Optimal. Leaf size=152 \[ \frac {32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac {16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac {4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 152, 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 = {2672, 2671} \[ \frac {32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac {16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac {4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac {2 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a}\\ &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac {4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac {8 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a^2}\\ &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac {4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac {16 (a+a \sin (c+d x))^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac {16 \int \frac {(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{11/2}} \, dx}{5 a^3}\\ &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac {4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac {16 (a+a \sin (c+d x))^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac {32 (a+a \sin (c+d x))^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 74, normalized size = 0.49 \[ \frac {2 \sec ^5(c+d x) (a (\sin (c+d x)+1))^{3/2} \sqrt {e \cos (c+d x)} (6 \sin (c+d x)-4 \sin (3 (c+d x))+12 \cos (2 (c+d x))+7)}{45 d e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 98, normalized size = 0.64 \[ -\frac {2 \, {\left (24 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (8 \, a \cos \left (d x + c\right )^{2} - 5 \, a\right )} \sin \left (d x + c\right ) - 5 \, a\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45 \, {\left (d e^{6} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d e^{6} \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 70, normalized size = 0.46 \[ -\frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-24 \left (\cos ^{2}\left (d x +c \right )\right )-10 \sin \left (d x +c \right )+5\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \cos \left (d x +c \right )}{45 d \left (e \cos \left (d x +c \right )\right )^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 357, normalized size = 2.35 \[ \frac {2 \, {\left (19 \, a^{\frac {3}{2}} \sqrt {e} - \frac {12 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {58 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {116 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {116 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {58 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {12 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {19 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{45 \, {\left (e^{6} + \frac {4 \, e^{6} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, e^{6} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, e^{6} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {e^{6} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.96, size = 261, normalized size = 1.72 \[ \frac {14\,a\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+12\,a\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+24\,a\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-8\,a\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {45\,d\,e^5\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {45\,d\,e^5\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}-\frac {45\,d\,e^5\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {45\,d\,e^5\,\sin \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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