Optimal. Leaf size=521 \[ \frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (45 a^5 A b-99 a^3 A b^3+72 a A b^5-105 a^6 B+223 a^4 b^2 B-128 a^2 b^4 B-8 b^6 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{12 b^5 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^5 (a+b)^3 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \]
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Rubi [A]
time = 0.96, antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3039, 4115,
4185, 4189, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \begin {gather*} \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}+\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{12 b^3 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}+\frac {\left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\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 b^5 d (a-b)^2 (a+b)^3}-\frac {\left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 b^5 d \left (a^2-b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 2884
Rule 3039
Rule 3856
Rule 3872
Rule 3934
Rule 4115
Rule 4185
Rule 4189
Rule 4191
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx &=\int \frac {B+A \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^3} \, dx\\ &=\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (-3 a A b+7 a^2 B-4 b^2 B\right )-2 b (A b-a B) \sec (c+d x)+\frac {5}{2} a (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (-15 a^3 A b+33 a A b^3+35 a^4 B-61 a^2 b^2 B+8 b^4 B\right )+b \left (a^2 A b+2 A b^3+a^3 B-4 a b^2 B\right ) \sec (c+d x)+\frac {3}{4} a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}-\frac {\int \frac {-\frac {3}{8} \left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right )-\frac {1}{2} b \left (3 a^3 A b-12 a A b^3-7 a^4 B+14 a^2 b^2 B+2 b^4 B\right ) \sec (c+d x)+\frac {1}{8} a \left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}-\frac {\int \frac {-\frac {3}{8} b \left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right )-\left (\frac {1}{2} b^2 \left (3 a^3 A b-12 a A b^3-7 a^4 B+14 a^2 b^2 B+2 b^4 B\right )-\frac {3}{8} a \left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 b^5 \left (a^2-b^2\right )^2}+\frac {\left (a^2 \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{8 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}+\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}-\frac {\left (45 a^5 A b-99 a^3 A b^3+72 a A b^5-105 a^6 B+223 a^4 b^2 B-128 a^2 b^4 B-8 b^6 B\right ) \int \sqrt {\sec (c+d x)} \, dx}{24 b^5 \left (a^2-b^2\right )^2}+\frac {\left (a^2 \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\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 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {a^2 \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\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-b)^2 b^5 (a+b)^3 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}+\frac {\left (\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}-\frac {\left (\left (45 a^5 A b-99 a^3 A b^3+72 a A b^5-105 a^6 B+223 a^4 b^2 B-128 a^2 b^4 B-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (45 a^5 A b-99 a^3 A b^3+72 a A b^5-105 a^6 B+223 a^4 b^2 B-128 a^2 b^4 B-8 b^6 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{12 b^5 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\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-b)^2 b^5 (a+b)^3 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 37.39, size = 865, normalized size = 1.66 \begin {gather*} -\frac {\frac {2 \left (-15 a^4 A b+21 a^2 A b^3-24 A b^5+35 a^5 B-73 a^3 b^2 B+56 a b^4 B\right ) \cos ^2(c+d x) \left (F\left (\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-\Pi \left (-\frac {a}{b};\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-24 a^3 A b^2+96 a A b^4+56 a^4 b B-112 a^2 b^3 B-16 b^5 B\right ) \cos ^2(c+d x) \Pi \left (-\frac {a}{b};\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-45 a^4 A b+87 a^2 A b^3-24 A b^5+105 a^5 B-195 a^3 b^2 B+72 a b^4 B\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b F\left (\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \Pi \left (-\frac {a}{b};\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \Pi \left (-\frac {a}{b};\left .\text {ArcSin}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 (a-b)^2 b^3 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {a^2 \left (-7 a^2 A b+13 A b^3+11 a^3 B-17 a b^2 B\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2}-\frac {-a^4 A b \sin (c+d x)+a^5 B \sin (c+d x)}{2 b^4 \left (-a^2+b^2\right ) (a+b \cos (c+d x))^2}+\frac {9 a^5 A b \sin (c+d x)-15 a^3 A b^3 \sin (c+d x)-13 a^6 B \sin (c+d x)+19 a^4 b^2 B \sin (c+d x)}{4 b^4 \left (-a^2+b^2\right )^2 (a+b \cos (c+d x))}+\frac {B \sin (2 (c+d x))}{3 b^3}\right )}{d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2194\) vs.
\(2(569)=1138\).
time = 1.91, size = 2195, normalized size = 4.21
method | result | size |
default | \(\text {Expression too large to display}\) | \(2195\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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