Optimal. Leaf size=415 \[ -\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \]
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Rubi [A]
time = 0.23, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3566, 726, 52,
59, 631, 210, 31} \begin {gather*} \frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {1}{4} x \sqrt [3]{a-\sqrt {-b^2}}-\frac {1}{4} x \sqrt [3]{a+\sqrt {-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 631
Rule 726
Rule 3566
Rubi steps
\begin {align*} \int \sqrt [3]{a+b \tan (c+d x)} \, dx &=\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{a+x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \text {Subst}\left (\int \left (\frac {\sqrt {-b^2} \sqrt [3]{a+x}}{2 b^2 \left (\sqrt {-b^2}-x\right )}+\frac {\sqrt {-b^2} \sqrt [3]{a+x}}{2 b^2 \left (\sqrt {-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{a+x}}{\sqrt {-b^2}-x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}-\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{a+x}}{\sqrt {-b^2}+x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}\\ &=-\frac {\left (b \left (a+\sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b^2}-x\right ) (a+x)^{2/3}} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}+\frac {\left (b^2+a \sqrt {-b^2}\right ) \text {Subst}\left (\int \frac {1}{(a+x)^{2/3} \left (\sqrt {-b^2}+x\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=-\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 b d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {\left (3 b \sqrt [3]{a+\sqrt {-b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+\sqrt {-b^2}}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {\left (3 b \left (a+\sqrt {-b^2}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (a+\sqrt {-b^2}\right )^{2/3}+\sqrt [3]{a+\sqrt {-b^2}} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {\left (3 \left (b^2+a \sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a-\sqrt {-b^2}}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b \left (a-\sqrt {-b^2}\right )^{2/3} d}-\frac {\left (3 \left (b^2+a \sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (a-\sqrt {-b^2}\right )^{2/3}+\sqrt [3]{a-\sqrt {-b^2}} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b \sqrt [3]{a-\sqrt {-b^2}} d}\\ &=-\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 b d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {3 \sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {\left (3 b \sqrt [3]{a+\sqrt {-b^2}}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}\right )}{2 \sqrt {-b^2} d}+\frac {\left (3 \left (b^2+a \sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}\right )}{2 b \left (a-\sqrt {-b^2}\right )^{2/3} d}\\ &=-\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x-\frac {\sqrt {3} \sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 b d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}+\frac {\sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 b d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {3 \sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 294, normalized size = 0.71 \begin {gather*} \frac {-i \sqrt [3]{a-i b} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a-i b)^{2/3}+\sqrt [3]{a-i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )+i \sqrt [3]{a+i b} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a+i b)^{2/3}+\sqrt [3]{a+i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )}{4 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 = 0.40, size = 60, normalized size = 0.14
method | result | size |
derivativedivides | \(\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
default | \(\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 3320 vs.
\(2 (325) = 650\).
time = 1.55, size = 3320, normalized size = 8.00 \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 \sqrt [3]{a + b \tan {\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 = 7.01, size = 863, normalized size = 2.08 \begin {gather*} \ln \left ({\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}+d\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}+\ln \left (-a\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}+b\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}+d^4\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{4/3}+2\,b\,d\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\right )\,{\left (\frac {-b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}-\ln \left (\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}-\frac {\left (\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}-3888\,a\,b^4\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (a^2+b^2\right )\right )}{4}+\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}+\ln \left (\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}-\frac {\left (\frac {\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}+3888\,a\,b^4\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (a^2+b^2\right )\right )}{4}-\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}-\ln \left (-\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}+\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}-3888\,a\,b^4\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a^2+b^2\right )\right )}{4}+\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {-b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3}+\ln \left (\frac {486\,\left (b^8-a^4\,b^4\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d^4}-\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3888\,b^5\,\left (a^2+b^2\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}+3888\,a\,b^4\,{\left (\frac {-b+a\,1{}\mathrm {i}}{d^3}\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a^2+b^2\right )\right )}{4}-\frac {1944\,a\,b^5\,\left (a^2+b^2\right )}{d^3}\right )}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {-b+a\,1{}\mathrm {i}}{8\,d^3}\right )}^{1/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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