Optimal. Leaf size=815 \[ \frac {3 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d}+\frac {3 \sqrt {2} a A F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}+\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}-\frac {15 \left (1+\sqrt {3}\right ) a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {15 \sqrt [3]{2} \sqrt [4]{3} a C E\left (\text {ArcCos}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}+\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) a C F\left (\text {ArcCos}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7\ 2^{2/3} d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
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Rubi [A]
time = 0.73, antiderivative size = 815, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4140, 4009,
3864, 3863, 141, 3913, 3912, 52, 65, 314, 231, 1895} \begin {gather*} \frac {3 C \tan (c+d x) (\sec (c+d x) a+a)^{4/3}}{7 d}+\frac {3 \sqrt {2} a A F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right ) (\sec (c+d x)+1) \tan (c+d x) \sqrt [3]{\sec (c+d x) a+a}}{11 d \sqrt {1-\sec (c+d x)}}+\frac {15 \sqrt [3]{2} \sqrt [4]{3} a C E\left (\text {ArcCos}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \tan (c+d x) \sqrt [3]{\sec (c+d x) a+a}}{7 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) a C F\left (\text {ArcCos}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \tan (c+d x) \sqrt [3]{\sec (c+d x) a+a}}{7\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac {3 a C \tan (c+d x) \sqrt [3]{\sec (c+d x) a+a}}{7 d}-\frac {15 \left (1+\sqrt {3}\right ) a C \tan (c+d x) \sqrt [3]{\sec (c+d x) a+a}}{7 d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 52
Rule 65
Rule 141
Rule 231
Rule 314
Rule 1895
Rule 3863
Rule 3864
Rule 3912
Rule 3913
Rule 4009
Rule 4140
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}+\frac {3 \int (a+a \sec (c+d x))^{4/3} \left (\frac {7 a A}{3}+\frac {4}{3} a C \sec (c+d x)\right ) \, dx}{7 a}\\ &=\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}+A \int (a+a \sec (c+d x))^{4/3} \, dx+\frac {1}{7} (4 C) \int \sec (c+d x) (a+a \sec (c+d x))^{4/3} \, dx\\ &=\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}+\frac {\left (a A \sqrt [3]{a+a \sec (c+d x)}\right ) \int (1+\sec (c+d x))^{4/3} \, dx}{\sqrt [3]{1+\sec (c+d x)}}+\frac {\left (4 a C \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sec (c+d x) (1+\sec (c+d x))^{4/3} \, dx}{7 \sqrt [3]{1+\sec (c+d x)}}\\ &=\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}-\frac {\left (a A \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(1+x)^{5/6}}{\sqrt {1-x} x} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}-\frac {\left (4 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(1+x)^{5/6}}{\sqrt {1-x}} \, dx,x,\sec (c+d x)\right )}{7 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d}+\frac {3 \sqrt {2} a A F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}+\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}-\frac {\left (5 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{7 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d}+\frac {3 \sqrt {2} a A F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}+\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}-\frac {\left (30 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d}+\frac {3 \sqrt {2} a A F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}+\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}+\frac {\left (15 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}+\frac {\left (15\ 2^{2/3} \left (1-\sqrt {3}\right ) a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d}+\frac {3 \sqrt {2} a A F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}+\frac {3 C (a+a \sec (c+d x))^{4/3} \tan (c+d x)}{7 d}-\frac {15 \left (1+\sqrt {3}\right ) a C \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {15 \sqrt [3]{2} \sqrt [4]{3} a C E\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}+\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) a C F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7\ 2^{2/3} d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}\\ \end {align*}
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Mathematica [F]
time = 27.64, size = 0, normalized size = 0.00 \begin {gather*} \int (a+a \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.94, size = 0, normalized size = 0.00 \[\int \left (a +a \sec \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx\]
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 \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}} \left (A + C \sec ^{2}{\left (c + d x \right )}\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 = -0.00 \begin {gather*} \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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