}{\left (c + d x \right )} \right )}}{6 a^{\frac {2}{3}} d} + \frac {\sqrt [3]{-1} \sqrt {3} \sqrt [3]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \sin {\left (c + d x \right )}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )}}{3 a^{\frac {2}{3}} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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