Optimal. Leaf size=254 \[ \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.41, antiderivative size = 254, 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, 3059,
2851, 2850} \begin {gather*} -\frac {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^4 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2841
Rule 2850
Rule 2851
Rule 3059
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}-\frac {(2 a) \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-15 d)-\frac {1}{2} a (3 c+11 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {\left (a^2 \left (3 c^2+22 c d+115 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 d^2 (c+d)^2}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {\left (2 a^2 \left (3 c^2+22 c d+115 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 d^2 (c+d)^3}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 2.73, size = 216, normalized size = 0.85 \begin {gather*} \frac {a^2 \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 (-623 c^3-495 c^2 d-977 c d^2-145 d^3+\left (21 c^3+157 c^2 d+827 c d^2+115 d^3\right ) \cos (2 (e+f x))-\left (196 c^3+1865 c^2 d+694 c d^2+465 d^3\right ) \sin (e+f x)+3 c^2 d \sin (3 (e+f x))+22 c d^2 \sin (3 (e+f x))+115 d^3 \sin (3 (e+f x))\right )}{105 (c+d)^4 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))^{7/2}} \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. \(1222\) vs.
\(2(230)=460\).
time = 7.29, size = 1223, normalized size = 4.81
method | result | size |
default | \(\text {Expression too large to display}\) | \(1223\) |
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. 731 vs.
\(2 (242) = 484\).
time = 0.74, size = 731, normalized size = 2.88 \begin {gather*} -\frac {2 \, {\left ({\left (301 \, c^{4} + 169 \, c^{3} d + 75 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {5}{2}} - \frac {3 \, {\left (35 \, c^{4} - 763 \, c^{3} d - 297 \, c^{2} d^{2} - 85 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {6 \, {\left (182 \, c^{4} - 127 \, c^{3} d + 1059 \, c^{2} d^{2} + 251 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, {\left (50 \, c^{4} - 421 \, c^{3} d + 201 \, c^{2} d^{2} - 535 \, c d^{3} - 55 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, {\left (11 \, c^{4} - 36 \, c^{3} d + 80 \, c^{2} d^{2} - 40 \, c d^{3} + 25 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {126 \, {\left (11 \, c^{4} - 36 \, c^{3} d + 80 \, c^{2} d^{2} - 40 \, c d^{3} + 25 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {14 \, {\left (50 \, c^{4} - 421 \, c^{3} d + 201 \, c^{2} d^{2} - 535 \, c d^{3} - 55 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {6 \, {\left (182 \, c^{4} - 127 \, c^{3} d + 1059 \, c^{2} d^{2} + 251 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {3 \, {\left (35 \, c^{4} - 763 \, c^{3} d - 297 \, c^{2} d^{2} - 85 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {{\left (301 \, c^{4} + 169 \, c^{3} d + 75 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{2}}{105 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} + \frac {2 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\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 {9}{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. 1053 vs.
\(2 (242) = 484\).
time = 0.41, size = 1053, normalized size = 4.15 \begin {gather*} -\frac {2 \, {\left (224 \, a^{2} c^{3} - 608 \, a^{2} c^{2} d + 544 \, a^{2} c d^{2} - 160 \, a^{2} d^{3} - 2 \, {\left (3 \, a^{2} c^{2} d + 22 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (21 \, a^{2} c^{3} + 157 \, a^{2} c^{2} d + 827 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (77 \, a^{2} c^{3} + 783 \, a^{2} c^{2} d - 425 \, a^{2} c d^{2} + 405 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (161 \, a^{2} c^{3} + 163 \, a^{2} c^{2} d + 451 \, a^{2} c d^{2} + 65 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) - {\left (224 \, a^{2} c^{3} - 608 \, a^{2} c^{2} d + 544 \, a^{2} c d^{2} - 160 \, a^{2} d^{3} + 2 \, {\left (3 \, a^{2} c^{2} d + 22 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (21 \, a^{2} c^{3} + 151 \, a^{2} c^{2} d + 783 \, a^{2} c d^{2} - 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (49 \, a^{2} c^{3} + 467 \, a^{2} c^{2} d + 179 \, a^{2} c d^{2} + 145 \, a^{2} d^{3}\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}}{105 \, {\left ({\left (c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{5} + {\left (4 \, c^{5} d^{3} + 17 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 22 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, c^{6} d^{2} + 12 \, c^{5} d^{3} + 19 \, c^{4} d^{4} + 16 \, c^{3} d^{5} + 9 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (2 \, c^{7} d + 11 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 43 \, c^{4} d^{4} + 42 \, c^{3} d^{5} + 25 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{8} + 4 \, c^{7} d + 12 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 38 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right ) + {\left (c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f + {\left ({\left (c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{4} - 4 \, {\left (c^{5} d^{3} + 4 \, c^{4} d^{4} + 6 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, c^{6} d^{2} + 14 \, c^{5} d^{3} + 27 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 17 \, c^{2} d^{6} + 6 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{2} + 4 \, {\left (c^{7} d + 4 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 8 \, c^{4} d^{4} + 7 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\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(-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 = 20.61, size = 862, normalized size = 3.39 \begin {gather*} \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {8\,a^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (3\,c^2+22\,c\,d+115\,d^2\right )}{105\,d^3\,f\,{\left (c+d\right )}^4}+\frac {8\,a^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,3{}\mathrm {i}+c\,d\,22{}\mathrm {i}+d^2\,115{}\mathrm {i}\right )}{105\,d^3\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (30\,c^3-25\,c^2\,d+36\,c\,d^2-5\,d^3\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,30{}\mathrm {i}-c^2\,d\,25{}\mathrm {i}+c\,d^2\,36{}\mathrm {i}-d^3\,5{}\mathrm {i}\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (25\,c^3+244\,c^2\,d-19\,c\,d^2+50\,d^3\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,25{}\mathrm {i}+c^2\,d\,244{}\mathrm {i}-c\,d^2\,19{}\mathrm {i}+d^3\,50{}\mathrm {i}\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}+\frac {8\,a^2\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (3\,c^2+22\,c\,d+115\,d^2\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}+\frac {8\,a^2\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,3{}\mathrm {i}+c\,d\,22{}\mathrm {i}+d^2\,115{}\mathrm {i}\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}\right )}{{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{{\left (c+d\right )}^4}-\frac {4\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+6\,c^2\,d+6\,c\,d^2+d^3\right )}{d^3}-\frac {4\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (6\,c^2+2\,c\,d+d^2\right )}{d^2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (8\,c+d\right )}{d}+\frac {2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (8\,c^4+16\,c^3\,d+24\,c^2\,d^2+12\,c\,d^3+3\,d^4\right )}{d^4}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (8\,c^3+6\,c^2\,d+6\,c\,d^2+d^3\right )\,4{}\mathrm {i}}{d^3\,{\left (c+d\right )}^4}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (6\,c^2+2\,c\,d+d^2\right )\,4{}\mathrm {i}}{d^2\,{\left (c+d\right )}^4}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (8\,c+d\right )\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{d\,{\left (c+d\right )}^4}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (8\,c^4+16\,c^3\,d+24\,c^2\,d^2+12\,c\,d^3+3\,d^4\right )\,2{}\mathrm {i}}{d^4\,{\left (c+d\right )}^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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