Optimal. Leaf size=218 \[ \frac {2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{7/2} \sqrt {c+d} f}+\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f} \]
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Rubi [A]
time = 0.60, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {3055, 3060,
2852, 214} \begin {gather*} \frac {2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{7/2} f \sqrt {c+d}}+\frac {2 a^3 \left (5 A d (3 c-7 d)-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (-5 A d+5 B c-8 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2852
Rule 3055
Rule 3060
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx &=-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}+\frac {2 \int \frac {(a+a \sin (e+f x))^{3/2} \left (\frac {1}{2} a (3 B c+5 A d)-\frac {1}{2} a (5 B c-5 A d-8 B d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{5 d}\\ &=\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}+\frac {4 \int \frac {\sqrt {a+a \sin (e+f x)} \left (-\frac {1}{4} a^2 (B c (5 c-17 d)-5 A d (c+3 d))-\frac {1}{4} a^2 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{15 d^2}\\ &=\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}-\frac {\left (a^2 (c-d)^2 (B c-A d)\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{d^3}\\ &=\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}+\frac {\left (2 a^3 (c-d)^2 (B c-A d)\right ) \text {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d^3 f}\\ &=\frac {2 a^{5/2} (c-d)^2 (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{7/2} \sqrt {c+d} f}+\frac {2 a^3 \left (5 A (3 c-7 d) d-B \left (15 c^2-35 c d+32 d^2\right )\right ) \cos (e+f x)}{15 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-5 A d-8 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 d f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(450\) vs. \(2(218)=436\).
time = 4.30, size = 450, normalized size = 2.06 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{5/2} \left (-30 \sqrt {d} \left (A d (-2 c+5 d)+B \left (2 c^2-5 c d+5 d^2\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-5 d^{3/2} (-2 B c+2 A d+5 B d) \cos \left (\frac {3}{2} (e+f x)\right )+3 B d^{5/2} \cos \left (\frac {5}{2} (e+f x)\right )+\frac {15 (c-d)^2 (B c-A d) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+2 \log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (c+d+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )}{\sqrt {c+d}}-\frac {15 (c-d)^2 (B c-A d) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+2 \log \left ((c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \left (-1+2 \tan \left (\frac {1}{4} (e+f x)\right )+\tan ^2\left (\frac {1}{4} (e+f x)\right )\right )\right )\right )}{\sqrt {c+d}}+30 \sqrt {d} \left (A d (-2 c+5 d)+B \left (2 c^2-5 c d+5 d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )-5 d^{3/2} (-2 B c+2 A d+5 B d) \sin \left (\frac {3}{2} (e+f x)\right )-3 B d^{5/2} \sin \left (\frac {5}{2} (e+f x)\right )\right )}{30 d^{7/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(542\) vs.
\(2(192)=384\).
time = 11.74, size = 543, normalized size = 2.49
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-3 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a \left (c +d \right ) d}\, d^{2}+5 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, a \,d^{2}-15 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {c d a +a \,d^{2}}}\right ) a^{3} c^{2} d +30 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {c d a +a \,d^{2}}}\right ) a^{3} c \,d^{2}-15 A \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {c d a +a \,d^{2}}}\right ) a^{3} d^{3}-5 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, a c d +20 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, a \,d^{2}+15 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {c d a +a \,d^{2}}}\right ) a^{3} c^{3}-30 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {c d a +a \,d^{2}}}\right ) a^{3} c^{2} d +15 B \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {c d a +a \,d^{2}}}\right ) a^{3} c \,d^{2}+15 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} c d -45 A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} d^{2}-15 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} c^{2}+45 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} c d -60 B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a^{2} d^{2}\right )}{15 d^{3} \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(543\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 513 vs.
