Optimal. Leaf size=299 \[ \frac {\left (2 a^2+5 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{4 \sqrt {2} b d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {a \left (a^2-b^2\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{2 \sqrt {2} b d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac {3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 d}+\frac {3 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{8 d} \]
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Rubi [A] time = 0.46, antiderivative size = 299, 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 = {3835, 4002, 4007, 3834, 139, 138} \[ \frac {\left (2 a^2+5 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{4 \sqrt {2} b d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {a \left (a^2-b^2\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{2 \sqrt {2} b d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac {3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 d}+\frac {3 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{8 d} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 3834
Rule 3835
Rule 4002
Rule 4007
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/3} \, dx &=\frac {3 (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{8 d}+\frac {5}{8} \int \sec (c+d x) (b+a \sec (c+d x)) (a+b \sec (c+d x))^{2/3} \, dx\\ &=\frac {3 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d}+\frac {3 (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{8 d}+\frac {3}{8} \int \frac {\sec (c+d x) \left (\frac {7 a b}{3}+\frac {1}{3} \left (2 a^2+5 b^2\right ) \sec (c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}} \, dx\\ &=\frac {3 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d}+\frac {3 (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{8 d}-\frac {\left (a \left (a^2-b^2\right )\right ) \int \frac {\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{4 b}+\frac {\left (2 a^2+5 b^2\right ) \int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx}{8 b}\\ &=\frac {3 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d}+\frac {3 (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{8 d}+\frac {\left (a \left (a^2-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 )}{4 b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\frac {\left (\left (2 a^2+5 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 )}{8 b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\\ &=\frac {3 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d}+\frac {3 (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{8 d}-\frac {\left (\left (2 a^2+5 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 )}{8 b 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 (a^2-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 )}{4 b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}}\\ &=\frac {3 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d}+\frac {3 (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{8 d}+\frac {\left (2 a^2+5 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)}{4 \sqrt {2} b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {a \left (a^2-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)}{2 \sqrt {2} b d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 27.28, size = 19016, normalized size = 63.60 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right )^{3} + a \sec \left (d x + c\right )^{2}\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}, x\right ) \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}} \sec \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.75, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{2}\left (d x +c \right )\right ) \left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{3}}\, dx \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}} \sec \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/3}}{{\cos \left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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