Optimal. Leaf size=313 \[ -\frac {a \left (9 a^2+11 b^2\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{10 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac {\left (18 a^2+25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{20 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d} \]
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Rubi [A] time = 0.49, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3865, 4082, 4007, 3834, 139, 138} \[ -\frac {a \left (9 a^2+11 b^2\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{10 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac {\left (18 a^2+25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{20 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 3834
Rule 3865
Rule 4007
Rule 4082
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) F_1\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 ) F_1\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*}
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Mathematica [B] time = 26.60, size = 19015, normalized size = 60.75 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}\left (d x +c \right )}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\sqrt [3]{a + b \sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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