Optimal. Leaf size=255 \[ \frac {\left (a^2-7 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b d \left (a^2-b^2\right )^2}+\frac {3 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}-\frac {a^2 \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 0.75, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4264, 3845, 4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac {\left (a^2-7 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b d \left (a^2-b^2\right )^2}+\frac {3 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}+\frac {3 \left (-2 a^2 b^2+a^4+5 b^4\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 d (a-b)^2 (a+b)^3}-\frac {a^2 \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)} (a+b \sec (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3771
Rule 3787
Rule 3845
Rule 3849
Rule 4098
Rule 4106
Rule 4264
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx\\ &=-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {a^2}{2}-2 a b \sec (c+d x)-\frac {1}{2} \left (3 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^2 \left (a^2-3 b^2\right )+a b \left (a^2-4 b^2\right ) \sec (c+d x)+\frac {1}{4} \left (3 a^4-5 a^2 b^2+8 b^4\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^3 \left (a^2-3 b^2\right )-\left (-a^2 b \left (a^2-4 b^2\right )+\frac {3}{4} a^2 b \left (a^2-3 b^2\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{2 a^2 b^2 \left (a^2-b^2\right )^2}+\frac {\left (3 \left (a^4-2 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (3 \left (a^4-2 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{8 b^2 \left (a^2-b^2\right )^2}+\frac {\left (\left (a^2-7 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{8 b \left (a^2-b^2\right )^2}+\frac {\left (3 a \left (a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{8 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 (a-b)^2 b^2 (a+b)^3 d}-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (a^2-7 b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{8 b \left (a^2-b^2\right )^2}+\frac {\left (3 a \left (a^2-3 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {3 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2-7 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b \left (a^2-b^2\right )^2 d}+\frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 (a-b)^2 b^2 (a+b)^3 d}-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A] time = 3.16, size = 297, normalized size = 1.16 \[ \frac {\frac {\frac {16 b \left (a^2-4 b^2\right ) \left ((a+b) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-b \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{a+b}+\frac {6 \left (a^2-3 b^2\right ) \sin (c+d x) \left (\left (a^2-2 b^2\right ) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{b \sqrt {\sin ^2(c+d x)}}+\frac {2 \left (9 a^4-19 a^2 b^2+16 b^4\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}}{(a-b)^2 (a+b)^2}-\frac {4 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (3 a \left (a^2-3 b^2\right ) \cos (c+d x)+5 a^2 b-11 b^3\right )}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}}{16 b^2 d} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.64, size = 1203, normalized size = 4.72 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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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