Optimal. Leaf size=841 \[ -\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {3 \sqrt {2} A F_1\left (-\frac {1}{6};\frac {1}{2},1;\frac {5}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac {3 \left (1+\sqrt {3}\right ) (2 A-5 C) \sqrt [3]{1+\sec (c+d x)} \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {3 \sqrt [3]{2} \sqrt [4]{3} (2 A-5 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]{1+\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 a d (1-\sec (c+d x)) (a+a \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 {3^{3/4} \left (1-\sqrt {3}\right ) (2 A-5 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]{1+\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} a d (1-\sec (c+d x)) (a+a \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 = 0.66, antiderivative size = 841, 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 = {4138, 4009,
3864, 3863, 141, 3913, 3912, 53, 65, 314, 231, 1895} \begin {gather*} \frac {3 \sqrt [3]{2} \sqrt [4]{3} (2 A-5 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 ) \sqrt [3]{\sec (c+d x)+1} \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)}{7 a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{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^{3/4} \left (1-\sqrt {3}\right ) (2 A-5 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 ) \sqrt [3]{\sec (c+d x)+1} \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)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{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 \sqrt {2} A F_1\left (-\frac {1}{6};\frac {1}{2},1;\frac {5}{6};\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right ) \tan (c+d x)}{a d \sqrt {1-\sec (c+d x)} (\sec (c+d x) a+a)^{2/3}}-\frac {3 (2 A-5 C) \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3}}-\frac {3 \left (1+\sqrt {3}\right ) (2 A-5 C) \sqrt [3]{\sec (c+d x)+1} \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}-\frac {3 (A+C) \tan (c+d x)}{7 d (\sec (c+d x) a+a)^{5/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 53
Rule 65
Rule 141
Rule 231
Rule 314
Rule 1895
Rule 3863
Rule 3864
Rule 3912
Rule 3913
Rule 4009
Rule 4138
Rubi steps
\begin {align*} \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {3 \int \frac {-\frac {7 a A}{3}+\frac {1}{3} a (2 A-5 C) \sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx}{7 a^2}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {A \int \frac {1}{(a+a \sec (c+d x))^{2/3}} \, dx}{a}-\frac {(2 A-5 C) \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx}{7 a}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {\left (A (1+\sec (c+d x))^{2/3}\right ) \int \frac {1}{(1+\sec (c+d x))^{2/3}} \, dx}{a (a+a \sec (c+d x))^{2/3}}-\frac {\left ((2 A-5 C) (1+\sec (c+d x))^{2/3}\right ) \int \frac {\sec (c+d x)}{(1+\sec (c+d x))^{2/3}} \, dx}{7 a (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {\left (A \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x (1+x)^{7/6}} \, dx,x,\sec (c+d x)\right )}{a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}+\frac {\left ((2 A-5 C) \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)^{7/6}} \, dx,x,\sec (c+d x)\right )}{7 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {3 \sqrt {2} A F_1\left (-\frac {1}{6};\frac {1}{2},1;\frac {5}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac {\left ((2 A-5 C) \sqrt [6]{1+\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 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {3 \sqrt {2} A F_1\left (-\frac {1}{6};\frac {1}{2},1;\frac {5}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac {\left (6 (2 A-5 C) \sqrt [6]{1+\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 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {3 \sqrt {2} A F_1\left (-\frac {1}{6};\frac {1}{2},1;\frac {5}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}+\frac {\left (3 (2 A-5 C) \sqrt [6]{1+\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 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}+\frac {\left (3\ 2^{2/3} \left (1-\sqrt {3}\right ) (2 A-5 C) \sqrt [6]{1+\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 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {3 \sqrt {2} A F_1\left (-\frac {1}{6};\frac {1}{2},1;\frac {5}{6};\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac {3 \left (1+\sqrt {3}\right ) (2 A-5 C) \sqrt [3]{1+\sec (c+d x)} \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {3 \sqrt [3]{2} \sqrt [4]{3} (2 A-5 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]{1+\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 a d (1-\sec (c+d x)) (a+a \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 {3^{3/4} \left (1-\sqrt {3}\right ) (2 A-5 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]{1+\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} a d (1-\sec (c+d x)) (a+a \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 = 10.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.93, size = 0, normalized size = 0.00 \[\int \frac {A +C \left (\sec ^{2}\left (d x +c \right )\right )}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{3}}}\, dx\]
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 {A + C \sec ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{3}}}\, 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 \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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