Optimal. Leaf size=455 \[ \frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{12 a^3 d \left (a^2-b^2\right )^2}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2} \]
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Rubi [A] time = 1.45, antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3238, 3845, 4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}+\frac {\left (-61 a^2 b^2+8 a^4+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{12 a^3 d \left (a^2-b^2\right )^2}-\frac {b \left (-65 a^2 b^2+24 a^4+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {\left (-61 a^2 b^2+8 a^4+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac {b \left (-65 a^2 b^2+24 a^4+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {b^2 \left (-86 a^2 b^2+63 a^4+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3238
Rule 3771
Rule 3787
Rule 3845
Rule 3849
Rule 4098
Rule 4102
Rule 4106
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=\int \frac {\sec ^{\frac {11}{2}}(c+d x)}{(b+a \sec (c+d x))^3} \, dx\\ &=\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {5 b^2}{2}-2 a b \sec (c+d x)+\frac {1}{2} \left (4 a^2-7 b^2\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\frac {3}{4} b^2 \left (13 a^2-7 b^2\right )-a b \left (4 a^2-b^2\right ) \sec (c+d x)+\frac {1}{4} \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{8} b \left (8 a^4-61 a^2 b^2+35 b^4\right )+\frac {1}{2} a \left (2 a^4+14 a^2 b^2-7 b^4\right ) \sec (c+d x)-\frac {3}{8} b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {2 \int \frac {\frac {3}{16} b^2 \left (24 a^4-65 a^2 b^2+35 b^4\right )+\frac {1}{4} a b \left (20 a^4-64 a^2 b^2+35 b^4\right ) \sec (c+d x)+\frac {1}{16} \left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {2 \int \frac {\frac {3}{16} b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right )-\left (\frac {3}{16} a b^2 \left (24 a^4-65 a^2 b^2+35 b^4\right )-\frac {1}{4} a b^2 \left (20 a^4-64 a^2 b^2+35 b^4\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 a^4 b^2 \left (a^2-b^2\right )^2}+\frac {\left (b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\left (b \left (24 a^4-65 a^2 b^2+35 b^4\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{8 a^4 \left (a^2-b^2\right )^2}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \int \sqrt {\sec (c+d x)} \, dx}{24 a^3 \left (a^2-b^2\right )^2}+\frac {\left (b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\left (b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}+\frac {\left (\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.83, size = 747, normalized size = 1.64 \[ \frac {\sqrt {\sec (c+d x)} \left (\frac {2 \tan (c+d x)}{3 a^3}-\frac {b^3 \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2}-\frac {3 \left (5 a^2 b^3 \sin (c+d x)-3 b^5 \sin (c+d x)\right )}{4 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}\right )}{d}+\frac {\frac {2 \left (160 a^5 b-512 a^3 b^3+280 a b^5\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac {\left (72 a^4 b^2-195 a^2 b^4+105 b^6\right ) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left (-4 a^2 \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+2 b^2 \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac {2 \left (16 a^6+328 a^4 b^2-641 a^2 b^4+315 b^6\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left (F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-\Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}}{48 a^4 d (a-b)^2 (a+b)^2} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.95, size = 2128, normalized size = 4.68 \[ \text {Expression too large to display} \]
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 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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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