Optimal. Leaf size=293 \[ \frac {(2 A-7 B) \text {ArcSin}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \]
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Rubi [A]
time = 0.67, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3056, 3062,
3061, 2861, 211, 2853, 222} \begin {gather*} \frac {(2 A-7 B) \text {ArcSin}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {7 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}+\frac {(79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac {(3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 222
Rule 2853
Rule 2861
Rule 3056
Rule 3061
Rule 3062
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx &=\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7}{2} a (A-B)-a (A-7 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {15}{4} a^2 (3 A-7 B)-\frac {1}{2} a^2 (17 A-77 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{8} a^3 (79 A-259 B)-\frac {21}{4} a^3 (7 A-27 B) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {-\frac {21}{8} a^4 (7 A-27 B)+24 a^4 (2 A-7 B) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^7}\\ &=\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {(177 A-637 B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}+\frac {(2 A-7 B) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{2 a^4}\\ &=\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {(177 A-637 B) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 a^2 d}-\frac {(2 A-7 B) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^4 d}\\ &=\frac {(2 A-7 B) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.66, size = 396, normalized size = 1.35 \begin {gather*} \frac {\cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\frac {3 \sqrt {2} e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (128 A d x-448 B d x-64 i (2 A-7 B) \sinh ^{-1}\left (e^{i (c+d x)}\right )+i \sqrt {2} (177 A-637 B) \log \left (1+e^{i (c+d x)}\right )+128 i A \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )-448 i B \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )-177 i \sqrt {2} A \log \left (1-e^{i (c+d x)}+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )+637 i \sqrt {2} B \log \left (1-e^{i (c+d x)}+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right )}{\sqrt {1+e^{2 i (c+d x)}}}+\frac {1}{4} \sqrt {\cos (c+d x)} (-541 A+2233 B+(-724 A+3172 B) \cos (c+d x)+(-247 A+1099 B) \cos (2 (c+d x))+96 B \cos (3 (c+d x))) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{48 d (a (1+\cos (c+d x)))^{7/2}} \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. \(886\) vs.
\(2(250)=500\).
time = 0.41, size = 887, normalized size = 3.03
method | result | size |
default | \(\text {Expression too large to display}\) | \(887\) |
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 [A]
time = 14.42, size = 368, normalized size = 1.26 \begin {gather*} \frac {3 \, \sqrt {2} {\left ({\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right ) + 177 \, A - 637 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + 2 \, {\left (192 \, B \cos \left (d x + c\right )^{3} - {\left (247 \, A - 1099 \, B\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (181 \, A - 721 \, B\right )} \cos \left (d x + c\right ) - 147 \, A + 567 \, B\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 384 \, {\left ({\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right ) + 2 \, A - 7 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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