Optimal. Leaf size=113 \[ \frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac {16 (a+a \sin (c+d x))^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750}
\begin {gather*} \frac {16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac {8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2751
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx &=\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {4 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a}\\ &=\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac {8 \int \frac {(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{11/2}} \, dx}{5 a^2}\\ &=\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac {8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac {16 (a+a \sin (c+d x))^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 64, normalized size = 0.57 \begin {gather*} \frac {2 \sqrt {e \cos (c+d x)} \sec ^5(c+d x) (a (1+\sin (c+d x)))^{5/2} \left (17-20 \sin (c+d x)+8 \sin ^2(c+d x)\right )}{45 d e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 54, normalized size = 0.48
method | result | size |
default | \(-\frac {2 \left (8 \left (\cos ^{2}\left (d x +c \right )\right )+20 \sin \left (d x +c \right )-25\right ) \cos \left (d x +c \right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}}{45 d \left (e \cos \left (d x +c \right )\right )^{\frac {11}{2}}}\) | \(54\) |
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. 255 vs.
\(2 (88) = 176\).
time = 0.54, size = 255, normalized size = 2.26 \begin {gather*} \frac {2 \, {\left (17 \, a^{\frac {5}{2}} - \frac {40 \, a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {49 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {49 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {40 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {17 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3} e^{\left (-\frac {11}{2}\right )}}{45 \, d \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 95, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{2} + 20 \, a^{2} \sin \left (d x + c\right ) - 25 \, a^{2}\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{45 \, {\left (d \cos \left (d x + c\right )^{3} e^{\frac {11}{2}} + 2 \, d \cos \left (d x + c\right ) e^{\frac {11}{2}} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right ) e^{\frac {11}{2}}\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 [B]
time = 6.69, size = 119, normalized size = 1.05 \begin {gather*} \frac {8\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (2\,\cos \left (4\,c+4\,d\,x\right )-73\,\cos \left (2\,c+2\,d\,x\right )-162\,\sin \left (c+d\,x\right )+18\,\sin \left (3\,c+3\,d\,x\right )+105\right )}{45\,d\,e^5\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (\cos \left (4\,c+4\,d\,x\right )-28\,\cos \left (2\,c+2\,d\,x\right )-56\,\sin \left (c+d\,x\right )+8\,\sin \left (3\,c+3\,d\,x\right )+35\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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