Optimal. Leaf size=193 \[ -\frac {256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {32 \sqrt {a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {16}{77 a d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac {256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {32 \sqrt {a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {16}{77 a d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac {2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}+\frac {8 \int \frac {1}{(e \cos (c+d x))^{7/2} \sqrt {a+a \sin (c+d x)}} \, dx}{11 a}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}+\frac {48 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^2}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {64 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^3}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {128 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^4}\\ &=-\frac {2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac {256 (a+a \sin (c+d x))^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 76, normalized size = 0.39 \[ \frac {2 (104 \sin (c+d x)+48 \sin (3 (c+d x))+8 \cos (2 (c+d x))-16 \cos (4 (c+d x))+45)}{385 d e (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 125, normalized size = 0.65 \[ \frac {2 \, {\left (128 \, \cos \left (d x + c\right )^{4} - 144 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (24 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 21\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{385 \, {\left (a^{2} d e^{4} \cos \left (d x + c\right )^{5} - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 80, normalized size = 0.41 \[ \frac {2 \left (-128 \left (\cos ^{4}\left (d x +c \right )\right )+192 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+144 \left (\cos ^{2}\left (d x +c \right )\right )+56 \sin \left (d x +c \right )+21\right ) \cos \left (d x +c \right )}{385 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 451, normalized size = 2.34 \[ \frac {2 \, {\left (37 \, \sqrt {a} \sqrt {e} + \frac {496 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {559 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {544 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1526 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1526 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {544 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {559 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {496 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {37 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{385 \, {\left (a^{2} e^{4} + \frac {5 \, a^{2} e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{2} e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{2} e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{2} e^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{2} e^{4} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.65, size = 413, normalized size = 2.14 \[ \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}\,\left (\frac {288\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}}{77\,a^2\,d\,e^3}+\frac {256\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )}{385\,a^2\,d\,e^3}-\frac {512\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )}{385\,a^2\,d\,e^3}+\frac {1536\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )}{385\,a^2\,d\,e^3}+\frac {3328\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )}{385\,a^2\,d\,e^3}\right )}{10\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}-2\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}+8\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}} \]
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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