Optimal. Leaf size=308 \[ -\frac {(c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{24 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (9 A c+15 B c+39 A d-95 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{48 a^3 f}-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.72, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3056, 3047,
3102, 2830, 2728, 212} \begin {gather*} -\frac {(c-d) \left (3 A \left (c^2+6 c d+25 d^2\right )+B \left (5 c^2+62 c d-163 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{48 a^3 f}+\frac {d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{24 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac {(3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a \sin (e+f x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2830
Rule 3047
Rule 3056
Rule 3102
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {(c+d \sin (e+f x))^2 \left (\frac {1}{2} a (3 A c+5 B c+6 A d-6 B d)-\frac {1}{2} a (3 A-11 B) d \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {(c+d \sin (e+f x)) \left (\frac {1}{4} a^2 \left (B \left (5 c^2+47 c d-68 d^2\right )+3 A \left (c^2+3 c d+12 d^2\right )\right )-\frac {1}{4} a^2 d (9 A c+15 B c+39 A d-95 B d) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{8 a^4}\\ &=-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {\frac {1}{4} a^2 c \left (B \left (5 c^2+47 c d-68 d^2\right )+3 A \left (c^2+3 c d+12 d^2\right )\right )+\left (-\frac {1}{4} a^2 c d (9 A c+15 B c+39 A d-95 B d)+\frac {1}{4} a^2 d \left (B \left (5 c^2+47 c d-68 d^2\right )+3 A \left (c^2+3 c d+12 d^2\right )\right )\right ) \sin (e+f x)-\frac {1}{4} a^2 d^2 (9 A c+15 B c+39 A d-95 B d) \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{8 a^4}\\ &=\frac {d^2 (9 A c+15 B c+39 A d-95 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{48 a^3 f}-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {\frac {1}{8} a^3 \left (3 A \left (3 c^3+9 c^2 d+33 c d^2-13 d^3\right )+B \left (15 c^3+141 c^2 d-219 c d^2+95 d^3\right )\right )-\frac {1}{4} a^3 d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{12 a^5}\\ &=\frac {d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{24 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (9 A c+15 B c+39 A d-95 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{48 a^3 f}-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\left ((c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2}\\ &=\frac {d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{24 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (9 A c+15 B c+39 A d-95 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{48 a^3 f}-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\left ((c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{16 a^2 f}\\ &=-\frac {(c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{24 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (9 A c+15 B c+39 A d-95 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{48 a^3 f}-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.76, size = 523, normalized size = 1.70 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (24 (A-B) (c-d)^3 \sin \left (\frac {1}{2} (e+f x)\right )-12 (A-B) (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+6 (c-d)^2 (B (5 c-29 d)+3 A (c+7 d)) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-3 (c-d)^2 (B (5 c-29 d)+3 A (c+7 d)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+(3+3 i) (-1)^{3/4} (c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right ) \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-16 B d^3 \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+(24+24 i) d^2 (-6 B c-2 A d+5 B d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+(24+24 i) d^2 (6 B c+2 A d-5 B d) \left (i \cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-16 B d^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{48 f (a (1+\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. \(1437\) vs.
\(2(281)=562\).
time = 12.45, size = 1438, normalized size = 4.67
method | result | size |
default | \(\text {Expression too large to display}\) | \(1438\) |
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. 1012 vs.
\(2 (294) = 588\).
time = 0.43, size = 1012, normalized size = 3.29 \begin {gather*} -\frac {3 \, \sqrt {2} {\left (4 \, {\left (3 \, A + 5 \, B\right )} c^{3} + 12 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 12 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - 4 \, {\left (75 \, A - 163 \, B\right )} d^{3} - {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 3 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - {\left (75 \, A - 163 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 3 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - {\left (75 \, A - 163 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 3 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - {\left (75 \, A - 163 \, B\right )} d^{3}\right )} \cos \left (f x + e\right ) + {\left (4 \, {\left (3 \, A + 5 \, B\right )} c^{3} + 12 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 12 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - 4 \, {\left (75 \, A - 163 \, B\right )} d^{3} - {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 3 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - {\left (75 \, A - 163 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A + 19 \, B\right )} c^{2} d + 3 \, {\left (19 \, A - 75 \, B\right )} c d^{2} - {\left (75 \, A - 163 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (32 \, B d^{3} \cos \left (f x + e\right )^{4} - 12 \, {\left (A - B\right )} c^{3} + 36 \, {\left (A - B\right )} c^{2} d - 36 \, {\left (A - B\right )} c d^{2} + 12 \, {\left (A - B\right )} d^{3} + 32 \, {\left (9 \, B c d^{2} + {\left (3 \, A - 5 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A - 13 \, B\right )} c^{2} d - 3 \, {\left (13 \, A - 53 \, B\right )} c d^{2} + {\left (53 \, A - 93 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left ({\left (7 \, A + B\right )} c^{3} + 3 \, {\left (A - 9 \, B\right )} c^{2} d - 27 \, {\left (A - 9 \, B\right )} c d^{2} + 9 \, {\left (9 \, A - 17 \, B\right )} d^{3}\right )} \cos \left (f x + e\right ) + {\left (32 \, B d^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (A - B\right )} c^{3} - 36 \, {\left (A - B\right )} c^{2} d + 36 \, {\left (A - B\right )} c d^{2} - 12 \, {\left (A - B\right )} d^{3} - 96 \, {\left (3 \, B c d^{2} + {\left (A - 2 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left ({\left (3 \, A + 5 \, B\right )} c^{3} + 3 \, {\left (5 \, A - 13 \, B\right )} c^{2} d - 3 \, {\left (13 \, A - 85 \, B\right )} c d^{2} + {\left (85 \, A - 157 \, B\right )} 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}}{192 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 778 vs.
\(2 (294) = 588\).
time = 0.80, size = 778, normalized size = 2.53 \begin {gather*} \frac {\frac {3 \, \sqrt {2} {\left (3 \, A \sqrt {a} c^{3} + 5 \, B \sqrt {a} c^{3} + 15 \, A \sqrt {a} c^{2} d + 57 \, B \sqrt {a} c^{2} d + 57 \, A \sqrt {a} c d^{2} - 225 \, B \sqrt {a} c d^{2} - 75 \, A \sqrt {a} d^{3} + 163 \, B \sqrt {a} d^{3}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (3 \, A \sqrt {a} c^{3} + 5 \, B \sqrt {a} c^{3} + 15 \, A \sqrt {a} c^{2} d + 57 \, B \sqrt {a} c^{2} d + 57 \, A \sqrt {a} c d^{2} - 225 \, B \sqrt {a} c d^{2} - 75 \, A \sqrt {a} d^{3} + 163 \, B \sqrt {a} d^{3}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sqrt {2} {\left (3 \, A \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, B \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, A \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 39 \, B \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 39 \, A \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 63 \, B \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 21 \, A \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 29 \, B \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, A \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, B \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 9 \, A \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 33 \, B \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 33 \, A \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 57 \, B \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 19 \, A \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 27 \, B \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {128 \, \sqrt {2} {\left (2 \, B a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, B a^{\frac {13}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, B a^{\frac {13}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{9} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{192 \, 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 (c+d\,\sin \left (e+f\,x\right )\right )}^3}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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