Integrand size = 23, antiderivative size = 766 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=-\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac {375 \left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{28 a^2 d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}-\frac {375 \sqrt [4]{3} E\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}-\frac {125\ 3^{3/4} \left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
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Time = 1.43 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3909, 4093, 4085, 3913, 3912, 65, 314, 231, 1895} \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=-\frac {125\ 3^{3/4} \left (1-\sqrt {3}\right ) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}-\frac {375 \sqrt [4]{3} \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} E\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac {375 \left (1+\sqrt {3}\right ) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{28 a^2 d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a \sec (c+d x)+a)^{2/3}}-\frac {33 \tan (c+d x)}{28 d (a \sec (c+d x)+a)^{5/3}} \]
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Rule 65
Rule 231
Rule 314
Rule 1895
Rule 3909
Rule 3912
Rule 3913
Rule 4085
Rule 4093
Rubi steps \begin{align*} \text {integral}& = \frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \int \frac {\sec ^2(c+d x) \left (2 a-\frac {5}{3} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^{5/3}} \, dx}{4 a} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}-\frac {9 \int \frac {\sec (c+d x) \left (-\frac {55 a^2}{9}+\frac {35}{9} a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^{2/3}} \, dx}{28 a^3} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}-\frac {125 \int \sec (c+d x) \sqrt [3]{a+a \sec (c+d x)} \, dx}{28 a^2} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}-\frac {\left (125 \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \, dx}{28 a^2 \sqrt [3]{1+\sec (c+d x)}} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac {\left (125 \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 )}{28 a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac {\left (375 \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 )}{14 a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}-\frac {\left (375 \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 )}{28 a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}-\frac {\left (375 \left (1-\sqrt {3}\right ) \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 )}{14 \sqrt [3]{2} a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}} \\ & = -\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac {375 \left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{28 a^2 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 {375 \sqrt [4]{3} E\left (\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)}{14\ 2^{2/3} a^2 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 {125\ 3^{3/4} \left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\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)}{28\ 2^{2/3} a^2 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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.59 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.14 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\frac {\left (-250 2^{5/6} \cos ^2\left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right ) \sec (c+d x) \sqrt [6]{1+\sec (c+d x)}+3 \left (79+90 \sec (c+d x)+7 \sec ^2(c+d x)\right )\right ) \tan (c+d x)}{28 d (a (1+\sec (c+d x)))^{5/3}} \]
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\[\int \frac {\sec \left (d x +c \right )^{4}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{3}}}d x\]
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\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{3}}}\, dx \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \]
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