Optimal. Leaf size=172 \[ \frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.22, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2841, 21, 2851,
2850} \begin {gather*} -\frac {4 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^3 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac {2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 2841
Rule 2850
Rule 2851
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {(2 a) \int \frac {-\frac {1}{2} a (c+9 d)-\frac {1}{2} a (c+9 d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}} \, dx}{5 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {(a (c+9 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {(2 a (c+9 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 d (c+d)^2}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^2 (c+9 d) \cos (e+f x)}{15 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 140, normalized size = 0.81 \begin {gather*} -\frac {2 a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (25 c^2+13 c d+12 d^2-d (c+9 d) \cos (2 (e+f x))+\left (5 c^2+46 c d+9 d^2\right ) \sin (e+f x)\right )}{15 (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs.
\(2(154)=308\).
time = 0.28, size = 625, normalized size = 3.63
method | result | size |
default | \(-\frac {2 \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \sqrt {c +d \sin \left (f x +e \right )}\, \left (40 c^{5}+24 d^{5}-80 c^{2} d^{3}+8 c \,d^{4}-48 c^{3} d^{2}+56 c^{4} d -40 c^{5} \sin \left (f x +e \right )+78 \left (\cos ^{4}\left (f x +e \right )\right ) d^{5}-27 \left (\cos ^{6}\left (f x +e \right )\right ) d^{5}-75 \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}-31 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d +63 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}+186 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+114 \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}+23 \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}-105 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d^{3}-19 \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}-57 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) d^{5}-9 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}-90 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}-146 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+15 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}-24 \sin \left (f x +e \right ) d^{5}-56 \sin \left (f x +e \right ) c^{4} d +48 \sin \left (f x +e \right ) c^{3} d^{2}+80 \sin \left (f x +e \right ) c^{2} d^{3}-8 \sin \left (f x +e \right ) c \,d^{4}-15 \left (\cos ^{2}\left (f x +e \right )\right ) c^{5}-13 \left (\cos ^{4}\left (f x +e \right )\right ) c^{4} d -5 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{5}+18 \sin \left (f x +e \right ) \left (\cos ^{6}\left (f x +e \right )\right ) d^{5}-\left (\cos ^{6}\left (f x +e \right )\right ) c^{2} d^{3}-63 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3} d^{2}-12 \left (\cos ^{6}\left (f x +e \right )\right ) c \,d^{4}+2 \sin \left (f x +e \right ) \left (\cos ^{6}\left (f x +e \right )\right ) c \,d^{4}+9 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c^{3} d^{2}+57 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d^{3}+3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d \right )}{15 f \cos \left (f x +e \right )^{3} \left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )^{3} \left (c +d \right )^{3}}\) | \(625\) |
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. 529 vs.
\(2 (163) = 326\).
time = 0.60, size = 529, normalized size = 3.08 \begin {gather*} -\frac {2 \, {\left ({\left (25 \, c^{3} + 12 \, c^{2} d + 3 \, c d^{2}\right )} a^{\frac {3}{2}} - \frac {{\left (15 \, c^{3} - 130 \, c^{2} d - 39 \, c d^{2} - 6 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (65 \, c^{3} - 78 \, c^{2} d + 223 \, c d^{2} + 30 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, {\left (11 \, c^{3} - 44 \, c^{2} d + 33 \, c d^{2} - 24 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, {\left (11 \, c^{3} - 44 \, c^{2} d + 33 \, c d^{2} - 24 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (65 \, c^{3} - 78 \, c^{2} d + 223 \, c d^{2} + 30 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {{\left (15 \, c^{3} - 130 \, c^{2} d - 39 \, c d^{2} - 6 \, d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {{\left (25 \, c^{3} + 12 \, c^{2} d + 3 \, c d^{2}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{2}}{15 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + \frac {2 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {7}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 614 vs.
\(2 (163) = 326\).
time = 0.38, size = 614, normalized size = 3.57 \begin {gather*} \frac {2 \, {\left (2 \, {\left (a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} - 20 \, a c^{2} + 32 \, a c d - 12 \, a d^{2} - {\left (5 \, a c^{2} + 44 \, a c d - 9 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (25 \, a c^{2} + 14 \, a c d + 21 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (20 \, a c^{2} - 32 \, a c d + 12 \, a d^{2} - 2 \, {\left (a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (5 \, a c^{2} + 46 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{15 \, {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{4} - 3 \, {\left (c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right )} f \cos \left (f x + e\right )^{3} - {\left (3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f - {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right ) - {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 17.48, size = 541, normalized size = 3.15 \begin {gather*} \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {8\,a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c+9\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^2\,f\,{\left (c+d\right )}^3}-\frac {8\,a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (9\,c^2-4\,c\,d+3\,d^2\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}+\frac {8\,a\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,9{}\mathrm {i}-c\,d\,4{}\mathrm {i}+d^2\,3{}\mathrm {i}\right )}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {8\,a\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,9{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^2\,f\,{\left (c+d\right )}^3}+\frac {8\,a\,c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,9{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d^3\,f\,{\left (c+d\right )}^3}-\frac {8\,a\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (c+9\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d^3\,f\,{\left (c+d\right )}^3}\right )}{{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3}{{\left (c+d\right )}^3}-\frac {3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (6\,c+d\right )}{d}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c\,6{}\mathrm {i}+d\,1{}\mathrm {i}\right )}{d}-\frac {3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3\,{\left (c+d\right )}^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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