Optimal. Leaf size=307 \[ -\frac {a \left (3 a^2-4 b^2\right ) \tan (c+d x) \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3} F_1\left (\frac {1}{2};\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{\sqrt {2} b^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac {\left (3 a^2-2 b^2\right ) \tan (c+d x) \sqrt [3]{a+b \sec (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{\sqrt {2} b^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)+1} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}-\frac {3 a^2 \tan (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.38, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3839, 4007, 3834, 139, 138} \[ -\frac {a \left (3 a^2-4 b^2\right ) \tan (c+d x) \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3} F_1\left (\frac {1}{2};\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{\sqrt {2} b^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac {\left (3 a^2-2 b^2\right ) \tan (c+d x) \sqrt [3]{a+b \sec (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{\sqrt {2} b^2 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)+1} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}-\frac {3 a^2 \tan (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 3834
Rule 3839
Rule 4007
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx &=-\frac {3 a^2 \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{2/3}}-\frac {3 \int \frac {\sec (c+d x) \left (-\frac {2 a b}{3}-\frac {1}{3} \left (3 a^2-2 b^2\right ) \sec (c+d x)\right )}{(a+b \sec (c+d x))^{2/3}} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {3 a^2 \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{2/3}}-\frac {\left (a \left (3 a^2-4 b^2\right )\right ) \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx}{2 b^2 \left (a^2-b^2\right )}+\frac {\left (3 a^2-2 b^2\right ) \int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=-\frac {3 a^2 \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{2/3}}+\frac {\left (a \left (3 a^2-4 b^2\right ) \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} (a+b x)^{2/3}} \, dx,x,\sec (c+d x)\right )}{2 b^2 \left (a^2-b^2\right ) d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\frac {\left (\left (3 a^2-2 b^2\right ) \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{2 b^2 \left (a^2-b^2\right ) d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\\ &=-\frac {3 a^2 \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{2/3}}-\frac {\left (\left (3 a^2-2 b^2\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{2 b^2 \left (a^2-b^2\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 {\left (a \left (3 a^2-4 b^2\right ) \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}} \, dx,x,\sec (c+d x)\right )}{2 b^2 \left (a^2-b^2\right ) d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}}\\ &=-\frac {3 a^2 \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{2/3}}+\frac {\left (3 a^2-2 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]{a+b \sec (c+d x)} \tan (c+d x)}{\sqrt {2} b^2 \left (a^2-b^2\right ) d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}}-\frac {a \left (3 a^2-4 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 ) \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3} \tan (c+d x)}{\sqrt {2} b^2 \left (a^2-b^2\right ) d \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [B] time = 26.70, size = 19126, normalized size = 62.30 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \sec \left (d x + c\right )^{3}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{3}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}\left (d x +c \right )}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \]
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 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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