Optimal. Leaf size=259 \[ \frac {x}{b}-\frac {2 \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b d}-\frac {2 \sqrt [3]{a} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b d}+\frac {2 \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b d} \]
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Rubi [A]
time = 0.33, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 3292,
2739, 632, 210} \begin {gather*} -\frac {2 \sqrt [3]{a} \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 d \sqrt {a^{2/3}-b^{2/3}}}-\frac {2 \sqrt [3]{a} \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 d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {2 \sqrt [3]{a} \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 d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 3292
Rule 3299
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (\frac {1}{b}-\frac {a}{b \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {x}{b}-\frac {a \int \frac {1}{a+b \sin ^3(c+d x)} \, dx}{b}\\ &=\frac {x}{b}-\frac {a \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}\\ &=\frac {x}{b}+\frac {\sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b}+\frac {\sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b}+\frac {\sqrt [3]{a} \int \frac {1}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b}\\ &=\frac {x}{b}+\frac {\left (2 \sqrt [3]{a}\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 d}+\frac {\left (2 \sqrt [3]{a}\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 d}+\frac {\left (2 \sqrt [3]{a}\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 d}\\ &=\frac {x}{b}-\frac {\left (4 \sqrt [3]{a}\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 d}-\frac {\left (4 \sqrt [3]{a}\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 d}-\frac {\left (4 \sqrt [3]{a}\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 d}\\ &=\frac {x}{b}+\frac {2 \sqrt [3]{a} \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 d}-\frac {2 \sqrt [3]{a} \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 d}-\frac {2 \sqrt [3]{a} \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 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 = 0.14, size = 140, normalized size = 0.54 \begin {gather*} \frac {3 c+3 d x+2 i a \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 ]}{3 b 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 = 0.58, size = 104, normalized size = 0.40
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}}{d}\) | \(104\) |
default | \(\frac {-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}}{d}\) | \(104\) |
risch | \(\frac {x}{b}+\frac {i \left (\munderset {\textit {\_R} =\RootOf \left (\left (729 a^{2} b^{6} d^{6}-729 b^{8} d^{6}\right ) \textit {\_Z}^{6}-15552 a^{2} b^{4} d^{4} \textit {\_Z}^{4}+110592 a^{2} b^{2} d^{2} \textit {\_Z}^{2}-262144 a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {243 i a \,b^{4} d^{5}}{16384}-\frac {243 i b^{6} d^{5}}{16384 a}\right ) \textit {\_R}^{5}+\left (-\frac {81 i a \,b^{3} d^{4}}{4096}+\frac {81 i b^{5} d^{4}}{4096 a}\right ) \textit {\_R}^{4}+\left (-\frac {135 i a \,b^{2} d^{3}}{512}-\frac {27 i b^{4} d^{3}}{512 a}\right ) \textit {\_R}^{3}+\frac {27 i a b \,d^{2} \textit {\_R}^{2}}{64}+\frac {9 i a d \textit {\_R}}{8}-\frac {2 i a}{b}\right )\right )}{8}\) | \(190\) |
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 = 1.44, size = 29221, normalized size = 112.82 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 14.90, size = 1672, normalized size = 6.46 \begin {gather*} \frac {\sum _{k=1}^6\ln \left (-1073741824\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^2\,a^7\,b\,268435456+\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )\,a^7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,134217728+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^2\,a^5\,b^3\,4831838208+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^3\,a^6\,b^3\,33722204160+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^4\,a^5\,b^5\,15703474176-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^4\,a^7\,b^3\,4831838208-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^5\,a^4\,b^7\,130459631616+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^5\,a^6\,b^5\,154014842880+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^6\,a^5\,b^7\,35332816896-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^6\,a^7\,b^5\,21743271936-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^7\,a^4\,b^9\,130459631616+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^7\,a^6\,b^7\,122305904640+\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )\,a^6\,b\,2013265920-\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )\,a^5\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3221225472-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^2\,a^6\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,18589155328-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^3\,a^5\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,17716740096+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^3\,a^7\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,2818572288+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^4\,a^4\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,86973087744-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^4\,a^6\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,88181047296-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^5\,a^5\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,30802968576+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^5\,a^7\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,18119393280+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^6\,a^4\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,86973087744-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^6\,a^6\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,70665633792-{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^7\,a^5\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,40768634880+{\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}^7\,a^7\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,32614907904\right )\,\mathrm {root}\left (729\,a^2\,b^6\,z^6-729\,b^8\,z^6+243\,a^2\,b^4\,z^4+27\,a^2\,b^2\,z^2+a^2,z,k\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{b\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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