Optimal. Leaf size=150 \[ -\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac {4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac {16 (a+a \sin (c+d x))^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac {32 (a+a \sin (c+d x))^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {32 (a \sin (c+d x)+a)^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}-\frac {16 (a \sin (c+d x)+a)^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac {4 (a \sin (c+d x)+a)^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}} \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))^{13/2}} \, dx &=-\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac {6 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{13/2}} \, dx}{a}\\ &=-\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac {4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac {8 \int \frac {(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{13/2}} \, dx}{a^2}\\ &=-\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac {4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac {16 (a+a \sin (c+d x))^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac {16 \int \frac {(a+a \sin (c+d x))^{11/2}}{(e \cos (c+d x))^{13/2}} \, dx}{7 a^3}\\ &=-\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac {4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac {16 (a+a \sin (c+d x))^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac {32 (a+a \sin (c+d x))^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 74, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {e \cos (c+d x)} \sec ^6(c+d x) (a (1+\sin (c+d x)))^{5/2} \left (5+26 \sin (c+d x)-40 \sin ^2(c+d x)+16 \sin ^3(c+d x)\right )}{77 d e^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 70, normalized size = 0.47
method | result | size |
default | \(-\frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-40 \left (\cos ^{2}\left (d x +c \right )\right )-42 \sin \left (d x +c \right )+35\right ) \cos \left (d x +c \right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}}{77 d \left (e \cos \left (d x +c \right )\right )^{\frac {13}{2}}}\) | \(70\) |
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. 321 vs.
\(2 (118) = 236\).
time = 0.55, size = 321, normalized size = 2.14 \begin {gather*} \frac {2 \, {\left (5 \, a^{\frac {5}{2}} + \frac {52 \, a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {150 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {180 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {180 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {150 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {52 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {5 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4} e^{\left (-\frac {13}{2}\right )}}{77 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (\frac {4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {\sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 115, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (40 \, a^{2} \cos \left (d x + c\right )^{2} - 35 \, a^{2} - 2 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{2} - 21 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{77 \, {\left (d \cos \left (d x + c\right )^{4} e^{\frac {13}{2}} + 2 \, d \cos \left (d x + c\right )^{2} e^{\frac {13}{2}} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{2} e^{\frac {13}{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 = 11.17, size = 232, normalized size = 1.55 \begin {gather*} \frac {30\,a^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-40\,a^2\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,a^2\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-76\,a^2\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {77\,d\,e^6\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}+77\,d\,e^6\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}-\frac {385\,d\,e^6\,\cos \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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