Integrand size = 23, antiderivative size = 313 \[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=-\frac {9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac {\left (18 a^2+25 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {a \left (9 a^2+11 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{10 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \]
[Out]
Time = 0.60 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3950, 4167, 4092, 3919, 144, 143} \[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=-\frac {a \left (9 a^2+11 b^2\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{10 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac {\left (18 a^2+25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{20 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d} \]
[In]
[Out]
Rule 143
Rule 144
Rule 3919
Rule 3950
Rule 4092
Rule 4167
Rubi steps \begin{align*} \text {integral}& = \frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac {3 \int \frac {\sec (c+d x) \left (a+\frac {5}{3} b \sec (c+d x)-2 a \sec ^2(c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{8 b} \\ & = -\frac {9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac {9 \int \frac {\sec (c+d x) \left (\frac {a b}{3}+\frac {1}{9} \left (18 a^2+25 b^2\right ) \sec (c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{40 b^2} \\ & = -\frac {9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}-\frac {\left (a \left (9 a^2+11 b^2\right )\right ) \int \frac {\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{20 b^3}+\frac {\left (18 a^2+25 b^2\right ) \int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx}{40 b^3} \\ & = -\frac {9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac {\left (a \left (9 a^2+11 b^2\right ) \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{a+b x}} \, dx,x,\sec (c+d x)\right )}{20 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\frac {\left (\left (18 a^2+25 b^2\right ) \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{40 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = -\frac {9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}-\frac {\left (\left (18 a^2+25 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{40 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3}}+\frac {\left (a \left (9 a^2+11 b^2\right ) \sqrt [3]{-\frac {a+b \sec (c+d x)}{-a-b}} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}} \, dx,x,\sec (c+d x)\right )}{20 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ & = -\frac {9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac {\left (18 a^2+25 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {a \left (9 a^2+11 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{10 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(19015\) vs. \(2(313)=626\).
Time = 45.82 (sec) , antiderivative size = 19015, normalized size of antiderivative = 60.75 \[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\text {Result too large to show} \]
[In]
[Out]
\[\int \frac {\sec \left (d x +c \right )^{4}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
[In]
[Out]
\[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sqrt [3]{a + b \sec {\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]
[In]
[Out]