Optimal. Leaf size=176 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b} d}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )} \]
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Rubi [A]
time = 0.08, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3302, 205, 206,
31, 648, 631, 210, 642} \begin {gather*} -\frac {2 \text {ArcTan}\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 {\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 {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 205
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 3302
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a d}\\ &=\frac {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {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 \text {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 {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )}+\frac {\text {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 {\text {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 {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {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 {\sin (c+d x)}{3 a d \left (a+b \sin ^3(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 152, normalized size = 0.86 \begin {gather*} \frac {-\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 )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{\sqrt [3]{b}}+\frac {3 a^{2/3} \sin (c+d x)}{a+b \sin ^3(c+d x)}}{9 a^{5/3} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 146, normalized size = 0.83
method | result | size |
risch | \(-\frac {4 \left ({\mathrm e}^{4 i \left (d x +c \right )}-{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a d \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (729 a^{5} b \,d^{3} \textit {\_Z}^{3}-8\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+9 i a^{2} d \textit {\_R} \,{\mathrm e}^{i \left (d x +c \right )}-1\right )\right )\) | \(134\) |
derivativedivides | \(\frac {\frac {\sin \left (d x +c \right )}{3 a \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}+\frac {\frac {2 \ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}}{d}\) | \(146\) |
default | \(\frac {\frac {\sin \left (d x +c \right )}{3 a \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}+\frac {\frac {2 \ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}}{d}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 155, normalized size = 0.88 \begin {gather*} \frac {\frac {3 \, \sin \left (d x + c\right )}{a b \sin \left (d x + c\right )^{3} + a^{2}} + \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs.
\(2 (137) = 274\).
time = 0.49, size = 655, normalized size = 3.72 \begin {gather*} \left [\frac {3 \, a^{2} b \sin \left (d x + c\right ) + 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 ) + 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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 617 vs.
\(2 (162) = 324\).
time = 122.54, size = 617, normalized size = 3.51 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x \cos {\left (c \right )}}{\sin ^{6}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {1}{5 b^{2} d \sin ^{5}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {\tilde {\infty } \sin {\left (c + d x \right )}}{d} & \text {for}\: b = - \frac {a}{\sin ^{3}{\left (c + d x \right )}} \\\frac {x \cos {\left (c \right )}}{\left (a + b \sin ^{3}{\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\- \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (- \sqrt [3]{- \frac {a}{b}} + \sin {\left (c + d x \right )} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {a \sqrt [3]{- \frac {a}{b}} \log {\left (4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {a}{b}} \sin {\left (c + d x \right )} + 4 \sin ^{2}{\left (c + d x \right )} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {2 \sqrt {3} a \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \sin {\left (c + d x \right )}}{3 \sqrt [3]{- \frac {a}{b}}} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {3 a \sin {\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} - \frac {2 b \sqrt [3]{- \frac {a}{b}} \log {\left (- \sqrt [3]{- \frac {a}{b}} + \sin {\left (c + d x \right )} \right )} \sin ^{3}{\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {b \sqrt [3]{- \frac {a}{b}} \log {\left (4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {a}{b}} \sin {\left (c + d x \right )} + 4 \sin ^{2}{\left (c + d x \right )} \right )} \sin ^{3}{\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {2 \sqrt {3} b \sqrt [3]{- \frac {a}{b}} \sin ^{3}{\left (c + d x \right )} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \sin {\left (c + d x \right )}}{3 \sqrt [3]{- \frac {a}{b}}} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} - \frac {2 b \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )} \sin ^{3}{\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 162, normalized size = 0.92 \begin {gather*} -\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 {3 \, \sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{3} + a\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^{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}}{9 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.00, size = 165, normalized size = 0.94 \begin {gather*} \frac {\sin \left (c+d\,x\right )}{3\,a\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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