Optimal. Leaf size=183 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b} d}+\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3223, 1854, 12, 200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b} d}+\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 1854
Rule 3223
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int -\frac {2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a d}\\ &=\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a d}\\ &=\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} d}+\frac {2 \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 )}{9 a^{5/3} d}\\ &=\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )}+\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 )}{3 a^{4/3} 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 )}{9 a^{5/3} \sqrt [3]{b} d}\\ &=\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \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 )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \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 )}{3 a^{5/3} \sqrt [3]{b} d}\\ &=-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b} d}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \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 )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {a+b \sin (c+d x)}{3 a b d \left (a+b \sin ^3(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 184, normalized size = 1.01 \[ \frac {\frac {-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3}}+\frac {2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3}}+\frac {3}{a+b \sin ^3(c+d x)}}{b}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b}}+\frac {3 \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}}{9 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 665, normalized size = 3.63 \[ \left [\frac {3 \, a^{2} b \sin \left (d x + c\right ) + 3 \, a^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (a^{2} b - {\left (a b^{2} \cos \left (d x + c\right )^{2} - a b^{2}\right )} \sin \left (d x + c\right )\right )} \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}} {\left ({\left (b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - a\right )} \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}} {\left ({\left (b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - a\right )} \log \left (a b \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{4} b d - {\left (a^{3} b^{2} d \cos \left (d x + c\right )^{2} - a^{3} b^{2} d\right )} \sin \left (d x + c\right )\right )}}, \frac {3 \, a^{2} b \sin \left (d x + c\right ) + 3 \, a^{3} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{2} b - {\left (a b^{2} \cos \left (d x + c\right )^{2} - a b^{2}\right )} \sin \left (d x + c\right )\right )} \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}} {\left ({\left (b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - a\right )} \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}} {\left ({\left (b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - a\right )} \log \left (a b \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{4} b d - {\left (a^{3} b^{2} d \cos \left (d x + c\right )^{2} - a^{3} b^{2} d\right )} \sin \left (d x + c\right )\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 169, normalized size = 0.92 \[ -\frac {\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} b} - \frac {3 \, {\left (b \sin \left (d x + c\right ) + a\right )}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )} a b}}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.93, size = 179, normalized size = 0.98 \[ \frac {\sin \left (d x +c \right )}{3 a d \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}+\frac {2 \ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 d b a \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 d b a \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 d b a \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {1}{3 d b \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 163, normalized size = 0.89 \[ \frac {\frac {3 \, {\left (b \sin \left (d x + c\right ) + a\right )}}{a b^{2} \sin \left (d x + c\right )^{3} + a^{2} b} + \frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 172, normalized size = 0.94 \[ \frac {\frac {\sin \left (c+d\,x\right )}{3\,a}+\frac {1}{3\,b}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^3+a\right )}+\frac {2\,\ln \left (\frac {2\,b^{5/3}}{a^{2/3}}+\frac {2\,b^2\,\sin \left (c+d\,x\right )}{a}\right )}{9\,a^{5/3}\,b^{1/3}\,d}+\frac {\ln \left (\frac {2\,b^2\,\sin \left (c+d\,x\right )}{a}+\frac {b^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{5/3}\,b^{1/3}\,d}-\frac {\ln \left (\frac {2\,b^2\,\sin \left (c+d\,x\right )}{a}-\frac {b^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{5/3}\,b^{1/3}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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