Optimal. Leaf size=335 \[ \frac {17}{4} \sqrt {1-x} \cos (x)-5 (1-x)^{3/2} \cos (x)+(1-x)^{5/2} \cos (x)+\frac {15}{4} \sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-4 \sqrt {2 \pi } \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {15}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+4 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {15}{2} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)+\frac {15}{4} \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {15}{2} \sqrt {1-x} \sin (x)+\frac {5}{2} (1-x)^{3/2} \sin (x) \]
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Rubi [A]
time = 0.34, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6264, 6874,
3434, 3433, 3432, 3466, 3435, 3467} \begin {gather*} -4 \sqrt {2 \pi } \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {15}{2} \sqrt {\frac {\pi }{2}} \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-4 \sqrt {2 \pi } \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {15}{4} \sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-4 \sqrt {2 \pi } \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {15}{4} \sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+4 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {15}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {5}{2} (1-x)^{3/2} \sin (x)-\frac {15}{2} \sqrt {1-x} \sin (x)+(1-x)^{5/2} \cos (x)-5 (1-x)^{3/2} \cos (x)+\frac {17}{4} \sqrt {1-x} \cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3434
Rule 3435
Rule 3466
Rule 3467
Rule 6264
Rule 6874
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} x (1+x)^{3/2} \sin (x) \, dx &=\int \frac {x (1+x)^2 \sin (x)}{\sqrt {1-x}} \, dx\\ &=2 \text {Subst}\left (\int \left (-2+x^2\right )^2 \left (-1+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=2 \text {Subst}\left (\int \left (-4 \sin \left (1-x^2\right )+8 x^2 \sin \left (1-x^2\right )-5 x^4 \sin \left (1-x^2\right )+x^6 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt {1-x}\right )\\ &=2 \text {Subst}\left (\int x^6 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-8 \text {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-10 \text {Subst}\left (\int x^4 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+16 \text {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=8 \sqrt {1-x} \cos (x)-5 (1-x)^{3/2} \cos (x)+(1-x)^{5/2} \cos (x)-5 \text {Subst}\left (\int x^4 \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-8 \text {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+15 \text {Subst}\left (\int x^2 \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+(8 \cos (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )-(8 \sin (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=8 \sqrt {1-x} \cos (x)-5 (1-x)^{3/2} \cos (x)+(1-x)^{5/2} \cos (x)+4 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-4 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {15}{2} \sqrt {1-x} \sin (x)+\frac {5}{2} (1-x)^{3/2} \sin (x)+\frac {15}{2} \text {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-\frac {15}{2} \text {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-(8 \cos (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )-(8 \sin (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=\frac {17}{4} \sqrt {1-x} \cos (x)-5 (1-x)^{3/2} \cos (x)+(1-x)^{5/2} \cos (x)-4 \sqrt {2 \pi } \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+4 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-4 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {15}{2} \sqrt {1-x} \sin (x)+\frac {5}{2} (1-x)^{3/2} \sin (x)+\frac {15}{4} \text {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-\frac {1}{2} (15 \cos (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )+\frac {1}{2} (15 \sin (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=\frac {17}{4} \sqrt {1-x} \cos (x)-5 (1-x)^{3/2} \cos (x)+(1-x)^{5/2} \cos (x)-4 \sqrt {2 \pi } \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {15}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+4 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {15}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {15}{2} \sqrt {1-x} \sin (x)+\frac {5}{2} (1-x)^{3/2} \sin (x)+\frac {1}{4} (15 \cos (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )+\frac {1}{4} (15 \sin (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=\frac {17}{4} \sqrt {1-x} \cos (x)-5 (1-x)^{3/2} \cos (x)+(1-x)^{5/2} \cos (x)+\frac {15}{4} \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-4 \sqrt {2 \pi } \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {15}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+4 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {15}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)+\frac {15}{4} \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-4 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {15}{2} \sqrt {1-x} \sin (x)+\frac {5}{2} (1-x)^{3/2} \sin (x)\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.76, size = 201, normalized size = 0.60 \begin {gather*} \frac {\left (\frac {1}{32}+\frac {i}{32}\right ) \sqrt {1+x} \left ((-2-17 i) \sqrt {2 \pi } \sqrt {-1+x} \text {Erfi}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (\cos (1)+i \sin (1))-(2-2 i) \left ((-1-20 i)-(11-10 i) x+(8+10 i) x^2+4 x^3\right ) (\cos (x)+i \sin (x))-(1+i) \left (2 \left ((-1+20 i)-(11+10 i) x+(8-10 i) x^2+4 x^3\right ) (-i \cos (1)+\sin (1))+(15+19 i) \sqrt {\frac {\pi }{2}} \sqrt {-1+x} \text {Erf}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (\cos (x)+i \sin (x))\right ) (\cos (1+x)-i \sin (1+x))\right )}{\sqrt {1-x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (1+x \right )^{\frac {5}{2}} x \sin \left (x \right )}{\sqrt {-x^{2}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.36, size = 983, normalized size = 2.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 [C] Result contains complex when optimal does not.
time = 0.42, size = 202, normalized size = 0.60 \begin {gather*} -\left (\frac {19}{32} i - \frac {15}{32}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{i} + \left (\frac {19}{32} i + \frac {15}{32}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{\left (-i\right )} - \frac {1}{8} i \, {\left (4 i \, {\left (x - 1\right )}^{2} \sqrt {-x + 1} - \left (12 i - 10\right ) \, {\left (-x + 1\right )}^{\frac {3}{2}} - \left (3 i + 18\right ) \, \sqrt {-x + 1}\right )} e^{\left (i \, x\right )} - \frac {1}{2} i \, {\left (-2 i \, {\left (-x + 1\right )}^{\frac {3}{2}} + \left (4 i - 3\right ) \, \sqrt {-x + 1}\right )} e^{\left (i \, x\right )} - \frac {1}{8} i \, {\left (4 i \, {\left (x - 1\right )}^{2} \sqrt {-x + 1} - \left (12 i + 10\right ) \, {\left (-x + 1\right )}^{\frac {3}{2}} - \left (3 i - 18\right ) \, \sqrt {-x + 1}\right )} e^{\left (-i \, x\right )} - \frac {1}{2} i \, {\left (-2 i \, {\left (-x + 1\right )}^{\frac {3}{2}} + \left (4 i + 3\right ) \, \sqrt {-x + 1}\right )} e^{\left (-i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (-i \, x\right )} + 3.25954715712000 \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 {x\,\sin \left (x\right )\,{\left (x+1\right )}^{5/2}}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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