\(2 (200) = 400\).
time = 1.89, size = 1356, normalized size = 6.22 \begin {gather*} \left [\frac {15 \, {\left (B a^{2} c^{3} - {\left (A + 2 \, B\right )} a^{2} c^{2} d + {\left (2 \, A + B\right )} a^{2} c d^{2} - A a^{2} d^{3} + {\left (B a^{2} c^{3} - {\left (A + 2 \, B\right )} a^{2} c^{2} d + {\left (2 \, A + B\right )} a^{2} c d^{2} - A a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left (B a^{2} c^{3} - {\left (A + 2 \, B\right )} a^{2} c^{2} d + {\left (2 \, A + B\right )} a^{2} c d^{2} - A a^{2} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{c d + d^{2}}} \log \left (\frac {a d^{2} \cos \left (f x + e\right )^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left (6 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left (c d^{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 {\frac {a}{c d + d^{2}}} - {\left (a c^{2} + 8 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (a d^{2} \cos \left (f x + e\right )^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left (3 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{3} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - c^{2} - 2 \, c d - d^{2} - {\left (c^{2} + d^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \cos \left (f x + e\right ) - c^{2} - 2 \, c d - d^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (3 \, B a^{2} d^{2} \cos \left (f x + e\right )^{3} - 15 \, B a^{2} c^{2} + 5 \, {\left (3 \, A + 7 \, B\right )} a^{2} c d - {\left (35 \, A + 32 \, B\right )} a^{2} d^{2} + {\left (5 \, B a^{2} c d - {\left (5 \, A + 11 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (15 \, B a^{2} c^{2} - 5 \, {\left (3 \, A + 8 \, B\right )} a^{2} c d + 2 \, {\left (20 \, A + 23 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, B a^{2} d^{2} \cos \left (f x + e\right )^{2} - 15 \, B a^{2} c^{2} + 5 \, {\left (3 \, A + 7 \, B\right )} a^{2} c d - {\left (35 \, A + 32 \, B\right )} a^{2} d^{2} - {\left (5 \, B a^{2} c d - {\left (5 \, A + 14 \, B\right )} a^{2} 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}}{30 \, {\left (d^{3} f \cos \left (f x + e\right ) + d^{3} f \sin \left (f x + e\right ) + d^{3} f\right )}}, \frac {15 \, {\left (B a^{2} c^{3} - {\left (A + 2 \, B\right )} a^{2} c^{2} d + {\left (2 \, A + B\right )} a^{2} c d^{2} - A a^{2} d^{3} + {\left (B a^{2} c^{3} - {\left (A + 2 \, B\right )} a^{2} c^{2} d + {\left (2 \, A + B\right )} a^{2} c d^{2} - A a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left (B a^{2} c^{3} - {\left (A + 2 \, B\right )} a^{2} c^{2} d + {\left (2 \, A + B\right )} a^{2} c d^{2} - A a^{2} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{c d + d^{2}}} \arctan \left (\frac {\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) - c - 2 \, d\right )} \sqrt {-\frac {a}{c d + d^{2}}}}{2 \, a \cos \left (f x + e\right )}\right ) + 2 \, {\left (3 \, B a^{2} d^{2} \cos \left (f x + e\right )^{3} - 15 \, B a^{2} c^{2} + 5 \, {\left (3 \, A + 7 \, B\right )} a^{2} c d - {\left (35 \, A + 32 \, B\right )} a^{2} d^{2} + {\left (5 \, B a^{2} c d - {\left (5 \, A + 11 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (15 \, B a^{2} c^{2} - 5 \, {\left (3 \, A + 8 \, B\right )} a^{2} c d + 2 \, {\left (20 \, A + 23 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, B a^{2} d^{2} \cos \left (f x + e\right )^{2} - 15 \, B a^{2} c^{2} + 5 \, {\left (3 \, A + 7 \, B\right )} a^{2} c d - {\left (35 \, A + 32 \, B\right )} a^{2} d^{2} - {\left (5 \, B a^{2} c d - {\left (5 \, A + 14 \, B\right )} a^{2} 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}}{15 \, {\left (d^{3} f \cos \left (f x + e\right ) + d^{3} f \sin \left (f x + e\right ) + d^{3} f\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs.
\(2 (200) = 400\).
time = 0.58, size = 548, normalized size = 2.51 \begin {gather*} \frac {\sqrt {2} \sqrt {a} {\left (\frac {15 \, \sqrt {2} {\left (B a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - A a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2 \, B a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, A a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - A a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c d - d^{2}}}\right )}{\sqrt {-c d - d^{2}} d^{3}} + \frac {2 \, {\left (12 \, B a^{2} d^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 10 \, B a^{2} c d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, A a^{2} d^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 40 \, B a^{2} d^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B a^{2} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, A a^{2} c d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 45 \, B a^{2} c d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 45 \, A a^{2} d^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, B a^{2} d^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{d^{5}}\right )}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{c+d\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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