Optimal. Leaf size=266 \[ \frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (112 A+143 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A]
time = 0.45, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {4172, 4102,
4100, 3890, 3889} \begin {gather*} \frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{1155 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a A \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rule 4172
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {3 a A}{2}+\frac {1}{2} a (6 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {2 a A \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {3}{4} a^2 (28 A+33 C)+\frac {9}{4} a^2 (8 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {1}{77} (a (112 A+143 C)) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {1}{385} (4 a (112 A+143 C)) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {(8 a (112 A+143 C)) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{1155}\\ &=\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (112 A+143 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 1.95, size = 125, normalized size = 0.47 \begin {gather*} \frac {a (18494 A+21736 C+2 (5789 A+5566 C) \cos (c+d x)+8 (581 A+429 C) \cos (2 (c+d x))+1645 A \cos (3 (c+d x))+660 C \cos (3 (c+d x))+490 A \cos (4 (c+d x))+105 A \cos (5 (c+d x))) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{9240 d \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 19.72, size = 152, normalized size = 0.57
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (105 A \left (\cos ^{5}\left (d x +c \right )\right )+245 A \left (\cos ^{4}\left (d x +c \right )\right )+280 A \left (\cos ^{3}\left (d x +c \right )\right )+165 C \left (\cos ^{3}\left (d x +c \right )\right )+336 A \left (\cos ^{2}\left (d x +c \right )\right )+429 C \left (\cos ^{2}\left (d x +c \right )\right )+448 A \cos \left (d x +c \right )+572 C \cos \left (d x +c \right )+896 A +1144 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}} \left (\cos ^{6}\left (d x +c \right )\right ) a}{1155 d \sin \left (d x +c \right )}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 794 vs.
\(2 (230) = 460\).
time = 0.66, size = 794, normalized size = 2.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.03, size = 144, normalized size = 0.54 \begin {gather*} \frac {2 \, {\left (105 \, A a \cos \left (d x + c\right )^{6} + 245 \, A a \cos \left (d x + c\right )^{5} + 5 \, {\left (56 \, A + 33 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{3} + 4 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right ) + d\right )} \sqrt {\cos \left (d x + c\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]
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 [B]
time = 9.14, size = 374, normalized size = 1.41 \begin {gather*} \frac {\sqrt {a-\frac {a}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}}\,\left (2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {a\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (9\,A+10\,C\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{12\,d}+\frac {a\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (7\,A+4\,C\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{56\,d}+\frac {a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (11\,A+14\,C\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{4\,d}+\frac {a\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (13\,A+12\,C\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{40\,d}+\frac {A\,a\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{24\,d}+\frac {A\,a\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{88\,d}\right )}{2\,\sqrt {-\frac {1}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}}\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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