Integrand size = 21, antiderivative size = 108 \[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\frac {\sqrt {2} (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}} \]
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Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3919, 144, 143} \[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\frac {\sqrt {2} (a+b) \tan (c+d x) \sqrt [3]{a+b \sec (c+d x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{d \sqrt {\sec (c+d x)+1} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}} \]
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Rule 143
Rule 144
Rule 3919
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan (c+d x) \text {Subst}\left (\int \frac {(a+b x)^{4/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = \frac {\left ((-a-b) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{4/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{-\frac {a+b \sec (c+d x)}{-a-b}}} \\ & = \frac {\sqrt {2} (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(7313\) vs. \(2(108)=216\).
Time = 48.68 (sec) , antiderivative size = 7313, normalized size of antiderivative = 67.71 \[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\text {Result too large to show} \]
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\[\int \sec \left (d x +c \right ) \left (a +b \sec \left (d x +c \right )\right )^{\frac {4}{3}}d x\]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sec \left (d x + c\right ) \,d x } \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {4}{3}} \sec {\left (c + d x \right )}\, dx \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sec \left (d x + c\right ) \,d x } \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}}{\cos \left (c+d\,x\right )} \,d x \]
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