Optimal. Leaf size=215 \[ -\frac {6 b B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^2 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.18, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {16, 3100,
2827, 2716, 2721, 2719, 2720} \begin {gather*} \frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {2 b^2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 b^2 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {6 b B E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2716
Rule 2719
Rule 2720
Rule 2721
Rule 2827
Rule 3100
Rubi steps
\begin {align*} \int (b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=b^6 \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx\\ &=\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {1}{7} \left (2 b^3\right ) \int \frac {\frac {7 b^2 B}{2}+\frac {1}{2} b^2 (5 A+7 C) \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\left (b^5 B\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx+\frac {1}{7} \left (b^4 (5 A+7 C)\right ) \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx\\ &=\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {1}{5} \left (3 b^3 B\right ) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx+\frac {1}{21} \left (b^2 (5 A+7 C)\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx\\ &=\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^2 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {1}{5} (3 b B) \int \sqrt {b \cos (c+d x)} \, dx+\frac {\left (b^2 (5 A+7 C) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}}\\ &=\frac {2 b^2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^2 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {\left (3 b B \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {6 b B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 b^2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 b^2 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.83, size = 134, normalized size = 0.62 \begin {gather*} \frac {(b \cos (c+d x))^{3/2} \sec ^5(c+d x) \left (-504 B \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 (5 A+7 C) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 (110 A+70 C+273 B \cos (c+d x)+10 (5 A+7 C) \cos (2 (c+d x))+63 B \cos (3 (c+d x))) \sin (c+d x)\right )}{420 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. \(726\) vs.
\(2(239)=478\).
time = 1.25, size = 727, normalized size = 3.38
method | result | size |
default | \(\text {Expression too large to display}\) | \(727\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 235, normalized size = 1.09 \begin {gather*} \frac {-5 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {3}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {3}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 63 i \, \sqrt {2} B b^{\frac {3}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 63 i \, \sqrt {2} B b^{\frac {3}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (63 \, B b \cos \left (d x + c\right )^{3} + 5 \, {\left (5 \, A + 7 \, C\right )} b \cos \left (d x + c\right )^{2} + 21 \, B b \cos \left (d x + c\right ) + 15 \, A b\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \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 {{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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