Optimal. Leaf size=250 \[ \frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {\sqrt {3} \log \left (1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\sqrt {3} \log \left (1+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2647, 2654,
301, 648, 632, 210, 642, 209} \begin {gather*} \frac {\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt {3}\right )}{2 b}-\frac {\text {ArcTan}\left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac {\sqrt {3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}{4 b}+\frac {\sqrt {3} \log \left (\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}{4 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 210
Rule 301
Rule 632
Rule 642
Rule 648
Rule 2647
Rule 2654
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {4}{3}}(a+b x)}{\sin ^{\frac {4}{3}}(a+b x)} \, dx &=-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\int \frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)} \, dx\\ &=-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac {3 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}\\ &=-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}-\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}+\frac {\sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {\sqrt {3} \log \left (1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\sqrt {3} \log \left (1+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}\\ &=\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac {\sqrt {3} \log \left (1-\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\sqrt {3} \log \left (1+\frac {\sqrt {3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac {\sin ^{\frac {2}{3}}(a+b x)}{\cos ^{\frac {2}{3}}(a+b x)}\right )}{4 b}-\frac {3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 55, normalized size = 0.22 \begin {gather*} -\frac {3 \cos ^2(a+b x)^{5/6} \, _2F_1\left (-\frac {1}{6},-\frac {1}{6};\frac {5}{6};\sin ^2(a+b x)\right )}{b \cos ^{\frac {5}{3}}(a+b x) \sqrt [3]{\sin (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{\frac {4}{3}}\left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{\frac {4}{3}}{\left (a + b x \right )}}{\sin ^{\frac {4}{3}}{\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.70, size = 44, normalized size = 0.18 \begin {gather*} -\frac {3\,{\cos \left (a+b\,x\right )}^{7/3}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{1/6}\,{{}}_2{\mathrm {F}}_1\left (\frac {7}{6},\frac {7}{6};\ \frac {13}{6};\ {\cos \left (a+b\,x\right )}^2\right )}{7\,b\,{\sin \left (a+b\,x\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________