Optimal. Leaf size=377 \[ -\frac {a^{5/2} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{128 d^{5/2} f}-\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{128 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f} \]
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Rubi [A] time = 0.85, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2763, 2981, 2770, 2775, 205} \[ -\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{128 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^{5/2} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{128 d^{5/2} f}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2763
Rule 2770
Rule 2775
Rule 2981
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a^2 (c+17 d)-\frac {3}{2} a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2} \, dx}{5 d}\\ &=\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\left (a^2 \left (3 c^2-34 c d+283 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx}{80 d^2}\\ &=-\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\left (a^2 (c+d) \left (3 c^2-34 c d+283 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{96 d^2}\\ &=-\frac {a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\left (a^2 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{128 d^2}\\ &=-\frac {a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{128 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\left (a^2 (c+d)^3 \left (3 c^2-34 c d+283 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{256 d^2}\\ &=-\frac {a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{128 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}-\frac {\left (a^3 (c+d)^3 \left (3 c^2-34 c d+283 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{128 d^2 f}\\ &=-\frac {a^{5/2} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{128 d^{5/2} f}-\frac {a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{128 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\\ \end {align*}
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Mathematica [A] time = 2.80, size = 395, normalized size = 1.05 \[ \frac {(a (\sin (e+f x)+1))^{5/2} \left (\frac {\left (3 c^2-34 c d+283 d^2\right ) (c+d)^3 \left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {c+d \sin (e+f x)}+\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )\right )}{d^{5/2}}+\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (45 c^4-30 c^3 d \sin (e+f x)-390 c^3 d-3322 c^2 d^2 \sin (e+f x)+4 d^2 \left (93 c^2+488 c d+331 d^2\right ) \cos (2 (e+f x))-8396 c^2 d^2-7774 c d^3 \sin (e+f x)+252 c d^3 \sin (3 (e+f x))-12762 c d^3-3874 d^4 \sin (e+f x)+348 d^4 \sin (3 (e+f x))-48 d^4 \cos (4 (e+f x))-5521 d^4\right )}{15 d^2}\right )}{256 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 2.06, size = 2083, normalized size = 5.53 \[ \text {result too large to display} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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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