Optimal. Leaf size=174 \[ -\frac {b F_1\left (\frac {1}{2};-\frac {1}{6},1;\frac {3}{2};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin (c+d x)}{\left (a^2-b^2\right ) d \sqrt [6]{\cos ^2(c+d x)} \sqrt [3]{\sec (c+d x)}}+\frac {a F_1\left (\frac {1}{2};-\frac {2}{3},1;\frac {3}{2};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt [3]{\cos ^2(c+d x)} \sec ^{\frac {2}{3}}(c+d x) \sin (c+d x)}{\left (a^2-b^2\right ) d} \]
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Rubi [A]
time = 0.18, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3954, 2902,
3268, 440} \begin {gather*} \frac {a \sin (c+d x) \sqrt [3]{\cos ^2(c+d x)} \sec ^{\frac {2}{3}}(c+d x) F_1\left (\frac {1}{2};-\frac {2}{3},1;\frac {3}{2};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{d \left (a^2-b^2\right )}-\frac {b \sin (c+d x) F_1\left (\frac {1}{2};-\frac {1}{6},1;\frac {3}{2};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{d \left (a^2-b^2\right ) \sqrt [6]{\cos ^2(c+d x)} \sqrt [3]{\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 2902
Rule 3268
Rule 3954
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))} \, dx &=\left (\cos ^{\frac {2}{3}}(c+d x) \sec ^{\frac {2}{3}}(c+d x)\right ) \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\left (\left (a \cos ^{\frac {2}{3}}(c+d x) \sec ^{\frac {2}{3}}(c+d x)\right ) \int \frac {\cos ^{\frac {7}{3}}(c+d x)}{b^2-a^2 \cos ^2(c+d x)} \, dx\right )+\left (b \cos ^{\frac {2}{3}}(c+d x) \sec ^{\frac {2}{3}}(c+d x)\right ) \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{b^2-a^2 \cos ^2(c+d x)} \, dx\\ &=\frac {b \text {Subst}\left (\int \frac {\sqrt [6]{1-x^2}}{-a^2+b^2+a^2 x^2} \, dx,x,\sin (c+d x)\right )}{d \sqrt [6]{\cos ^2(c+d x)} \sqrt [3]{\sec (c+d x)}}-\frac {\left (a \sqrt [3]{\cos ^2(c+d x)} \sec ^{\frac {2}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{2/3}}{-a^2+b^2+a^2 x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {b F_1\left (\frac {1}{2};-\frac {1}{6},1;\frac {3}{2};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin (c+d x)}{\left (a^2-b^2\right ) d \sqrt [6]{\cos ^2(c+d x)} \sqrt [3]{\sec (c+d x)}}+\frac {a F_1\left (\frac {1}{2};-\frac {2}{3},1;\frac {3}{2};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt [3]{\cos ^2(c+d x)} \sec ^{\frac {2}{3}}(c+d x) \sin (c+d x)}{\left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(7542\) vs. \(2(174)=348\).
time = 126.71, size = 7542, normalized size = 43.34 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\sec \left (d x +c \right )^{\frac {1}{3}} \left (a +b \sec \left (d x +c \right )\right )}\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right ) \sqrt [3]{\sec {\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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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