Optimal. Leaf size=128 \[ \frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b d}+\frac {2 a^2 \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}+\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}} \]
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Rubi [A]
time = 0.37, antiderivative size = 128, 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 = {4349, 3936,
4187, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \begin {gather*} \frac {2 a^2 \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}+\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b d}+\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 2884
Rule 3856
Rule 3872
Rule 3934
Rule 3936
Rule 4187
Rule 4191
Rule 4349
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, 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)} \, dx\\ &=\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {a}{2}+\frac {1}{2} b \sec (c+d x)-\frac {3}{2} a \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b}\\ &=\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3 a^2}{4}+a b \sec (c+d x)+\frac {1}{4} \left (3 a^2+b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 b^2}\\ &=\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3 a^3}{4}+\frac {1}{4} a^2 b \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 a^2 b^2}+\frac {\left (a^2 \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}{b^2}\\ &=\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {a^2 \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{b^2}+\frac {\left (a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{b^2}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{3 b}\\ &=\frac {2 a^2 \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}+\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {a \int \sqrt {\cos (c+d x)} \, dx}{b^2}+\frac {\int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b}\\ &=\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b d}+\frac {2 a^2 \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}+\frac {2 \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 a \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 12.62, size = 210, normalized size = 1.64 \begin {gather*} \frac {\frac {2 \left (9 a^2+2 b^2\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+8 b \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 b \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )+\frac {4 (b-3 a \cos (c+d x)) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {6 \left (-2 a b E\left (\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) F\left (\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+\left (a^2-2 b^2\right ) \Pi \left (-\frac {a}{b};\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right ) \sin (c+d x)}{b \sqrt {\sin ^2(c+d x)}}}{6 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(422\) vs.
\(2(198)=396\).
time = 0.33, size = 423, normalized size = 3.30
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\frac {-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{3 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{2}}+\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}}{b}-\frac {2 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 a}{a -b}, \sqrt {2}\right )}{b^{2} \left (a^{2}-b a \right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}-\frac {2 a \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(423\) |
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 = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^{7/2}\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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