Optimal. Leaf size=293 \[ -\frac {a x}{b^2}+\frac {2 a^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^2 d}+\frac {2 a^{4/3} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b^2 d}-\frac {2 a^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b^2 d}-\frac {\cos (c+d x)}{b d}+\frac {\cos ^3(c+d x)}{3 b d} \]
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Rubi [A]
time = 0.30, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3299, 2713,
3292, 2739, 632, 210} \begin {gather*} \frac {2 a^{4/3} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 b^2 d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 a^{4/3} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 b^2 d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}-\frac {2 a^{4/3} \text {ArcTan}\left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 b^2 d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {a x}{b^2}+\frac {\cos ^3(c+d x)}{3 b d}-\frac {\cos (c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2713
Rule 2739
Rule 3292
Rule 3299
Rubi steps
\begin {align*} \int \frac {\sin ^6(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (-\frac {a}{b^2}+\frac {\sin ^3(c+d x)}{b}+\frac {a^2}{b^2 \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=-\frac {a x}{b^2}+\frac {a^2 \int \frac {1}{a+b \sin ^3(c+d x)} \, dx}{b^2}+\frac {\int \sin ^3(c+d x) \, dx}{b}\\ &=-\frac {a x}{b^2}+\frac {a^2 \int \left (-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{b^2}-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{b d}\\ &=-\frac {a x}{b^2}-\frac {\cos (c+d x)}{b d}+\frac {\cos ^3(c+d x)}{3 b d}-\frac {a^{4/3} \int \frac {1}{-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^2}-\frac {a^{4/3} \int \frac {1}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^2}-\frac {a^{4/3} \int \frac {1}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^2}\\ &=-\frac {a x}{b^2}-\frac {\cos (c+d x)}{b d}+\frac {\cos ^3(c+d x)}{3 b d}-\frac {\left (2 a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{a}-2 \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^2 d}-\frac {\left (2 a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^2 d}-\frac {\left (2 a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{a}-2 (-1)^{2/3} \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^2 d}\\ &=-\frac {a x}{b^2}-\frac {\cos (c+d x)}{b d}+\frac {\cos ^3(c+d x)}{3 b d}+\frac {\left (4 a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^2 d}+\frac {\left (4 a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^2 d}+\frac {\left (4 a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^2 d}\\ &=-\frac {a x}{b^2}-\frac {2 a^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b^2 d}+\frac {2 a^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^2 d}+\frac {2 a^{4/3} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b^2 d}-\frac {\cos (c+d x)}{b d}+\frac {\cos ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.20, size = 164, normalized size = 0.56 \begin {gather*} -\frac {12 a c+12 a d x+9 b \cos (c+d x)-b \cos (3 (c+d x))+8 i a^2 \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]}{12 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.71, size = 143, normalized size = 0.49
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b^{2}}-\frac {2 \left (\frac {2 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 b}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{2}}}{d}\) | \(143\) |
default | \(\frac {\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b^{2}}-\frac {2 \left (\frac {2 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 b}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{2}}}{d}\) | \(143\) |
risch | \(-\frac {a x}{b^{2}}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{8 b d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b d}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (729 a^{2} b^{12} d^{6}-729 b^{14} d^{6}\right ) \textit {\_Z}^{6}+995328 a^{4} b^{8} d^{4} \textit {\_Z}^{4}+452984832 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+68719476736 a^{8}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {243 b^{9} d^{5}}{536870912 a^{4}}-\frac {243 b^{11} d^{5}}{536870912 a^{6}}\right ) \textit {\_R}^{5}+\left (-\frac {81 i d^{4} b^{7}}{16777216 a^{3}}+\frac {81 i d^{4} b^{9}}{16777216 a^{5}}\right ) \textit {\_R}^{4}+\left (\frac {135 b^{5} d^{3}}{262144 a^{2}}+\frac {27 b^{7} d^{3}}{262144 a^{4}}\right ) \textit {\_R}^{3}-\frac {27 i d^{2} b^{3} \textit {\_R}^{2}}{4096 a}+\frac {9 b d \textit {\_R}}{64}-\frac {2 i a}{b}\right )\right )}{64}+\frac {\cos \left (3 d x +3 c \right )}{12 b d}\) | \(247\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.53, size = 29350, normalized size = 100.17 \begin {gather*} \text {too large to display} \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 {\sin ^{6}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, 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 [B]
time = 14.58, size = 1800, normalized size = 6.14 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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